BIOMETRIC SECURITY AND IDENTITY MANAGEMENT
1 1. FALSE ACCEPTANCE RATES IN FINGERPRINT RECOGNITION SYSTEMS
Problem 1. In a fingerprint recognition system, the False Acceptance Rate (FAR) is defined as
the probability of incorrectly identifying an unauthorized person as an authorized user.
Given that a certain fingerprint recognition system has an FAR of 0.05, and it incorrectly iden-
tifies an unauthorized person as an authorized user:
a) What is the probability that this system correctly identifies the authorized user? b) If the
system is used 100 times, what is the probability of it correctly identifying the authorized user at
least 95 times? c) If the system is used 50 times, what is the probability of it correctly identifying
the authorized user exactly 47 times?
Solution 1. a) The Correct Acceptance Rate (CAR) is the complement of the FAR. Therefore,
the probability that the system correctly identifies the authorized user is:
CAR = 1 −FAR = 1 −0.05 = 0.95
b) The number of times the authorized user is correctly identified in 100 trials follows a binomial
distribution with n= 100 and p= 0.95. We want to find the probability of correctly identifying the
authorized user at least 95 times, which is:
P(X≥95) = 1 −P(X < 95)
Using the binomial probability formula:
P(X < 95) =
94
X
k=0 100
k(0.95)k(0.05)100−k
Using a computer or statistical software, we can calculate this probability.
c) Similarly, the probability of correctly identifying the authorized user exactly 47 times in 50
trials can be calculated using the binomial distribution formula:
P(X= 47) = 50
47(0.95)47(0.05)3
Calculating this probability will give us the answer.
2 2. VULNERABILITIES IN FACIAL RECOGNITION TECHNOLOGY
Problem 2. Consider a facial recognition system that uses a machine learning algorithm to
identify individuals. The system has an accuracy of 95% when tested on a dataset of 500 indi-
viduals. However, a vulnerability is discovered where the system can be tricked by presenting a
specially crafted image that fools the algorithm into misclassifying the individual.
Let’s assume that the vulnerability allows an attacker to successfully trick the system into mis-
classifying the individual with a probability of 10%. If the attacker tries this trick 4 times in a row,
what is the probability that at least one of these attempts will be successful?
Solution 2. Let’s calculate the probability of none of the 4 attempts being successful, and then
subtract this from 1 to find the probability that at least one attempt is successful.
Let pbe the probability of success (tricking the system) on a single attempt, which is 10% or
0.1.
The probability of failure on a single attempt is 1−p= 1 −0.1=0.9.
The probability of none of the 4 attempts being successful is:
(0.9)4= 0.6561
Therefore, the probability of at least one attempt being successful is:
1−0.6561 = 0.3439
So, the probability that at least one of the 4 attempts will be successful is approximately 34.39%.
3 3. ISSUES WITH IRIS AUTHENTICATION IN LOW LIGHTING CONDITIONS
Problem 3. The iris recognition system at a high-security facility struggled to authenticate
individuals in low lighting conditions due to excessive noise in the captured images. One of the
authentication methods used is based on the comparison of feature vectors extracted from the iris
image.
Given two feature vectors X= [1.2,3.4,−2.1,0.9] and Y= [0.8,3.9,−2.3,1.1], calculate the
Euclidean distance between these feature vectors as a measure of similarity.
Solution 3.
1. Let’s calculate the Euclidean distance between the two feature vectors Xand Y, given by:
euclidean distance =v
u
u
t
n
X
i=1
(Xi−Yi)2
where Xiand Yiare the elements of the feature vectors Xand Yrespectively.
2. Plugging in the values for Xand Y:
euclidean distance =p(1.2−0.8)2+ (3.4−3.9)2+ (−2.1+2.3)2+ (0.9−1.1)2
euclidean distance =√0.16 + 0.25 + 0.04 + 0.04
euclidean distance =√0.49 = 0.7
Hence, the Euclidean distance between the feature vectors Xand Yis 0.7 units.
4 Biometric Security and Identity Management
Problem: Consider a biometric system that uses fingerprint recognition for authentication. The
system has an equal error rate of 0.05. If there are 1000 users attempting to authenticate using this
system, calculate the number of users that will be incorrectly authenticated due to false rejection
and false acceptance.
Solution: Given the equal error rate (EER) is 0.05, this means that the false rejection rate
(FRR) is equal to the false acceptance rate (FAR) at the threshold where they intersect.
Let Nbe the total number of users attempting to authenticate, which is 1000.
1. Finding the number of users incorrectly rejected:
The false rejection rate (FRR) is 0.05. This means that 5
Number of users incorrectly rejected = FRR ×N Number of users incorrectly rejected = 0.05 ×
1000 Number of users incorrectly rejected = 50 users
So, 50 users will be incorrectly rejected.
2. Finding the number of users incorrectly accepted:
The false acceptance rate (FAR) is also 0.05. This means that 5
Number of impostors incorrectly accepted = FAR ×N Number of impostors incorrectly accepted
= 0.05 ×1000 Number of impostors incorrectly accepted = 50 users
Therefore, 50 impostors will be incorrectly accepted during authentication process.
5 5. LACK OF STANDARDIZATION IN BIOMETRIC DATA STORAGE AND ENCRYPTION
Problem 5. Many biometric systems lack standardization in storing and encrypting biometric
data. As a result, a company is considering implementing a biometric access system for their
employees which uses fingerprint recognition technology. The system saves the fingerprint data in
an encrypted format using a unique encryption algorithm. The system has a false acceptance rate
(FAR) of 0.01 and a false rejection rate (FRR) of 0.05.
If the company has 1000 employees and each employee tries to access the system twice a day
on average, calculate the following:
a) The expected number of false acceptances per day.
b) The expected number of false rejections per day.
c) The total expected number of authentication errors per day.
Solution 5.
a) To calculate the expected number of false acceptances per day, we need to consider the
FAR and the total number of attempted accesses by the 1000 employees.
The FAR represents the probability of the system incorrectly accepting a false fingerprint as a
match. Since FAR = 0.01, the probability of a false acceptance is 0.01.
Let pF A be the probability of false acceptance, then:
pF A = 0.01
The company has 1000 employees trying to access the system twice a day on average, which
results in 2000 attempted accesses per day. Therefore, the expected number of false acceptances
per day can be calculated as:
Expected number of false acceptances per day =pF A ×2000
Expected number of false acceptances per day = 0.01 ×2000 = 20
Therefore, the expected number of false acceptances per day is 20.
b) To calculate the expected number of false rejections per day, we need to consider the FRR
and the total number of attempted accesses by the 1000 employees.
Similar to the FAR, the FRR represents the probability of the system incorrectly rejecting a valid
fingerprint as a non-match. Since FRR = 0.05, the probability of a false rejection is 0.05.
Let pF R be the probability of false rejection, then:
pF R = 0.05
The expected number of false rejections per day can be calculated as:
Expected number of false rejections per day =pF R ×2000
Expected number of false rejections per day = 0.05 ×2000 = 100
Therefore, the expected number of false rejections per day is 100.
c) The total expected number of authentication errors per day is the sum of false acceptances
and false rejections. Therefore:
Total expected number of authentication errors per day =Number of false acceptances+Number of false rejections
Total expected number of authentication errors per day = 20 + 100 = 120
Thus, the total expected number of authentication errors per day is 120.
6 6. BIOMETRIC SPOOFING TECHNIQUES AND THE NEED FOR ANTI-SPOOFING MEA-
SURES
Problem 6. A biometric security system uses fingerprint recognition for authentication. In
testing the system, it is found that the False Acceptance Rate (FAR) is 0.1% and the False Rejection
Rate (FRR) is 1%. Calculate the Equal Error Rate (EER) for this system.
Solution 6. The Equal Error Rate (EER) is the point at which the False Acceptance Rate (FAR)
is equal to the False Rejection Rate (FRR).
Given FAR = 0.1% and FRR = 1%, we need to find the EER.
Let the threshold be denoted as T. The EER occurs when FAR = FRR.
From the definitions:
FAR =P(reject |genuine)
FRR =P(accept |imposter)
Therefore, the EER is the average of FAR and FRR.
EER =FAR +FRR
2
EER =0.1+1
2= 0.55%
So, the Equal Error Rate (EER) for this system is 0.55%.
By analyzing the EER, we can see how well the system balances between rejecting valid users
and accepting imposters.
7 Biometric Security and Identity Management
Problem: Biometric systems often use multiple modalities to enhance security. Consider a
biometric system that combines fingerprint and iris recognition technologies. The false acceptance
rates for fingerprint and iris recognition are 0.1% and 0.01%, respectively.
a) What is the probability of falsely accepting an impostor using only fingerprint recognition?
b) What is the probability of falsely accepting an impostor using only iris recognition?
c) If the system requires both modalities to match in order to authenticate, what is the overall
false acceptance rate of the system?
Solution:
a) The probability of falsely accepting an impostor using only fingerprint recognition is given by
the false acceptance rate of the fingerprint modality, which is 0.1% or 0.001.
b) The probability of falsely accepting an impostor using only iris recognition is given by the
false acceptance rate of the iris modality, which is 0.01% or 0.0001.
c) Since the system requires both modalities to match for authentication, the overall false ac-
ceptance rate of the system can be calculated by multiplying the false acceptance rates of each
modality:
Overall False Acceptance Rate =False Acceptance Rate of Fingerprint×False Acceptance Rate of Iris
= 0.001 ×0.0001 = 0.0000001
Therefore, the overall false acceptance rate of the system when using both fingerprint and iris
recognition modalities is 0.00001%.
8 8. AUTHENTICATION FAILURES DUE TO AGING OR CHANGING BIOMETRIC CHARAC-
TERISTICS
Problem 8. In a biometric security system, the false acceptance rate (FAR) is measured to
be 0.02 and the false rejection rate (FRR) is 0.03. Calculate the Equal Error Rate (EER) for this
system.
Solution 8. The Equal Error Rate (EER) is the point where the false acceptance rate (FAR)
and the false rejection rate (FRR) are equal. Mathematically, EER is given by the average of FAR
and FRR when they are equal:
EER = (FAR + FRR) / 2
Given: FAR = 0.02 FRR = 0.03
Substitute the values into the formula: EER = (0.02 + 0.03) / 2 EER = 0.05 / 2 EER = 0.025
Therefore, the Equal Error Rate (EER) for this biometric security system is 0.025.
9 9. PROBLEMS WITH BIOMETRIC MATCHING ALGORITHMS AND ACCURACY RATES
Problem 9.
A security system that uses fingerprint recognition has an accuracy rate of 99.5%. If a random
sample of 500 fingerprints are scanned, calculate the probability that:
a) Exactly 3 fingerprints are incorrectly recognized.
b) At most 5 fingerprints are incorrectly recognized.
c) More than 10 fingerprints are incorrectly recognized.
Solution 9.
a) To find the probability that exactly 3 fingerprints are incorrectly recognized, we can use the
binomial probability formula:
P(X=k) = n
k·pk·(1 −p)n−k
where n= 500 (number of fingerprint scans), k= 3 (number of incorrectly recognized finger-
prints), and p= 1 −0.995 = 0.005 (probability of incorrectly recognizing a fingerprint).
P(X= 3) = 500
3·0.0053·0.995497
Calculating this probability gives us the answer for part a.
b) To find the probability that at most 5 fingerprints are incorrectly recognized, we need to
calculate
P(X≤5) = P(X= 0) + P(X= 1) + P(X= 2) + P(X= 3) + P(X= 4) + P(X= 5)
Using the binomial probability formula for each value of k, we can add up the probabilities to
find the answer for part b.
c) To find the probability that more than 10 fingerprints are incorrectly recognized, we can cal-
culate the complement of the probability that at most 10 are incorrectly recognized:
P(X > 10) = 1 −P(X≤10)
Using the binomial probability formula as in part b, we can find P(X≤10) and subtract it from
1 to get the answer for part c.
10 10. PRIVACY CONCERNS AND DATA PROTECTION IN BIOMETRIC DATABASES
Problem 10. A biometric system has an FAR (False Acceptance Rate) of 0.1% and an FRR
(False Rejection Rate) of 5%. If an individual tries to authenticate using this system 50 times, what
is the probability that they will be falsely rejected at least once?
Solution 10. Given that the FRR (False Rejection Rate) is 5%, the probability of a successful
authentication (1 - FRR) for one attempt is 95% or 0.95. Let pbe the probability of being falsely
rejected at least once in 50 attempts.
The probability of being falsely rejected at least once in ntrials is given by 1−(1 −FRR)n.
Therefore, the probability of being falsely rejected at least once in 50 attempts is:
p= 1 −(1 −0.05)50 = 1 −0.9550 ≈0.9231
Therefore, the probability that the individual will be falsely rejected at least once when attempt-
ing to authenticate 50 times using this biometric system is approximately 92.31%.
I. Problem: Biometric System Performance Degradation Over Time
An iris recognition system has an initial false acceptance rate (FAR) of 0.01
Solution
Given: Initial FAR = 0.01Initial FRR = 0.05Degraded FAR = 0.03Degraded FRR = 0.08
EER is the point where FAR = FRR.
Initial system: EER = (Initial FAR + Initial FRR) / 2 EER = (0.0001 + 0.0005) / 2 EER = 0.0003
Degraded system: EER = (Degraded FAR + Degraded FRR) / 2 EER = (0.0003 + 0.0008) / 2
EER = 0.00055
Therefore, the EER for the initial system is 0.0003 and for the degraded system is 0.00055.
11 12. ISSUES WITH BIOMETRIC IDENTIFICATION IN LARGE-SCALE SYSTEMS
Problem 12. In a large-scale biometric identification system, the false acceptance rate (FAR) is
0.1% and the false rejection rate (FRR) is 5%. If 1000 individuals attempt to authenticate themselves
using the system, calculate:
a) The number of individuals falsely accepted by the system.
b) The number of individuals falsely rejected by the system.
c) The overall accuracy of the system.
Solution 12.
a) The false acceptance rate (FAR) is 0.1%, which means 0.001 or 1
1000 of attempts are falsely
accepted. For 1000 individuals, the number falsely accepted can be calculated as:
False Accepted = 1000 ×0.001 = 1 individual
Therefore, the system falsely accepts 1individual.
b) The false rejection rate (FRR) is 5%, which means 5% or 0.05 of attempts are falsely rejected.
For 1000 individuals, the number falsely rejected can be calculated as:
False Rejected = 1000 ×0.05 = 50 individuals
Therefore, the system falsely rejects 50 individuals.
c) The overall accuracy of the system can be calculated as:
Accuracy = (1 −FAR)×100% = (1 −0.001) ×100% = 99.9%
Thus, the overall accuracy of the system is 99.9%.
12 13. CHALLENGES IN SECURING BIOMETRIC DATA DURING TRANSMISSION AND
STORAGE
Problem 13. Biometric data of a user is being transmitted over a network for authentication
purposes. The biometric data consists of a fingerprint image of size 500 ×500 pixels, where each
pixel is represented by an 8-bit grayscale value. During transmission, encryption is applied using
a symmetric key algorithm that increases the size of the data by 20%. If the encryption key used
is 128 bits long, calculate the size of the encrypted biometric data in kilobytes.
Solution 13. a) The size of the original fingerprint image in bytes can be calculated as:
Image size =Width ×Height ×Bit depth
Since each pixel is represented by an 8-bit grayscale value, the bit depth is 8. Given the image
size is 500 ×500 pixels:
Image size = 500 ×500 ×8bits
b) To convert bits to bytes, we divide by 8:
Image size in bytes =500 ×500 ×8
8bytes
c) After encryption, the size of the data increases by 20%. The new size of the encrypted data
can be calculated as:
New encrypted data size = 1.2×Original data size (in bytes)
d) Given that the encryption key is 128 bits long, to convert it to bytes, we divide by 8:
Key size =128
8bytes
e) The total size of the encrypted biometric data is the sum of the encrypted data size and the
key size:
Total size =New encrypted data size +Key size
Now, substitute the values calculated in parts a, b, c, and d to find the total size of the encrypted
biometric data in kilobytes.
13 14. IMPLICATIONS OF BIOMETRIC DATA BREACHES AND IDENTITY THEFT
Problem 14. A company’s biometric security system has been breached, and the fingerprint
data of 1000 employees has been stolen. If each fingerprint template contains 500 bytes of data and
the company offers a reward of 500percompromisedfingerprinttemplate, calculatethetotalrewardamountthecompanywillneedtopaytoaf f ectedemployees.
Solution 14. Given that: - Number of compromised fingerprint templates = 1000 - Size of each
fingerprint template = 500 bytes - Reward offered per compromised template = 500
Step 1: Calculate the total amount of data compromised Total amount of data compromised
= Number of compromised templates ×Size of each template Total amount of data compromised
= 1000 ×500 bytes Total amount of data compromised = 500,000 bytes
Step 2: Calculate the total reward amount Total reward amount = Number of compromised
templates ×Reward offered per template Total reward amount = 1000 ×500T otalrewardamount =500,000
Therefore, the company will need to pay a total reward amount of 500,000totheaffectedemployees.
14 15. CULTURAL BIASES AND LIMITATIONS IN BIOMETRIC RECOGNITION SYSTEMS
Problem 15. A facial recognition system has been trained on a dataset composed mainly of
individuals of Asian descent. When tested on a diverse population that includes individuals of
African descent, the system shows a significantly higher error rate for this subgroup.
The error rates for the system on the test set are as follows:
•Overall error rate: 5
•Error rate for individuals of Asian descent: 3
•Error rate for individuals of African descent: 12
a) What is the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent?
b) If the system correctly identifies 90
c) Considering these results, discuss the potential impact of cultural biases on the effectiveness
of biometric recognition systems.
Solution 15.
a) The odds ratio of misidentification for individuals of African descent compared to individuals
of Asian descent can be calculated using the error rates:
Odds ratio =Error rate for African descent
Error rate for Asian descent =0.12
0.03 = 4
Therefore, the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent is 4.
b) Given that the system correctly identifies 90
Accuracy rate for African descent = 1 −Error rate for African descent = 1 −0.12 = 0.88 = 88%
Therefore, the accuracy rate for individuals of African descent is 88
c) The significantly higher error rate for individuals of African descent in the facial recognition
system compared to individuals of Asian descent highlights a bias that may have been introduced
during the training phase, where the dataset was skewed towards a specific ethnicity. This cultural
bias can lead to inequitable outcomes, with certain demographic groups facing higher risks of
misidentification and potential discrimination. Addressing cultural biases in biometric recognition
systems is crucial to ensure fair and accurate identification across diverse populations.
15 16. LEGAL AND ETHICAL IMPLICATIONS OF BIOMETRIC DATA COLLECTION AND
USE
Problem 16. A company is implementing a biometric authentication system for its employees
using fingerprint recognition. The system is expected to process 1000 fingerprint scans per day. If
each fingerprint scan produces a file size of 500 KB, calculate the total amount of data collected
by the system in one month.
Solution 16. Given that the system processes 1000 fingerprint scans per day and each scan
produces a file size of 500 KB, we can calculate the total data collected by the system in one day
as follows:
Total data collected per day = Number of scans per day x File size per scan Total data collected
per day = 1000 x 500 KB Total data collected per day = 500,000 KB
To convert KB to GB, we use the conversion factor: 1 GB = 1024 KB
Total data collected per day in GB = 500,000 KB / 1024 = 488.28 GB
Since there are typically 30 days in a month, we can calculate the total amount of data collected
by the system in one month as follows:
Total data collected per month = Total data collected per day x Number of days in a month Total
data collected per month = 488.28 GB x 30 Total data collected per month = 14,648.4 GB
Therefore, the total amount of data collected by the system in one month is 14,648.4 GB.
16 17. CHALLENGES IN IMPLEMENTING BIOMETRIC AUTHENTICATION IN IOT DEVICES
Problem 17. A company is considering implementing biometric authentication for their IoT
devices to enhance security. They estimate that the false acceptance rate (FAR) of the biometric
system is 0.1% and the false rejection rate (FRR) is 1%. If an unauthorized user attempts to access
the device 50 times, calculate the probability that the system will grant access to this unauthorized
user at least once.
Solution 17. To calculate the probability of the system granting access to the unauthorized
user at least once, we need to consider the probability of false acceptance for each attempt. Let’s
denote the probability of a false acceptance as P(F A).
Given that FAR is 0.1%, we have:
P(F A) = 0.001
Therefore, the probability of rejecting a false attempt is:
P(¬F A) = 1 −P(F A) = 1 −0.001 = 0.999
The probability that the system will grant access to the unauthorized user at least once in 50
attempts is the complement of the probability that it will reject all 50 attempts. Thus, the probability
is:
P(Access granted at least once)=1−P(Access not granted at all)
Since each attempt is independent, the probability of the system not granting access to the
unauthorized user in a single attempt is P(¬F A). Therefore, the probability that the system will not
grant access in 50 attempts is:
P(Access not granted at all)=(P(¬F A))n= 0.99950
Calculating this probability:
P(Access not granted at all)=0.99950 ≈0.950 = 95%
So, the probability that the system will grant access to the unauthorized user at least once is
approximately:
P(Access granted at least once)=1−0.950 = 0.050 = 5%
Therefore, the system has a 5% chance of granting access to the unauthorized user at least
once out of 50 attempts.
17 18. USER ACCEPTANCE AND TRUST ISSUES WITH BIOMETRIC SECURITY MEASURES
Problem 18.
A study is conducted to analyze the user acceptance of a new biometric authentication system.
The study involved 100 participants, and the results showed that 65 participants were willing to use
the biometric system, while the remaining participants expressed concerns about privacy and did
not want to adopt it.
a) Calculate the user acceptance rate in percentage.
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same, how many participants would be willing to use the biometric system?
c) Discuss why user acceptance and trust are essential factors in the implementation of bio-
metric security measures.
Solution 18.
a) The user acceptance rate can be calculated as follows:
User acceptance rate =Number of participants willing to use the system
Total number of participants ×100%
In this case, the number of participants willing to use the system is 65, and the total number of
participants is 100. Therefore,
User acceptance rate =65
100 ×100% = 65%
So, the user acceptance rate is 65
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same at 65
Number of participants willing to use the system =Acceptance rate×Total number of participants
Number of participants willing to use the system = 0.65 ×150 = 97.5
Since we cannot have a fraction of a participant, we can estimate that approximately 98 partic-
ipants would be willing to use the biometric system.
c) User acceptance and trust are crucial factors in the implementation of biometric security
measures because:
1. If users do not trust the system, they might not use it properly or try to bypass it, leading to
security vulnerabilities. 2. Acceptance by users is essential for the success of the system, as low
adoption rates can render the biometric security measures ineffective. 3. Ensuring user acceptance
and trust can help improve the overall user experience and satisfaction with the security system. 4.
Addressing privacy concerns and building trust can help mitigate potential legal and ethical issues
related to biometric data collection and storage.
18 19. CHALLENGES IN SECURELY SHARING BIOMETRIC DATA ACROSS DIFFERENT
PLATFORMS
Problem 19. Consider a scenario where a company wants to securely share biometric data
(such as fingerprint scans) across different platforms for employee authentication. However, due to
security concerns, the company decides to implement a secure encryption scheme before sharing
the data.
The encryption scheme they choose involves using a symmetric key system where each pixel
value of the fingerprint image is replaced by the result of applying a modular operation. If the
original pixel value is p= 125 and the symmetric key is k= 7, determine the encrypted pixel value.
Solution 19.
To determine the encrypted pixel value, we need to apply the modular operation using the
symmetric key. The encryption process is represented by the formula:
Encrypted pixel value = (p+k) mod 256
Given p= 125 and k= 7, we can substitute these values into the formula:
Encrypted pixel value = (125 + 7) mod 256
Encrypted pixel value = 132 mod 256
Encrypted pixel value = 132 −256 ×132
256
Encrypted pixel value = 132 −256 ×0
Encrypted pixel value = 132
Therefore, the encrypted pixel value is 132.
19 20. BIOMETRIC IDENTIFICATION ERRORS AND THEIR IMPACT ON SECURITY MEA-
SURES
Problem 20. A biometric system is being evaluated for its performance in identifying individuals
based on facial recognition. In a test dataset of 500 individuals, the system produced the following
results:
- True positives: 450 individuals correctly identified - False positives: 20 individuals incorrectly
identified - False negatives: 10 individuals not identified
Calculate the following metrics for the system:
a) The accuracy of the system (in percentage).
b) The false acceptance rate (FAR) of the system (in percentage).
c) The false rejection rate (FRR) of the system (in percentage).
Solution 20.
a) The accuracy of the system is given by the formula:
Accuracy =True Positives +True Negatives
Total Population =450
500 = 0.9
Converting this to a percentage:
Accuracy = 0.9×100% = 90%
Therefore, the accuracy of the system is 90
b) The false acceptance rate (FAR) of the system is calculated as:
FAR =False Positives
False Positives + True Negatives =20
20 + 500 −450 =20
70 = 0.2857
Converting this to a percentage:
FAR = 0.2857 ×100% = 28.57%
Therefore, the false acceptance rate (FAR) of the system is 28.57
c) The false rejection rate (FRR) of the system is calculated as:
FRR =False Negatives
False Negatives + True Positives =10
10 + 450 =10
460 ≈0.0217
Converting this to a percentage:
FRR = 0.0217 ×100% ≈2.17%
Therefore, the false rejection rate (FRR) of the system is approximately 2.17
Solution 2. Let’s calculate the probability of none of the 4 attempts being successful, and then
subtract this from 1 to find the probability that at least one attempt is successful.
Let pbe the probability of success (tricking the system) on a single attempt, which is 10% or
0.1.
The probability of failure on a single attempt is 1−p= 1 −0.1=0.9.
The probability of none of the 4 attempts being successful is:
(0.9)4= 0.6561
Therefore, the probability of at least one attempt being successful is:
1−0.6561 = 0.3439
So, the probability that at least one of the 4 attempts will be successful is approximately 34.39%.
3 3. ISSUES WITH IRIS AUTHENTICATION IN LOW LIGHTING CONDITIONS
Problem 3. The iris recognition system at a high-security facility struggled to authenticate
individuals in low lighting conditions due to excessive noise in the captured images. One of the
authentication methods used is based on the comparison of feature vectors extracted from the iris
image.
Given two feature vectors X= [1.2,3.4,−2.1,0.9] and Y= [0.8,3.9,−2.3,1.1], calculate the
Euclidean distance between these feature vectors as a measure of similarity.
Solution 3.
1. Let’s calculate the Euclidean distance between the two feature vectors Xand Y, given by:
euclidean distance =v
u
u
t
n
X
i=1
(Xi−Yi)2
where Xiand Yiare the elements of the feature vectors Xand Yrespectively.
2. Plugging in the values for Xand Y:
euclidean distance =p(1.2−0.8)2+ (3.4−3.9)2+ (−2.1+2.3)2+ (0.9−1.1)2
euclidean distance =√0.16 + 0.25 + 0.04 + 0.04
euclidean distance =√0.49 = 0.7
Hence, the Euclidean distance between the feature vectors Xand Yis 0.7 units.
4 Biometric Security and Identity Management
Problem: Consider a biometric system that uses fingerprint recognition for authentication. The
system has an equal error rate of 0.05. If there are 1000 users attempting to authenticate using this
system, calculate the number of users that will be incorrectly authenticated due to false rejection
and false acceptance.
Solution: Given the equal error rate (EER) is 0.05, this means that the false rejection rate
(FRR) is equal to the false acceptance rate (FAR) at the threshold where they intersect.
Let Nbe the total number of users attempting to authenticate, which is 1000.
1. Finding the number of users incorrectly rejected:
The false rejection rate (FRR) is 0.05. This means that 5
Number of users incorrectly rejected = FRR ×N Number of users incorrectly rejected = 0.05 ×
1000 Number of users incorrectly rejected = 50 users
So, 50 users will be incorrectly rejected.
2. Finding the number of users incorrectly accepted:
The false acceptance rate (FAR) is also 0.05. This means that 5
Number of impostors incorrectly accepted = FAR ×N Number of impostors incorrectly accepted
= 0.05 ×1000 Number of impostors incorrectly accepted = 50 users
Therefore, 50 impostors will be incorrectly accepted during authentication process.
5 5. LACK OF STANDARDIZATION IN BIOMETRIC DATA STORAGE AND ENCRYPTION
Problem 5. Many biometric systems lack standardization in storing and encrypting biometric
data. As a result, a company is considering implementing a biometric access system for their
employees which uses fingerprint recognition technology. The system saves the fingerprint data in
an encrypted format using a unique encryption algorithm. The system has a false acceptance rate
(FAR) of 0.01 and a false rejection rate (FRR) of 0.05.
If the company has 1000 employees and each employee tries to access the system twice a day
on average, calculate the following:
a) The expected number of false acceptances per day.
b) The expected number of false rejections per day.
c) The total expected number of authentication errors per day.
Solution 5.
a) To calculate the expected number of false acceptances per day, we need to consider the
FAR and the total number of attempted accesses by the 1000 employees.
The FAR represents the probability of the system incorrectly accepting a false fingerprint as a
match. Since FAR = 0.01, the probability of a false acceptance is 0.01.
Let pF A be the probability of false acceptance, then:
pF A = 0.01
The company has 1000 employees trying to access the system twice a day on average, which
results in 2000 attempted accesses per day. Therefore, the expected number of false acceptances
per day can be calculated as:
Expected number of false acceptances per day =pF A ×2000
Expected number of false acceptances per day = 0.01 ×2000 = 20
Therefore, the expected number of false acceptances per day is 20.
b) To calculate the expected number of false rejections per day, we need to consider the FRR
and the total number of attempted accesses by the 1000 employees.
Similar to the FAR, the FRR represents the probability of the system incorrectly rejecting a valid
fingerprint as a non-match. Since FRR = 0.05, the probability of a false rejection is 0.05.
Let pF R be the probability of false rejection, then:
pF R = 0.05
The expected number of false rejections per day can be calculated as:
Expected number of false rejections per day =pF R ×2000
Expected number of false rejections per day = 0.05 ×2000 = 100
Therefore, the expected number of false rejections per day is 100.
c) The total expected number of authentication errors per day is the sum of false acceptances
and false rejections. Therefore:
Total expected number of authentication errors per day =Number of false acceptances+Number of false rejections
Total expected number of authentication errors per day = 20 + 100 = 120
Thus, the total expected number of authentication errors per day is 120.
6 6. BIOMETRIC SPOOFING TECHNIQUES AND THE NEED FOR ANTI-SPOOFING MEA-
SURES
Problem 6. A biometric security system uses fingerprint recognition for authentication. In
testing the system, it is found that the False Acceptance Rate (FAR) is 0.1% and the False Rejection
Rate (FRR) is 1%. Calculate the Equal Error Rate (EER) for this system.
Solution 6. The Equal Error Rate (EER) is the point at which the False Acceptance Rate (FAR)
is equal to the False Rejection Rate (FRR).
Given FAR = 0.1% and FRR = 1%, we need to find the EER.
Let the threshold be denoted as T. The EER occurs when FAR = FRR.
From the definitions:
FAR =P(reject |genuine)
FRR =P(accept |imposter)
Therefore, the EER is the average of FAR and FRR.
EER =FAR +FRR
2
EER =0.1+1
2= 0.55%
So, the Equal Error Rate (EER) for this system is 0.55%.
By analyzing the EER, we can see how well the system balances between rejecting valid users
and accepting imposters.
7 Biometric Security and Identity Management
Problem: Biometric systems often use multiple modalities to enhance security. Consider a
biometric system that combines fingerprint and iris recognition technologies. The false acceptance
rates for fingerprint and iris recognition are 0.1% and 0.01%, respectively.
a) What is the probability of falsely accepting an impostor using only fingerprint recognition?
b) What is the probability of falsely accepting an impostor using only iris recognition?
c) If the system requires both modalities to match in order to authenticate, what is the overall
false acceptance rate of the system?
Solution:
a) The probability of falsely accepting an impostor using only fingerprint recognition is given by
the false acceptance rate of the fingerprint modality, which is 0.1% or 0.001.
b) The probability of falsely accepting an impostor using only iris recognition is given by the
false acceptance rate of the iris modality, which is 0.01% or 0.0001.
c) Since the system requires both modalities to match for authentication, the overall false ac-
ceptance rate of the system can be calculated by multiplying the false acceptance rates of each
modality:
Overall False Acceptance Rate =False Acceptance Rate of Fingerprint×False Acceptance Rate of Iris
= 0.001 ×0.0001 = 0.0000001
Therefore, the overall false acceptance rate of the system when using both fingerprint and iris
recognition modalities is 0.00001%.
8 8. AUTHENTICATION FAILURES DUE TO AGING OR CHANGING BIOMETRIC CHARAC-
TERISTICS
Problem 8. In a biometric security system, the false acceptance rate (FAR) is measured to
be 0.02 and the false rejection rate (FRR) is 0.03. Calculate the Equal Error Rate (EER) for this
system.
Solution 8. The Equal Error Rate (EER) is the point where the false acceptance rate (FAR)
and the false rejection rate (FRR) are equal. Mathematically, EER is given by the average of FAR
and FRR when they are equal:
EER = (FAR + FRR) / 2
Given: FAR = 0.02 FRR = 0.03
Substitute the values into the formula: EER = (0.02 + 0.03) / 2 EER = 0.05 / 2 EER = 0.025
Therefore, the Equal Error Rate (EER) for this biometric security system is 0.025.
9 9. PROBLEMS WITH BIOMETRIC MATCHING ALGORITHMS AND ACCURACY RATES
Problem 9.
A security system that uses fingerprint recognition has an accuracy rate of 99.5%. If a random
sample of 500 fingerprints are scanned, calculate the probability that:
a) Exactly 3 fingerprints are incorrectly recognized.
b) At most 5 fingerprints are incorrectly recognized.
c) More than 10 fingerprints are incorrectly recognized.
Solution 9.
a) To find the probability that exactly 3 fingerprints are incorrectly recognized, we can use the
binomial probability formula:
P(X=k) = n
k·pk·(1 −p)n−k
where n= 500 (number of fingerprint scans), k= 3 (number of incorrectly recognized finger-
prints), and p= 1 −0.995 = 0.005 (probability of incorrectly recognizing a fingerprint).
P(X= 3) = 500
3·0.0053·0.995497
Calculating this probability gives us the answer for part a.
b) To find the probability that at most 5 fingerprints are incorrectly recognized, we need to
calculate
P(X≤5) = P(X= 0) + P(X= 1) + P(X= 2) + P(X= 3) + P(X= 4) + P(X= 5)
Using the binomial probability formula for each value of k, we can add up the probabilities to
find the answer for part b.
c) To find the probability that more than 10 fingerprints are incorrectly recognized, we can cal-
culate the complement of the probability that at most 10 are incorrectly recognized:
P(X > 10) = 1 −P(X≤10)
Using the binomial probability formula as in part b, we can find P(X≤10) and subtract it from
1 to get the answer for part c.
10 10. PRIVACY CONCERNS AND DATA PROTECTION IN BIOMETRIC DATABASES
Problem 10. A biometric system has an FAR (False Acceptance Rate) of 0.1% and an FRR
(False Rejection Rate) of 5%. If an individual tries to authenticate using this system 50 times, what
is the probability that they will be falsely rejected at least once?
Solution 10. Given that the FRR (False Rejection Rate) is 5%, the probability of a successful
authentication (1 - FRR) for one attempt is 95% or 0.95. Let pbe the probability of being falsely
rejected at least once in 50 attempts.
The probability of being falsely rejected at least once in ntrials is given by 1−(1 −FRR)n.
Therefore, the probability of being falsely rejected at least once in 50 attempts is:
p= 1 −(1 −0.05)50 = 1 −0.9550 ≈0.9231
Therefore, the probability that the individual will be falsely rejected at least once when attempt-
ing to authenticate 50 times using this biometric system is approximately 92.31%.
I. Problem: Biometric System Performance Degradation Over Time
An iris recognition system has an initial false acceptance rate (FAR) of 0.01
Solution
Given: Initial FAR = 0.01Initial FRR = 0.05Degraded FAR = 0.03Degraded FRR = 0.08
EER is the point where FAR = FRR.
Initial system: EER = (Initial FAR + Initial FRR) / 2 EER = (0.0001 + 0.0005) / 2 EER = 0.0003
Degraded system: EER = (Degraded FAR + Degraded FRR) / 2 EER = (0.0003 + 0.0008) / 2
EER = 0.00055
Therefore, the EER for the initial system is 0.0003 and for the degraded system is 0.00055.
11 12. ISSUES WITH BIOMETRIC IDENTIFICATION IN LARGE-SCALE SYSTEMS
Problem 12. In a large-scale biometric identification system, the false acceptance rate (FAR) is
0.1% and the false rejection rate (FRR) is 5%. If 1000 individuals attempt to authenticate themselves
using the system, calculate:
a) The number of individuals falsely accepted by the system.
b) The number of individuals falsely rejected by the system.
c) The overall accuracy of the system.
Solution 12.
a) The false acceptance rate (FAR) is 0.1%, which means 0.001 or 1
1000 of attempts are falsely
accepted. For 1000 individuals, the number falsely accepted can be calculated as:
False Accepted = 1000 ×0.001 = 1 individual
Therefore, the system falsely accepts 1individual.
b) The false rejection rate (FRR) is 5%, which means 5% or 0.05 of attempts are falsely rejected.
For 1000 individuals, the number falsely rejected can be calculated as:
False Rejected = 1000 ×0.05 = 50 individuals
Therefore, the system falsely rejects 50 individuals.
c) The overall accuracy of the system can be calculated as:
Accuracy = (1 −FAR)×100% = (1 −0.001) ×100% = 99.9%
Thus, the overall accuracy of the system is 99.9%.
12 13. CHALLENGES IN SECURING BIOMETRIC DATA DURING TRANSMISSION AND
STORAGE
Problem 13. Biometric data of a user is being transmitted over a network for authentication
purposes. The biometric data consists of a fingerprint image of size 500 ×500 pixels, where each
pixel is represented by an 8-bit grayscale value. During transmission, encryption is applied using
a symmetric key algorithm that increases the size of the data by 20%. If the encryption key used
is 128 bits long, calculate the size of the encrypted biometric data in kilobytes.
Solution 13. a) The size of the original fingerprint image in bytes can be calculated as:
Image size =Width ×Height ×Bit depth
Since each pixel is represented by an 8-bit grayscale value, the bit depth is 8. Given the image
size is 500 ×500 pixels:
Image size = 500 ×500 ×8bits
b) To convert bits to bytes, we divide by 8:
Image size in bytes =500 ×500 ×8
8bytes
c) After encryption, the size of the data increases by 20%. The new size of the encrypted data
can be calculated as:
New encrypted data size = 1.2×Original data size (in bytes)
d) Given that the encryption key is 128 bits long, to convert it to bytes, we divide by 8:
Key size =128
8bytes
e) The total size of the encrypted biometric data is the sum of the encrypted data size and the
key size:
Total size =New encrypted data size +Key size
Now, substitute the values calculated in parts a, b, c, and d to find the total size of the encrypted
biometric data in kilobytes.
13 14. IMPLICATIONS OF BIOMETRIC DATA BREACHES AND IDENTITY THEFT
Problem 14. A company’s biometric security system has been breached, and the fingerprint
data of 1000 employees has been stolen. If each fingerprint template contains 500 bytes of data and
the company offers a reward of 500percompromisedfingerprinttemplate, calculatethetotalrewardamountthecompanywillneedtopaytoaf f ectedemployees.
Solution 14. Given that: - Number of compromised fingerprint templates = 1000 - Size of each
fingerprint template = 500 bytes - Reward offered per compromised template = 500
Step 1: Calculate the total amount of data compromised Total amount of data compromised
= Number of compromised templates ×Size of each template Total amount of data compromised
= 1000 ×500 bytes Total amount of data compromised = 500,000 bytes
Step 2: Calculate the total reward amount Total reward amount = Number of compromised
templates ×Reward offered per template Total reward amount = 1000 ×500T otalrewardamount =500,000
Therefore, the company will need to pay a total reward amount of 500,000totheaffectedemployees.
14 15. CULTURAL BIASES AND LIMITATIONS IN BIOMETRIC RECOGNITION SYSTEMS
Problem 15. A facial recognition system has been trained on a dataset composed mainly of
individuals of Asian descent. When tested on a diverse population that includes individuals of
African descent, the system shows a significantly higher error rate for this subgroup.
The error rates for the system on the test set are as follows:
•Overall error rate: 5
•Error rate for individuals of Asian descent: 3
•Error rate for individuals of African descent: 12
a) What is the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent?
b) If the system correctly identifies 90
c) Considering these results, discuss the potential impact of cultural biases on the effectiveness
of biometric recognition systems.
Solution 15.
a) The odds ratio of misidentification for individuals of African descent compared to individuals
of Asian descent can be calculated using the error rates:
Odds ratio =Error rate for African descent
Error rate for Asian descent =0.12
0.03 = 4
Therefore, the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent is 4.
b) Given that the system correctly identifies 90
Accuracy rate for African descent = 1 −Error rate for African descent = 1 −0.12 = 0.88 = 88%
Therefore, the accuracy rate for individuals of African descent is 88
c) The significantly higher error rate for individuals of African descent in the facial recognition
system compared to individuals of Asian descent highlights a bias that may have been introduced
during the training phase, where the dataset was skewed towards a specific ethnicity. This cultural
bias can lead to inequitable outcomes, with certain demographic groups facing higher risks of
misidentification and potential discrimination. Addressing cultural biases in biometric recognition
systems is crucial to ensure fair and accurate identification across diverse populations.
15 16. LEGAL AND ETHICAL IMPLICATIONS OF BIOMETRIC DATA COLLECTION AND
USE
Problem 16. A company is implementing a biometric authentication system for its employees
using fingerprint recognition. The system is expected to process 1000 fingerprint scans per day. If
each fingerprint scan produces a file size of 500 KB, calculate the total amount of data collected
by the system in one month.
Solution 16. Given that the system processes 1000 fingerprint scans per day and each scan
produces a file size of 500 KB, we can calculate the total data collected by the system in one day
as follows:
Total data collected per day = Number of scans per day x File size per scan Total data collected
per day = 1000 x 500 KB Total data collected per day = 500,000 KB
To convert KB to GB, we use the conversion factor: 1 GB = 1024 KB
Total data collected per day in GB = 500,000 KB / 1024 = 488.28 GB
Since there are typically 30 days in a month, we can calculate the total amount of data collected
by the system in one month as follows:
Total data collected per month = Total data collected per day x Number of days in a month Total
data collected per month = 488.28 GB x 30 Total data collected per month = 14,648.4 GB
Therefore, the total amount of data collected by the system in one month is 14,648.4 GB.
16 17. CHALLENGES IN IMPLEMENTING BIOMETRIC AUTHENTICATION IN IOT DEVICES
Problem 17. A company is considering implementing biometric authentication for their IoT
devices to enhance security. They estimate that the false acceptance rate (FAR) of the biometric
system is 0.1% and the false rejection rate (FRR) is 1%. If an unauthorized user attempts to access
the device 50 times, calculate the probability that the system will grant access to this unauthorized
user at least once.
Solution 17. To calculate the probability of the system granting access to the unauthorized
user at least once, we need to consider the probability of false acceptance for each attempt. Let’s
denote the probability of a false acceptance as P(F A).
Given that FAR is 0.1%, we have:
P(F A) = 0.001
Therefore, the probability of rejecting a false attempt is:
P(¬F A) = 1 −P(F A) = 1 −0.001 = 0.999
The probability that the system will grant access to the unauthorized user at least once in 50
attempts is the complement of the probability that it will reject all 50 attempts. Thus, the probability
is:
P(Access granted at least once)=1−P(Access not granted at all)
Since each attempt is independent, the probability of the system not granting access to the
unauthorized user in a single attempt is P(¬F A). Therefore, the probability that the system will not
grant access in 50 attempts is:
P(Access not granted at all)=(P(¬F A))n= 0.99950
Calculating this probability:
P(Access not granted at all)=0.99950 ≈0.950 = 95%
So, the probability that the system will grant access to the unauthorized user at least once is
approximately:
P(Access granted at least once)=1−0.950 = 0.050 = 5%
Therefore, the system has a 5% chance of granting access to the unauthorized user at least
once out of 50 attempts.
17 18. USER ACCEPTANCE AND TRUST ISSUES WITH BIOMETRIC SECURITY MEASURES
Problem 18.
A study is conducted to analyze the user acceptance of a new biometric authentication system.
The study involved 100 participants, and the results showed that 65 participants were willing to use
the biometric system, while the remaining participants expressed concerns about privacy and did
not want to adopt it.
a) Calculate the user acceptance rate in percentage.
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same, how many participants would be willing to use the biometric system?
c) Discuss why user acceptance and trust are essential factors in the implementation of bio-
metric security measures.
Solution 18.
a) The user acceptance rate can be calculated as follows:
User acceptance rate =Number of participants willing to use the system
Total number of participants ×100%
In this case, the number of participants willing to use the system is 65, and the total number of
participants is 100. Therefore,
User acceptance rate =65
100 ×100% = 65%
So, the user acceptance rate is 65
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same at 65
Number of participants willing to use the system =Acceptance rate×Total number of participants
Number of participants willing to use the system = 0.65 ×150 = 97.5
Since we cannot have a fraction of a participant, we can estimate that approximately 98 partic-
ipants would be willing to use the biometric system.
c) User acceptance and trust are crucial factors in the implementation of biometric security
measures because:
1. If users do not trust the system, they might not use it properly or try to bypass it, leading to
security vulnerabilities. 2. Acceptance by users is essential for the success of the system, as low
adoption rates can render the biometric security measures ineffective. 3. Ensuring user acceptance
and trust can help improve the overall user experience and satisfaction with the security system. 4.
Addressing privacy concerns and building trust can help mitigate potential legal and ethical issues
related to biometric data collection and storage.
18 19. CHALLENGES IN SECURELY SHARING BIOMETRIC DATA ACROSS DIFFERENT
PLATFORMS
Problem 19. Consider a scenario where a company wants to securely share biometric data
(such as fingerprint scans) across different platforms for employee authentication. However, due to
security concerns, the company decides to implement a secure encryption scheme before sharing
the data.
The encryption scheme they choose involves using a symmetric key system where each pixel
value of the fingerprint image is replaced by the result of applying a modular operation. If the
original pixel value is p= 125 and the symmetric key is k= 7, determine the encrypted pixel value.
Solution 19.
To determine the encrypted pixel value, we need to apply the modular operation using the
symmetric key. The encryption process is represented by the formula:
Encrypted pixel value = (p+k) mod 256
Given p= 125 and k= 7, we can substitute these values into the formula:
Encrypted pixel value = (125 + 7) mod 256
Encrypted pixel value = 132 mod 256
Encrypted pixel value = 132 −256 ×132
256
Encrypted pixel value = 132 −256 ×0
Encrypted pixel value = 132
Therefore, the encrypted pixel value is 132.
19 20. BIOMETRIC IDENTIFICATION ERRORS AND THEIR IMPACT ON SECURITY MEA-
SURES
Problem 20. A biometric system is being evaluated for its performance in identifying individuals
based on facial recognition. In a test dataset of 500 individuals, the system produced the following
results:
- True positives: 450 individuals correctly identified - False positives: 20 individuals incorrectly
identified - False negatives: 10 individuals not identified
Calculate the following metrics for the system:
a) The accuracy of the system (in percentage).
b) The false acceptance rate (FAR) of the system (in percentage).
c) The false rejection rate (FRR) of the system (in percentage).
Solution 20.
a) The accuracy of the system is given by the formula:
Accuracy =True Positives +True Negatives
Total Population =450
500 = 0.9
Converting this to a percentage:
Accuracy = 0.9×100% = 90%
Therefore, the accuracy of the system is 90
b) The false acceptance rate (FAR) of the system is calculated as:
FAR =False Positives
False Positives + True Negatives =20
20 + 500 −450 =20
70 = 0.2857
Converting this to a percentage:
FAR = 0.2857 ×100% = 28.57%
Therefore, the false acceptance rate (FAR) of the system is 28.57
c) The false rejection rate (FRR) of the system is calculated as:
FRR =False Negatives
False Negatives + True Positives =10
10 + 450 =10
460 ≈0.0217
Converting this to a percentage:
FRR = 0.0217 ×100% ≈2.17%
Therefore, the false rejection rate (FRR) of the system is approximately 2.17
Solution 2. Let’s calculate the probability of none of the 4 attempts being successful, and then
subtract this from 1 to find the probability that at least one attempt is successful.
Let pbe the probability of success (tricking the system) on a single attempt, which is 10% or
0.1.
The probability of failure on a single attempt is 1−p= 1 −0.1=0.9.
The probability of none of the 4 attempts being successful is:
(0.9)4= 0.6561
Therefore, the probability of at least one attempt being successful is:
1−0.6561 = 0.3439
So, the probability that at least one of the 4 attempts will be successful is approximately 34.39%.
3 3. ISSUES WITH IRIS AUTHENTICATION IN LOW LIGHTING CONDITIONS
Problem 3. The iris recognition system at a high-security facility struggled to authenticate
individuals in low lighting conditions due to excessive noise in the captured images. One of the
authentication methods used is based on the comparison of feature vectors extracted from the iris
image.
Given two feature vectors X= [1.2,3.4,−2.1,0.9] and Y= [0.8,3.9,−2.3,1.1], calculate the
Euclidean distance between these feature vectors as a measure of similarity.
Solution 3.
1. Let’s calculate the Euclidean distance between the two feature vectors Xand Y, given by:
euclidean distance =v
u
u
t
n
X
i=1
(Xi−Yi)2
where Xiand Yiare the elements of the feature vectors Xand Yrespectively.
2. Plugging in the values for Xand Y:
euclidean distance =p(1.2−0.8)2+ (3.4−3.9)2+ (−2.1+2.3)2+ (0.9−1.1)2
euclidean distance =√0.16 + 0.25 + 0.04 + 0.04
euclidean distance =√0.49 = 0.7
Hence, the Euclidean distance between the feature vectors Xand Yis 0.7 units.
4 Biometric Security and Identity Management
Problem: Consider a biometric system that uses fingerprint recognition for authentication. The
system has an equal error rate of 0.05. If there are 1000 users attempting to authenticate using this
system, calculate the number of users that will be incorrectly authenticated due to false rejection
and false acceptance.
Solution: Given the equal error rate (EER) is 0.05, this means that the false rejection rate
(FRR) is equal to the false acceptance rate (FAR) at the threshold where they intersect.
Let Nbe the total number of users attempting to authenticate, which is 1000.
1. Finding the number of users incorrectly rejected:
The false rejection rate (FRR) is 0.05. This means that 5
Number of users incorrectly rejected = FRR ×N Number of users incorrectly rejected = 0.05 ×
1000 Number of users incorrectly rejected = 50 users
So, 50 users will be incorrectly rejected.
2. Finding the number of users incorrectly accepted:
The false acceptance rate (FAR) is also 0.05. This means that 5
Number of impostors incorrectly accepted = FAR ×N Number of impostors incorrectly accepted
= 0.05 ×1000 Number of impostors incorrectly accepted = 50 users
Therefore, 50 impostors will be incorrectly accepted during authentication process.
5 5. LACK OF STANDARDIZATION IN BIOMETRIC DATA STORAGE AND ENCRYPTION
Problem 5. Many biometric systems lack standardization in storing and encrypting biometric
data. As a result, a company is considering implementing a biometric access system for their
employees which uses fingerprint recognition technology. The system saves the fingerprint data in
an encrypted format using a unique encryption algorithm. The system has a false acceptance rate
(FAR) of 0.01 and a false rejection rate (FRR) of 0.05.
If the company has 1000 employees and each employee tries to access the system twice a day
on average, calculate the following:
a) The expected number of false acceptances per day.
b) The expected number of false rejections per day.
c) The total expected number of authentication errors per day.
Solution 5.
a) To calculate the expected number of false acceptances per day, we need to consider the
FAR and the total number of attempted accesses by the 1000 employees.
The FAR represents the probability of the system incorrectly accepting a false fingerprint as a
match. Since FAR = 0.01, the probability of a false acceptance is 0.01.
Let pF A be the probability of false acceptance, then:
pF A = 0.01
The company has 1000 employees trying to access the system twice a day on average, which
results in 2000 attempted accesses per day. Therefore, the expected number of false acceptances
per day can be calculated as:
Expected number of false acceptances per day =pF A ×2000
Expected number of false acceptances per day = 0.01 ×2000 = 20
Therefore, the expected number of false acceptances per day is 20.
b) To calculate the expected number of false rejections per day, we need to consider the FRR
and the total number of attempted accesses by the 1000 employees.
Similar to the FAR, the FRR represents the probability of the system incorrectly rejecting a valid
fingerprint as a non-match. Since FRR = 0.05, the probability of a false rejection is 0.05.
Let pF R be the probability of false rejection, then:
pF R = 0.05
The expected number of false rejections per day can be calculated as:
Expected number of false rejections per day =pF R ×2000
Expected number of false rejections per day = 0.05 ×2000 = 100
Therefore, the expected number of false rejections per day is 100.
c) The total expected number of authentication errors per day is the sum of false acceptances
and false rejections. Therefore:
Total expected number of authentication errors per day =Number of false acceptances+Number of false rejections
Total expected number of authentication errors per day = 20 + 100 = 120
Thus, the total expected number of authentication errors per day is 120.
6 6. BIOMETRIC SPOOFING TECHNIQUES AND THE NEED FOR ANTI-SPOOFING MEA-
SURES
Problem 6. A biometric security system uses fingerprint recognition for authentication. In
testing the system, it is found that the False Acceptance Rate (FAR) is 0.1% and the False Rejection
Rate (FRR) is 1%. Calculate the Equal Error Rate (EER) for this system.
Solution 6. The Equal Error Rate (EER) is the point at which the False Acceptance Rate (FAR)
is equal to the False Rejection Rate (FRR).
Given FAR = 0.1% and FRR = 1%, we need to find the EER.
Let the threshold be denoted as T. The EER occurs when FAR = FRR.
From the definitions:
FAR =P(reject |genuine)
FRR =P(accept |imposter)
Therefore, the EER is the average of FAR and FRR.
EER =FAR +FRR
2
EER =0.1+1
2= 0.55%
So, the Equal Error Rate (EER) for this system is 0.55%.
By analyzing the EER, we can see how well the system balances between rejecting valid users
and accepting imposters.
7 Biometric Security and Identity Management
Problem: Biometric systems often use multiple modalities to enhance security. Consider a
biometric system that combines fingerprint and iris recognition technologies. The false acceptance
rates for fingerprint and iris recognition are 0.1% and 0.01%, respectively.
a) What is the probability of falsely accepting an impostor using only fingerprint recognition?
b) What is the probability of falsely accepting an impostor using only iris recognition?
c) If the system requires both modalities to match in order to authenticate, what is the overall
false acceptance rate of the system?
Solution:
a) The probability of falsely accepting an impostor using only fingerprint recognition is given by
the false acceptance rate of the fingerprint modality, which is 0.1% or 0.001.
b) The probability of falsely accepting an impostor using only iris recognition is given by the
false acceptance rate of the iris modality, which is 0.01% or 0.0001.
c) Since the system requires both modalities to match for authentication, the overall false ac-
ceptance rate of the system can be calculated by multiplying the false acceptance rates of each
modality:
Overall False Acceptance Rate =False Acceptance Rate of Fingerprint×False Acceptance Rate of Iris
= 0.001 ×0.0001 = 0.0000001
Therefore, the overall false acceptance rate of the system when using both fingerprint and iris
recognition modalities is 0.00001%.
8 8. AUTHENTICATION FAILURES DUE TO AGING OR CHANGING BIOMETRIC CHARAC-
TERISTICS
Problem 8. In a biometric security system, the false acceptance rate (FAR) is measured to
be 0.02 and the false rejection rate (FRR) is 0.03. Calculate the Equal Error Rate (EER) for this
system.
Solution 8. The Equal Error Rate (EER) is the point where the false acceptance rate (FAR)
and the false rejection rate (FRR) are equal. Mathematically, EER is given by the average of FAR
and FRR when they are equal:
EER = (FAR + FRR) / 2
Given: FAR = 0.02 FRR = 0.03
Substitute the values into the formula: EER = (0.02 + 0.03) / 2 EER = 0.05 / 2 EER = 0.025
Therefore, the Equal Error Rate (EER) for this biometric security system is 0.025.
9 9. PROBLEMS WITH BIOMETRIC MATCHING ALGORITHMS AND ACCURACY RATES
Problem 9.
A security system that uses fingerprint recognition has an accuracy rate of 99.5%. If a random
sample of 500 fingerprints are scanned, calculate the probability that:
a) Exactly 3 fingerprints are incorrectly recognized.
b) At most 5 fingerprints are incorrectly recognized.
c) More than 10 fingerprints are incorrectly recognized.
Solution 9.
a) To find the probability that exactly 3 fingerprints are incorrectly recognized, we can use the
binomial probability formula:
P(X=k) = n
k·pk·(1 −p)n−k
where n= 500 (number of fingerprint scans), k= 3 (number of incorrectly recognized finger-
prints), and p= 1 −0.995 = 0.005 (probability of incorrectly recognizing a fingerprint).
P(X= 3) = 500
3·0.0053·0.995497
Calculating this probability gives us the answer for part a.
b) To find the probability that at most 5 fingerprints are incorrectly recognized, we need to
calculate
P(X≤5) = P(X= 0) + P(X= 1) + P(X= 2) + P(X= 3) + P(X= 4) + P(X= 5)
Using the binomial probability formula for each value of k, we can add up the probabilities to
find the answer for part b.
c) To find the probability that more than 10 fingerprints are incorrectly recognized, we can cal-
culate the complement of the probability that at most 10 are incorrectly recognized:
P(X > 10) = 1 −P(X≤10)
Using the binomial probability formula as in part b, we can find P(X≤10) and subtract it from
1 to get the answer for part c.
10 10. PRIVACY CONCERNS AND DATA PROTECTION IN BIOMETRIC DATABASES
Problem 10. A biometric system has an FAR (False Acceptance Rate) of 0.1% and an FRR
(False Rejection Rate) of 5%. If an individual tries to authenticate using this system 50 times, what
is the probability that they will be falsely rejected at least once?
Solution 10. Given that the FRR (False Rejection Rate) is 5%, the probability of a successful
authentication (1 - FRR) for one attempt is 95% or 0.95. Let pbe the probability of being falsely
rejected at least once in 50 attempts.
The probability of being falsely rejected at least once in ntrials is given by 1−(1 −FRR)n.
Therefore, the probability of being falsely rejected at least once in 50 attempts is:
p= 1 −(1 −0.05)50 = 1 −0.9550 ≈0.9231
Therefore, the probability that the individual will be falsely rejected at least once when attempt-
ing to authenticate 50 times using this biometric system is approximately 92.31%.
I. Problem: Biometric System Performance Degradation Over Time
An iris recognition system has an initial false acceptance rate (FAR) of 0.01
Solution
Given: Initial FAR = 0.01Initial FRR = 0.05Degraded FAR = 0.03Degraded FRR = 0.08
EER is the point where FAR = FRR.
Initial system: EER = (Initial FAR + Initial FRR) / 2 EER = (0.0001 + 0.0005) / 2 EER = 0.0003
Degraded system: EER = (Degraded FAR + Degraded FRR) / 2 EER = (0.0003 + 0.0008) / 2
EER = 0.00055
Therefore, the EER for the initial system is 0.0003 and for the degraded system is 0.00055.
11 12. ISSUES WITH BIOMETRIC IDENTIFICATION IN LARGE-SCALE SYSTEMS
Problem 12. In a large-scale biometric identification system, the false acceptance rate (FAR) is
0.1% and the false rejection rate (FRR) is 5%. If 1000 individuals attempt to authenticate themselves
using the system, calculate:
a) The number of individuals falsely accepted by the system.
b) The number of individuals falsely rejected by the system.
c) The overall accuracy of the system.
Solution 12.
a) The false acceptance rate (FAR) is 0.1%, which means 0.001 or 1
1000 of attempts are falsely
accepted. For 1000 individuals, the number falsely accepted can be calculated as:
False Accepted = 1000 ×0.001 = 1 individual
Therefore, the system falsely accepts 1individual.
b) The false rejection rate (FRR) is 5%, which means 5% or 0.05 of attempts are falsely rejected.
For 1000 individuals, the number falsely rejected can be calculated as:
False Rejected = 1000 ×0.05 = 50 individuals
Therefore, the system falsely rejects 50 individuals.
c) The overall accuracy of the system can be calculated as:
Accuracy = (1 −FAR)×100% = (1 −0.001) ×100% = 99.9%
Thus, the overall accuracy of the system is 99.9%.
12 13. CHALLENGES IN SECURING BIOMETRIC DATA DURING TRANSMISSION AND
STORAGE
Problem 13. Biometric data of a user is being transmitted over a network for authentication
purposes. The biometric data consists of a fingerprint image of size 500 ×500 pixels, where each
pixel is represented by an 8-bit grayscale value. During transmission, encryption is applied using
a symmetric key algorithm that increases the size of the data by 20%. If the encryption key used
is 128 bits long, calculate the size of the encrypted biometric data in kilobytes.
Solution 13. a) The size of the original fingerprint image in bytes can be calculated as:
Image size =Width ×Height ×Bit depth
Since each pixel is represented by an 8-bit grayscale value, the bit depth is 8. Given the image
size is 500 ×500 pixels:
Image size = 500 ×500 ×8bits
b) To convert bits to bytes, we divide by 8:
Image size in bytes =500 ×500 ×8
8bytes
c) After encryption, the size of the data increases by 20%. The new size of the encrypted data
can be calculated as:
New encrypted data size = 1.2×Original data size (in bytes)
d) Given that the encryption key is 128 bits long, to convert it to bytes, we divide by 8:
Key size =128
8bytes
e) The total size of the encrypted biometric data is the sum of the encrypted data size and the
key size:
Total size =New encrypted data size +Key size
Now, substitute the values calculated in parts a, b, c, and d to find the total size of the encrypted
biometric data in kilobytes.
13 14. IMPLICATIONS OF BIOMETRIC DATA BREACHES AND IDENTITY THEFT
Problem 14. A company’s biometric security system has been breached, and the fingerprint
data of 1000 employees has been stolen. If each fingerprint template contains 500 bytes of data and
the company offers a reward of 500percompromisedfingerprinttemplate, calculatethetotalrewardamountthecompanywillneedtopaytoaf f ectedemployees.
Solution 14. Given that: - Number of compromised fingerprint templates = 1000 - Size of each
fingerprint template = 500 bytes - Reward offered per compromised template = 500
Step 1: Calculate the total amount of data compromised Total amount of data compromised
= Number of compromised templates ×Size of each template Total amount of data compromised
= 1000 ×500 bytes Total amount of data compromised = 500,000 bytes
Step 2: Calculate the total reward amount Total reward amount = Number of compromised
templates ×Reward offered per template Total reward amount = 1000 ×500T otalrewardamount =500,000
Therefore, the company will need to pay a total reward amount of 500,000totheaffectedemployees.
14 15. CULTURAL BIASES AND LIMITATIONS IN BIOMETRIC RECOGNITION SYSTEMS
Problem 15. A facial recognition system has been trained on a dataset composed mainly of
individuals of Asian descent. When tested on a diverse population that includes individuals of
African descent, the system shows a significantly higher error rate for this subgroup.
The error rates for the system on the test set are as follows:
•Overall error rate: 5
•Error rate for individuals of Asian descent: 3
•Error rate for individuals of African descent: 12
a) What is the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent?
b) If the system correctly identifies 90
c) Considering these results, discuss the potential impact of cultural biases on the effectiveness
of biometric recognition systems.
Solution 15.
a) The odds ratio of misidentification for individuals of African descent compared to individuals
of Asian descent can be calculated using the error rates:
Odds ratio =Error rate for African descent
Error rate for Asian descent =0.12
0.03 = 4
Therefore, the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent is 4.
b) Given that the system correctly identifies 90
Accuracy rate for African descent = 1 −Error rate for African descent = 1 −0.12 = 0.88 = 88%
Therefore, the accuracy rate for individuals of African descent is 88
c) The significantly higher error rate for individuals of African descent in the facial recognition
system compared to individuals of Asian descent highlights a bias that may have been introduced
during the training phase, where the dataset was skewed towards a specific ethnicity. This cultural
bias can lead to inequitable outcomes, with certain demographic groups facing higher risks of
misidentification and potential discrimination. Addressing cultural biases in biometric recognition
systems is crucial to ensure fair and accurate identification across diverse populations.
15 16. LEGAL AND ETHICAL IMPLICATIONS OF BIOMETRIC DATA COLLECTION AND
USE
Problem 16. A company is implementing a biometric authentication system for its employees
using fingerprint recognition. The system is expected to process 1000 fingerprint scans per day. If
each fingerprint scan produces a file size of 500 KB, calculate the total amount of data collected
by the system in one month.
Solution 16. Given that the system processes 1000 fingerprint scans per day and each scan
produces a file size of 500 KB, we can calculate the total data collected by the system in one day
as follows:
Total data collected per day = Number of scans per day x File size per scan Total data collected
per day = 1000 x 500 KB Total data collected per day = 500,000 KB
To convert KB to GB, we use the conversion factor: 1 GB = 1024 KB
Total data collected per day in GB = 500,000 KB / 1024 = 488.28 GB
Since there are typically 30 days in a month, we can calculate the total amount of data collected
by the system in one month as follows:
Total data collected per month = Total data collected per day x Number of days in a month Total
data collected per month = 488.28 GB x 30 Total data collected per month = 14,648.4 GB
Therefore, the total amount of data collected by the system in one month is 14,648.4 GB.
16 17. CHALLENGES IN IMPLEMENTING BIOMETRIC AUTHENTICATION IN IOT DEVICES
Problem 17. A company is considering implementing biometric authentication for their IoT
devices to enhance security. They estimate that the false acceptance rate (FAR) of the biometric
system is 0.1% and the false rejection rate (FRR) is 1%. If an unauthorized user attempts to access
the device 50 times, calculate the probability that the system will grant access to this unauthorized
user at least once.
Solution 17. To calculate the probability of the system granting access to the unauthorized
user at least once, we need to consider the probability of false acceptance for each attempt. Let’s
denote the probability of a false acceptance as P(F A).
Given that FAR is 0.1%, we have:
P(F A) = 0.001
Therefore, the probability of rejecting a false attempt is:
P(¬F A) = 1 −P(F A) = 1 −0.001 = 0.999
The probability that the system will grant access to the unauthorized user at least once in 50
attempts is the complement of the probability that it will reject all 50 attempts. Thus, the probability
is:
P(Access granted at least once)=1−P(Access not granted at all)
Since each attempt is independent, the probability of the system not granting access to the
unauthorized user in a single attempt is P(¬F A). Therefore, the probability that the system will not
grant access in 50 attempts is:
P(Access not granted at all)=(P(¬F A))n= 0.99950
Calculating this probability:
P(Access not granted at all)=0.99950 ≈0.950 = 95%
So, the probability that the system will grant access to the unauthorized user at least once is
approximately:
P(Access granted at least once)=1−0.950 = 0.050 = 5%
Therefore, the system has a 5% chance of granting access to the unauthorized user at least
once out of 50 attempts.
17 18. USER ACCEPTANCE AND TRUST ISSUES WITH BIOMETRIC SECURITY MEASURES
Problem 18.
A study is conducted to analyze the user acceptance of a new biometric authentication system.
The study involved 100 participants, and the results showed that 65 participants were willing to use
the biometric system, while the remaining participants expressed concerns about privacy and did
not want to adopt it.
a) Calculate the user acceptance rate in percentage.
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same, how many participants would be willing to use the biometric system?
c) Discuss why user acceptance and trust are essential factors in the implementation of bio-
metric security measures.
Solution 18.
a) The user acceptance rate can be calculated as follows:
User acceptance rate =Number of participants willing to use the system
Total number of participants ×100%
In this case, the number of participants willing to use the system is 65, and the total number of
participants is 100. Therefore,
User acceptance rate =65
100 ×100% = 65%
So, the user acceptance rate is 65
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same at 65
Number of participants willing to use the system =Acceptance rate×Total number of participants
Number of participants willing to use the system = 0.65 ×150 = 97.5
Since we cannot have a fraction of a participant, we can estimate that approximately 98 partic-
ipants would be willing to use the biometric system.
c) User acceptance and trust are crucial factors in the implementation of biometric security
measures because:
1. If users do not trust the system, they might not use it properly or try to bypass it, leading to
security vulnerabilities. 2. Acceptance by users is essential for the success of the system, as low
adoption rates can render the biometric security measures ineffective. 3. Ensuring user acceptance
and trust can help improve the overall user experience and satisfaction with the security system. 4.
Addressing privacy concerns and building trust can help mitigate potential legal and ethical issues
related to biometric data collection and storage.
18 19. CHALLENGES IN SECURELY SHARING BIOMETRIC DATA ACROSS DIFFERENT
PLATFORMS
Problem 19. Consider a scenario where a company wants to securely share biometric data
(such as fingerprint scans) across different platforms for employee authentication. However, due to
security concerns, the company decides to implement a secure encryption scheme before sharing
the data.
The encryption scheme they choose involves using a symmetric key system where each pixel
value of the fingerprint image is replaced by the result of applying a modular operation. If the
original pixel value is p= 125 and the symmetric key is k= 7, determine the encrypted pixel value.
Solution 19.
To determine the encrypted pixel value, we need to apply the modular operation using the
symmetric key. The encryption process is represented by the formula:
Encrypted pixel value = (p+k) mod 256
Given p= 125 and k= 7, we can substitute these values into the formula:
Encrypted pixel value = (125 + 7) mod 256
Encrypted pixel value = 132 mod 256
Encrypted pixel value = 132 −256 ×132
256
Encrypted pixel value = 132 −256 ×0
Encrypted pixel value = 132
Therefore, the encrypted pixel value is 132.
19 20. BIOMETRIC IDENTIFICATION ERRORS AND THEIR IMPACT ON SECURITY MEA-
SURES
Problem 20. A biometric system is being evaluated for its performance in identifying individuals
based on facial recognition. In a test dataset of 500 individuals, the system produced the following
results:
- True positives: 450 individuals correctly identified - False positives: 20 individuals incorrectly
identified - False negatives: 10 individuals not identified
Calculate the following metrics for the system:
a) The accuracy of the system (in percentage).
b) The false acceptance rate (FAR) of the system (in percentage).
c) The false rejection rate (FRR) of the system (in percentage).
Solution 20.
a) The accuracy of the system is given by the formula:
Accuracy =True Positives +True Negatives
Total Population =450
500 = 0.9
Converting this to a percentage:
Accuracy = 0.9×100% = 90%
Therefore, the accuracy of the system is 90
b) The false acceptance rate (FAR) of the system is calculated as:
FAR =False Positives
False Positives + True Negatives =20
20 + 500 −450 =20
70 = 0.2857
Converting this to a percentage:
FAR = 0.2857 ×100% = 28.57%
Therefore, the false acceptance rate (FAR) of the system is 28.57
c) The false rejection rate (FRR) of the system is calculated as:
FRR =False Negatives
False Negatives + True Positives =10
10 + 450 =10
460 ≈0.0217
Converting this to a percentage:
FRR = 0.0217 ×100% ≈2.17%
Therefore, the false rejection rate (FRR) of the system is approximately 2.17
Solution 2. Let’s calculate the probability of none of the 4 attempts being successful, and then
subtract this from 1 to find the probability that at least one attempt is successful.
Let pbe the probability of success (tricking the system) on a single attempt, which is 10% or
0.1.
The probability of failure on a single attempt is 1−p= 1 −0.1=0.9.
The probability of none of the 4 attempts being successful is:
(0.9)4= 0.6561
Therefore, the probability of at least one attempt being successful is:
1−0.6561 = 0.3439
So, the probability that at least one of the 4 attempts will be successful is approximately 34.39%.
3 3. ISSUES WITH IRIS AUTHENTICATION IN LOW LIGHTING CONDITIONS
Problem 3. The iris recognition system at a high-security facility struggled to authenticate
individuals in low lighting conditions due to excessive noise in the captured images. One of the
authentication methods used is based on the comparison of feature vectors extracted from the iris
image.
Given two feature vectors X= [1.2,3.4,−2.1,0.9] and Y= [0.8,3.9,−2.3,1.1], calculate the
Euclidean distance between these feature vectors as a measure of similarity.
Solution 3.
1. Let’s calculate the Euclidean distance between the two feature vectors Xand Y, given by:
euclidean distance =v
u
u
t
n
X
i=1
(Xi−Yi)2
where Xiand Yiare the elements of the feature vectors Xand Yrespectively.
2. Plugging in the values for Xand Y:
euclidean distance =p(1.2−0.8)2+ (3.4−3.9)2+ (−2.1+2.3)2+ (0.9−1.1)2
euclidean distance =√0.16 + 0.25 + 0.04 + 0.04
euclidean distance =√0.49 = 0.7
Hence, the Euclidean distance between the feature vectors Xand Yis 0.7 units.
4 Biometric Security and Identity Management
Problem: Consider a biometric system that uses fingerprint recognition for authentication. The
system has an equal error rate of 0.05. If there are 1000 users attempting to authenticate using this
system, calculate the number of users that will be incorrectly authenticated due to false rejection
and false acceptance.
Solution: Given the equal error rate (EER) is 0.05, this means that the false rejection rate
(FRR) is equal to the false acceptance rate (FAR) at the threshold where they intersect.
Let Nbe the total number of users attempting to authenticate, which is 1000.
1. Finding the number of users incorrectly rejected:
The false rejection rate (FRR) is 0.05. This means that 5
Number of users incorrectly rejected = FRR ×N Number of users incorrectly rejected = 0.05 ×
1000 Number of users incorrectly rejected = 50 users
So, 50 users will be incorrectly rejected.
2. Finding the number of users incorrectly accepted:
The false acceptance rate (FAR) is also 0.05. This means that 5
Number of impostors incorrectly accepted = FAR ×N Number of impostors incorrectly accepted
= 0.05 ×1000 Number of impostors incorrectly accepted = 50 users
Therefore, 50 impostors will be incorrectly accepted during authentication process.
5 5. LACK OF STANDARDIZATION IN BIOMETRIC DATA STORAGE AND ENCRYPTION
Problem 5. Many biometric systems lack standardization in storing and encrypting biometric
data. As a result, a company is considering implementing a biometric access system for their
employees which uses fingerprint recognition technology. The system saves the fingerprint data in
an encrypted format using a unique encryption algorithm. The system has a false acceptance rate
(FAR) of 0.01 and a false rejection rate (FRR) of 0.05.
If the company has 1000 employees and each employee tries to access the system twice a day
on average, calculate the following:
a) The expected number of false acceptances per day.
b) The expected number of false rejections per day.
c) The total expected number of authentication errors per day.
Solution 5.
a) To calculate the expected number of false acceptances per day, we need to consider the
FAR and the total number of attempted accesses by the 1000 employees.
The FAR represents the probability of the system incorrectly accepting a false fingerprint as a
match. Since FAR = 0.01, the probability of a false acceptance is 0.01.
Let pF A be the probability of false acceptance, then:
pF A = 0.01
The company has 1000 employees trying to access the system twice a day on average, which
results in 2000 attempted accesses per day. Therefore, the expected number of false acceptances
per day can be calculated as:
Expected number of false acceptances per day =pF A ×2000
Expected number of false acceptances per day = 0.01 ×2000 = 20
Therefore, the expected number of false acceptances per day is 20.
b) To calculate the expected number of false rejections per day, we need to consider the FRR
and the total number of attempted accesses by the 1000 employees.
Similar to the FAR, the FRR represents the probability of the system incorrectly rejecting a valid
fingerprint as a non-match. Since FRR = 0.05, the probability of a false rejection is 0.05.
Let pF R be the probability of false rejection, then:
pF R = 0.05
The expected number of false rejections per day can be calculated as:
Expected number of false rejections per day =pF R ×2000
Expected number of false rejections per day = 0.05 ×2000 = 100
Therefore, the expected number of false rejections per day is 100.
c) The total expected number of authentication errors per day is the sum of false acceptances
and false rejections. Therefore:
Total expected number of authentication errors per day =Number of false acceptances+Number of false rejections
Total expected number of authentication errors per day = 20 + 100 = 120
Thus, the total expected number of authentication errors per day is 120.
6 6. BIOMETRIC SPOOFING TECHNIQUES AND THE NEED FOR ANTI-SPOOFING MEA-
SURES
Problem 6. A biometric security system uses fingerprint recognition for authentication. In
testing the system, it is found that the False Acceptance Rate (FAR) is 0.1% and the False Rejection
Rate (FRR) is 1%. Calculate the Equal Error Rate (EER) for this system.
Solution 6. The Equal Error Rate (EER) is the point at which the False Acceptance Rate (FAR)
is equal to the False Rejection Rate (FRR).
Given FAR = 0.1% and FRR = 1%, we need to find the EER.
Let the threshold be denoted as T. The EER occurs when FAR = FRR.
From the definitions:
FAR =P(reject |genuine)
FRR =P(accept |imposter)
Therefore, the EER is the average of FAR and FRR.
EER =FAR +FRR
2
EER =0.1+1
2= 0.55%
So, the Equal Error Rate (EER) for this system is 0.55%.
By analyzing the EER, we can see how well the system balances between rejecting valid users
and accepting imposters.
7 Biometric Security and Identity Management
Problem: Biometric systems often use multiple modalities to enhance security. Consider a
biometric system that combines fingerprint and iris recognition technologies. The false acceptance
rates for fingerprint and iris recognition are 0.1% and 0.01%, respectively.
a) What is the probability of falsely accepting an impostor using only fingerprint recognition?
b) What is the probability of falsely accepting an impostor using only iris recognition?
c) If the system requires both modalities to match in order to authenticate, what is the overall
false acceptance rate of the system?
Solution:
a) The probability of falsely accepting an impostor using only fingerprint recognition is given by
the false acceptance rate of the fingerprint modality, which is 0.1% or 0.001.
b) The probability of falsely accepting an impostor using only iris recognition is given by the
false acceptance rate of the iris modality, which is 0.01% or 0.0001.
c) Since the system requires both modalities to match for authentication, the overall false ac-
ceptance rate of the system can be calculated by multiplying the false acceptance rates of each
modality:
Overall False Acceptance Rate =False Acceptance Rate of Fingerprint×False Acceptance Rate of Iris
= 0.001 ×0.0001 = 0.0000001
Therefore, the overall false acceptance rate of the system when using both fingerprint and iris
recognition modalities is 0.00001%.
8 8. AUTHENTICATION FAILURES DUE TO AGING OR CHANGING BIOMETRIC CHARAC-
TERISTICS
Problem 8. In a biometric security system, the false acceptance rate (FAR) is measured to
be 0.02 and the false rejection rate (FRR) is 0.03. Calculate the Equal Error Rate (EER) for this
system.
Solution 8. The Equal Error Rate (EER) is the point where the false acceptance rate (FAR)
and the false rejection rate (FRR) are equal. Mathematically, EER is given by the average of FAR
and FRR when they are equal:
EER = (FAR + FRR) / 2
Given: FAR = 0.02 FRR = 0.03
Substitute the values into the formula: EER = (0.02 + 0.03) / 2 EER = 0.05 / 2 EER = 0.025
Therefore, the Equal Error Rate (EER) for this biometric security system is 0.025.
9 9. PROBLEMS WITH BIOMETRIC MATCHING ALGORITHMS AND ACCURACY RATES
Problem 9.
A security system that uses fingerprint recognition has an accuracy rate of 99.5%. If a random
sample of 500 fingerprints are scanned, calculate the probability that:
a) Exactly 3 fingerprints are incorrectly recognized.
b) At most 5 fingerprints are incorrectly recognized.
c) More than 10 fingerprints are incorrectly recognized.
Solution 9.
a) To find the probability that exactly 3 fingerprints are incorrectly recognized, we can use the
binomial probability formula:
P(X=k) = n
k·pk·(1 −p)n−k
where n= 500 (number of fingerprint scans), k= 3 (number of incorrectly recognized finger-
prints), and p= 1 −0.995 = 0.005 (probability of incorrectly recognizing a fingerprint).
P(X= 3) = 500
3·0.0053·0.995497
Calculating this probability gives us the answer for part a.
b) To find the probability that at most 5 fingerprints are incorrectly recognized, we need to
calculate
P(X≤5) = P(X= 0) + P(X= 1) + P(X= 2) + P(X= 3) + P(X= 4) + P(X= 5)
Using the binomial probability formula for each value of k, we can add up the probabilities to
find the answer for part b.
c) To find the probability that more than 10 fingerprints are incorrectly recognized, we can cal-
culate the complement of the probability that at most 10 are incorrectly recognized:
P(X > 10) = 1 −P(X≤10)
Using the binomial probability formula as in part b, we can find P(X≤10) and subtract it from
1 to get the answer for part c.
10 10. PRIVACY CONCERNS AND DATA PROTECTION IN BIOMETRIC DATABASES
Problem 10. A biometric system has an FAR (False Acceptance Rate) of 0.1% and an FRR
(False Rejection Rate) of 5%. If an individual tries to authenticate using this system 50 times, what
is the probability that they will be falsely rejected at least once?
Solution 10. Given that the FRR (False Rejection Rate) is 5%, the probability of a successful
authentication (1 - FRR) for one attempt is 95% or 0.95. Let pbe the probability of being falsely
rejected at least once in 50 attempts.
The probability of being falsely rejected at least once in ntrials is given by 1−(1 −FRR)n.
Therefore, the probability of being falsely rejected at least once in 50 attempts is:
p= 1 −(1 −0.05)50 = 1 −0.9550 ≈0.9231
Therefore, the probability that the individual will be falsely rejected at least once when attempt-
ing to authenticate 50 times using this biometric system is approximately 92.31%.
I. Problem: Biometric System Performance Degradation Over Time
An iris recognition system has an initial false acceptance rate (FAR) of 0.01
Solution
Given: Initial FAR = 0.01Initial FRR = 0.05Degraded FAR = 0.03Degraded FRR = 0.08
EER is the point where FAR = FRR.
Initial system: EER = (Initial FAR + Initial FRR) / 2 EER = (0.0001 + 0.0005) / 2 EER = 0.0003
Degraded system: EER = (Degraded FAR + Degraded FRR) / 2 EER = (0.0003 + 0.0008) / 2
EER = 0.00055
Therefore, the EER for the initial system is 0.0003 and for the degraded system is 0.00055.
11 12. ISSUES WITH BIOMETRIC IDENTIFICATION IN LARGE-SCALE SYSTEMS
Problem 12. In a large-scale biometric identification system, the false acceptance rate (FAR) is
0.1% and the false rejection rate (FRR) is 5%. If 1000 individuals attempt to authenticate themselves
using the system, calculate:
a) The number of individuals falsely accepted by the system.
b) The number of individuals falsely rejected by the system.
c) The overall accuracy of the system.
Solution 12.
a) The false acceptance rate (FAR) is 0.1%, which means 0.001 or 1
1000 of attempts are falsely
accepted. For 1000 individuals, the number falsely accepted can be calculated as:
False Accepted = 1000 ×0.001 = 1 individual
Therefore, the system falsely accepts 1individual.
b) The false rejection rate (FRR) is 5%, which means 5% or 0.05 of attempts are falsely rejected.
For 1000 individuals, the number falsely rejected can be calculated as:
False Rejected = 1000 ×0.05 = 50 individuals
Therefore, the system falsely rejects 50 individuals.
c) The overall accuracy of the system can be calculated as:
Accuracy = (1 −FAR)×100% = (1 −0.001) ×100% = 99.9%
Thus, the overall accuracy of the system is 99.9%.
12 13. CHALLENGES IN SECURING BIOMETRIC DATA DURING TRANSMISSION AND
STORAGE
Problem 13. Biometric data of a user is being transmitted over a network for authentication
purposes. The biometric data consists of a fingerprint image of size 500 ×500 pixels, where each
pixel is represented by an 8-bit grayscale value. During transmission, encryption is applied using
a symmetric key algorithm that increases the size of the data by 20%. If the encryption key used
is 128 bits long, calculate the size of the encrypted biometric data in kilobytes.
Solution 13. a) The size of the original fingerprint image in bytes can be calculated as:
Image size =Width ×Height ×Bit depth
Since each pixel is represented by an 8-bit grayscale value, the bit depth is 8. Given the image
size is 500 ×500 pixels:
Image size = 500 ×500 ×8bits
b) To convert bits to bytes, we divide by 8:
Image size in bytes =500 ×500 ×8
8bytes
c) After encryption, the size of the data increases by 20%. The new size of the encrypted data
can be calculated as:
New encrypted data size = 1.2×Original data size (in bytes)
d) Given that the encryption key is 128 bits long, to convert it to bytes, we divide by 8:
Key size =128
8bytes
e) The total size of the encrypted biometric data is the sum of the encrypted data size and the
key size:
Total size =New encrypted data size +Key size
Now, substitute the values calculated in parts a, b, c, and d to find the total size of the encrypted
biometric data in kilobytes.
13 14. IMPLICATIONS OF BIOMETRIC DATA BREACHES AND IDENTITY THEFT
Problem 14. A company’s biometric security system has been breached, and the fingerprint
data of 1000 employees has been stolen. If each fingerprint template contains 500 bytes of data and
the company offers a reward of 500percompromisedfingerprinttemplate, calculatethetotalrewardamountthecompanywillneedtopaytoaf f ectedemployees.
Solution 14. Given that: - Number of compromised fingerprint templates = 1000 - Size of each
fingerprint template = 500 bytes - Reward offered per compromised template = 500
Step 1: Calculate the total amount of data compromised Total amount of data compromised
= Number of compromised templates ×Size of each template Total amount of data compromised
= 1000 ×500 bytes Total amount of data compromised = 500,000 bytes
Step 2: Calculate the total reward amount Total reward amount = Number of compromised
templates ×Reward offered per template Total reward amount = 1000 ×500T otalrewardamount =500,000
Therefore, the company will need to pay a total reward amount of 500,000totheaffectedemployees.
14 15. CULTURAL BIASES AND LIMITATIONS IN BIOMETRIC RECOGNITION SYSTEMS
Problem 15. A facial recognition system has been trained on a dataset composed mainly of
individuals of Asian descent. When tested on a diverse population that includes individuals of
African descent, the system shows a significantly higher error rate for this subgroup.
The error rates for the system on the test set are as follows:
•Overall error rate: 5
•Error rate for individuals of Asian descent: 3
•Error rate for individuals of African descent: 12
a) What is the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent?
b) If the system correctly identifies 90
c) Considering these results, discuss the potential impact of cultural biases on the effectiveness
of biometric recognition systems.
Solution 15.
a) The odds ratio of misidentification for individuals of African descent compared to individuals
of Asian descent can be calculated using the error rates:
Odds ratio =Error rate for African descent
Error rate for Asian descent =0.12
0.03 = 4
Therefore, the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent is 4.
b) Given that the system correctly identifies 90
Accuracy rate for African descent = 1 −Error rate for African descent = 1 −0.12 = 0.88 = 88%
Therefore, the accuracy rate for individuals of African descent is 88
c) The significantly higher error rate for individuals of African descent in the facial recognition
system compared to individuals of Asian descent highlights a bias that may have been introduced
during the training phase, where the dataset was skewed towards a specific ethnicity. This cultural
bias can lead to inequitable outcomes, with certain demographic groups facing higher risks of
misidentification and potential discrimination. Addressing cultural biases in biometric recognition
systems is crucial to ensure fair and accurate identification across diverse populations.
15 16. LEGAL AND ETHICAL IMPLICATIONS OF BIOMETRIC DATA COLLECTION AND
USE
Problem 16. A company is implementing a biometric authentication system for its employees
using fingerprint recognition. The system is expected to process 1000 fingerprint scans per day. If
each fingerprint scan produces a file size of 500 KB, calculate the total amount of data collected
by the system in one month.
Solution 16. Given that the system processes 1000 fingerprint scans per day and each scan
produces a file size of 500 KB, we can calculate the total data collected by the system in one day
as follows:
Total data collected per day = Number of scans per day x File size per scan Total data collected
per day = 1000 x 500 KB Total data collected per day = 500,000 KB
To convert KB to GB, we use the conversion factor: 1 GB = 1024 KB
Total data collected per day in GB = 500,000 KB / 1024 = 488.28 GB
Since there are typically 30 days in a month, we can calculate the total amount of data collected
by the system in one month as follows:
Total data collected per month = Total data collected per day x Number of days in a month Total
data collected per month = 488.28 GB x 30 Total data collected per month = 14,648.4 GB
Therefore, the total amount of data collected by the system in one month is 14,648.4 GB.
16 17. CHALLENGES IN IMPLEMENTING BIOMETRIC AUTHENTICATION IN IOT DEVICES
Problem 17. A company is considering implementing biometric authentication for their IoT
devices to enhance security. They estimate that the false acceptance rate (FAR) of the biometric
system is 0.1% and the false rejection rate (FRR) is 1%. If an unauthorized user attempts to access
the device 50 times, calculate the probability that the system will grant access to this unauthorized
user at least once.
Solution 17. To calculate the probability of the system granting access to the unauthorized
user at least once, we need to consider the probability of false acceptance for each attempt. Let’s
denote the probability of a false acceptance as P(F A).
Given that FAR is 0.1%, we have:
P(F A) = 0.001
Therefore, the probability of rejecting a false attempt is:
P(¬F A) = 1 −P(F A) = 1 −0.001 = 0.999
The probability that the system will grant access to the unauthorized user at least once in 50
attempts is the complement of the probability that it will reject all 50 attempts. Thus, the probability
is:
P(Access granted at least once)=1−P(Access not granted at all)
Since each attempt is independent, the probability of the system not granting access to the
unauthorized user in a single attempt is P(¬F A). Therefore, the probability that the system will not
grant access in 50 attempts is:
P(Access not granted at all)=(P(¬F A))n= 0.99950
Calculating this probability:
P(Access not granted at all)=0.99950 ≈0.950 = 95%
So, the probability that the system will grant access to the unauthorized user at least once is
approximately:
P(Access granted at least once)=1−0.950 = 0.050 = 5%
Therefore, the system has a 5% chance of granting access to the unauthorized user at least
once out of 50 attempts.
17 18. USER ACCEPTANCE AND TRUST ISSUES WITH BIOMETRIC SECURITY MEASURES
Problem 18.
A study is conducted to analyze the user acceptance of a new biometric authentication system.
The study involved 100 participants, and the results showed that 65 participants were willing to use
the biometric system, while the remaining participants expressed concerns about privacy and did
not want to adopt it.
a) Calculate the user acceptance rate in percentage.
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same, how many participants would be willing to use the biometric system?
c) Discuss why user acceptance and trust are essential factors in the implementation of bio-
metric security measures.
Solution 18.
a) The user acceptance rate can be calculated as follows:
User acceptance rate =Number of participants willing to use the system
Total number of participants ×100%
In this case, the number of participants willing to use the system is 65, and the total number of
participants is 100. Therefore,
User acceptance rate =65
100 ×100% = 65%
So, the user acceptance rate is 65
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same at 65
Number of participants willing to use the system =Acceptance rate×Total number of participants
Number of participants willing to use the system = 0.65 ×150 = 97.5
Since we cannot have a fraction of a participant, we can estimate that approximately 98 partic-
ipants would be willing to use the biometric system.
c) User acceptance and trust are crucial factors in the implementation of biometric security
measures because:
1. If users do not trust the system, they might not use it properly or try to bypass it, leading to
security vulnerabilities. 2. Acceptance by users is essential for the success of the system, as low
adoption rates can render the biometric security measures ineffective. 3. Ensuring user acceptance
and trust can help improve the overall user experience and satisfaction with the security system. 4.
Addressing privacy concerns and building trust can help mitigate potential legal and ethical issues
related to biometric data collection and storage.
18 19. CHALLENGES IN SECURELY SHARING BIOMETRIC DATA ACROSS DIFFERENT
PLATFORMS
Problem 19. Consider a scenario where a company wants to securely share biometric data
(such as fingerprint scans) across different platforms for employee authentication. However, due to
security concerns, the company decides to implement a secure encryption scheme before sharing
the data.
The encryption scheme they choose involves using a symmetric key system where each pixel
value of the fingerprint image is replaced by the result of applying a modular operation. If the
original pixel value is p= 125 and the symmetric key is k= 7, determine the encrypted pixel value.
Solution 19.
To determine the encrypted pixel value, we need to apply the modular operation using the
symmetric key. The encryption process is represented by the formula:
Encrypted pixel value = (p+k) mod 256
Given p= 125 and k= 7, we can substitute these values into the formula:
Encrypted pixel value = (125 + 7) mod 256
Encrypted pixel value = 132 mod 256
Encrypted pixel value = 132 −256 ×132
256
Encrypted pixel value = 132 −256 ×0
Encrypted pixel value = 132
Therefore, the encrypted pixel value is 132.
19 20. BIOMETRIC IDENTIFICATION ERRORS AND THEIR IMPACT ON SECURITY MEA-
SURES
Problem 20. A biometric system is being evaluated for its performance in identifying individuals
based on facial recognition. In a test dataset of 500 individuals, the system produced the following
results:
- True positives: 450 individuals correctly identified - False positives: 20 individuals incorrectly
identified - False negatives: 10 individuals not identified
Calculate the following metrics for the system:
a) The accuracy of the system (in percentage).
b) The false acceptance rate (FAR) of the system (in percentage).
c) The false rejection rate (FRR) of the system (in percentage).
Solution 20.
a) The accuracy of the system is given by the formula:
Accuracy =True Positives +True Negatives
Total Population =450
500 = 0.9
Converting this to a percentage:
Accuracy = 0.9×100% = 90%
Therefore, the accuracy of the system is 90
b) The false acceptance rate (FAR) of the system is calculated as:
FAR =False Positives
False Positives + True Negatives =20
20 + 500 −450 =20
70 = 0.2857
Converting this to a percentage:
FAR = 0.2857 ×100% = 28.57%
Therefore, the false acceptance rate (FAR) of the system is 28.57
c) The false rejection rate (FRR) of the system is calculated as:
FRR =False Negatives
False Negatives + True Positives =10
10 + 450 =10
460 ≈0.0217
Converting this to a percentage:
FRR = 0.0217 ×100% ≈2.17%
Therefore, the false rejection rate (FRR) of the system is approximately 2.17
Solution 2. Let’s calculate the probability of none of the 4 attempts being successful, and then
subtract this from 1 to find the probability that at least one attempt is successful.
Let pbe the probability of success (tricking the system) on a single attempt, which is 10% or
0.1.
The probability of failure on a single attempt is 1−p= 1 −0.1=0.9.
The probability of none of the 4 attempts being successful is:
(0.9)4= 0.6561
Therefore, the probability of at least one attempt being successful is:
1−0.6561 = 0.3439
So, the probability that at least one of the 4 attempts will be successful is approximately 34.39%.
3 3. ISSUES WITH IRIS AUTHENTICATION IN LOW LIGHTING CONDITIONS
Problem 3. The iris recognition system at a high-security facility struggled to authenticate
individuals in low lighting conditions due to excessive noise in the captured images. One of the
authentication methods used is based on the comparison of feature vectors extracted from the iris
image.
Given two feature vectors X= [1.2,3.4,−2.1,0.9] and Y= [0.8,3.9,−2.3,1.1], calculate the
Euclidean distance between these feature vectors as a measure of similarity.
Solution 3.
1. Let’s calculate the Euclidean distance between the two feature vectors Xand Y, given by:
euclidean distance =v
u
u
t
n
X
i=1
(Xi−Yi)2
where Xiand Yiare the elements of the feature vectors Xand Yrespectively.
2. Plugging in the values for Xand Y:
euclidean distance =p(1.2−0.8)2+ (3.4−3.9)2+ (−2.1+2.3)2+ (0.9−1.1)2
euclidean distance =√0.16 + 0.25 + 0.04 + 0.04
euclidean distance =√0.49 = 0.7
Hence, the Euclidean distance between the feature vectors Xand Yis 0.7 units.
4 Biometric Security and Identity Management
Problem: Consider a biometric system that uses fingerprint recognition for authentication. The
system has an equal error rate of 0.05. If there are 1000 users attempting to authenticate using this
system, calculate the number of users that will be incorrectly authenticated due to false rejection
and false acceptance.
Solution: Given the equal error rate (EER) is 0.05, this means that the false rejection rate
(FRR) is equal to the false acceptance rate (FAR) at the threshold where they intersect.
Let Nbe the total number of users attempting to authenticate, which is 1000.
1. Finding the number of users incorrectly rejected:
The false rejection rate (FRR) is 0.05. This means that 5
Number of users incorrectly rejected = FRR ×N Number of users incorrectly rejected = 0.05 ×
1000 Number of users incorrectly rejected = 50 users
So, 50 users will be incorrectly rejected.
2. Finding the number of users incorrectly accepted:
The false acceptance rate (FAR) is also 0.05. This means that 5
Number of impostors incorrectly accepted = FAR ×N Number of impostors incorrectly accepted
= 0.05 ×1000 Number of impostors incorrectly accepted = 50 users
Therefore, 50 impostors will be incorrectly accepted during authentication process.
5 5. LACK OF STANDARDIZATION IN BIOMETRIC DATA STORAGE AND ENCRYPTION
Problem 5. Many biometric systems lack standardization in storing and encrypting biometric
data. As a result, a company is considering implementing a biometric access system for their
employees which uses fingerprint recognition technology. The system saves the fingerprint data in
an encrypted format using a unique encryption algorithm. The system has a false acceptance rate
(FAR) of 0.01 and a false rejection rate (FRR) of 0.05.
If the company has 1000 employees and each employee tries to access the system twice a day
on average, calculate the following:
a) The expected number of false acceptances per day.
b) The expected number of false rejections per day.
c) The total expected number of authentication errors per day.
Solution 5.
a) To calculate the expected number of false acceptances per day, we need to consider the
FAR and the total number of attempted accesses by the 1000 employees.
The FAR represents the probability of the system incorrectly accepting a false fingerprint as a
match. Since FAR = 0.01, the probability of a false acceptance is 0.01.
Let pF A be the probability of false acceptance, then:
pF A = 0.01
The company has 1000 employees trying to access the system twice a day on average, which
results in 2000 attempted accesses per day. Therefore, the expected number of false acceptances
per day can be calculated as:
Expected number of false acceptances per day =pF A ×2000
Expected number of false acceptances per day = 0.01 ×2000 = 20
Therefore, the expected number of false acceptances per day is 20.
b) To calculate the expected number of false rejections per day, we need to consider the FRR
and the total number of attempted accesses by the 1000 employees.
Similar to the FAR, the FRR represents the probability of the system incorrectly rejecting a valid
fingerprint as a non-match. Since FRR = 0.05, the probability of a false rejection is 0.05.
Let pF R be the probability of false rejection, then:
pF R = 0.05
The expected number of false rejections per day can be calculated as:
Expected number of false rejections per day =pF R ×2000
Expected number of false rejections per day = 0.05 ×2000 = 100
Therefore, the expected number of false rejections per day is 100.
c) The total expected number of authentication errors per day is the sum of false acceptances
and false rejections. Therefore:
Total expected number of authentication errors per day =Number of false acceptances+Number of false rejections
Total expected number of authentication errors per day = 20 + 100 = 120
Thus, the total expected number of authentication errors per day is 120.
6 6. BIOMETRIC SPOOFING TECHNIQUES AND THE NEED FOR ANTI-SPOOFING MEA-
SURES
Problem 6. A biometric security system uses fingerprint recognition for authentication. In
testing the system, it is found that the False Acceptance Rate (FAR) is 0.1% and the False Rejection
Rate (FRR) is 1%. Calculate the Equal Error Rate (EER) for this system.
Solution 6. The Equal Error Rate (EER) is the point at which the False Acceptance Rate (FAR)
is equal to the False Rejection Rate (FRR).
Given FAR = 0.1% and FRR = 1%, we need to find the EER.
Let the threshold be denoted as T. The EER occurs when FAR = FRR.
From the definitions:
FAR =P(reject |genuine)
FRR =P(accept |imposter)
Therefore, the EER is the average of FAR and FRR.
EER =FAR +FRR
2
EER =0.1+1
2= 0.55%
So, the Equal Error Rate (EER) for this system is 0.55%.
By analyzing the EER, we can see how well the system balances between rejecting valid users
and accepting imposters.
7 Biometric Security and Identity Management
Problem: Biometric systems often use multiple modalities to enhance security. Consider a
biometric system that combines fingerprint and iris recognition technologies. The false acceptance
rates for fingerprint and iris recognition are 0.1% and 0.01%, respectively.
a) What is the probability of falsely accepting an impostor using only fingerprint recognition?
b) What is the probability of falsely accepting an impostor using only iris recognition?
c) If the system requires both modalities to match in order to authenticate, what is the overall
false acceptance rate of the system?
Solution:
a) The probability of falsely accepting an impostor using only fingerprint recognition is given by
the false acceptance rate of the fingerprint modality, which is 0.1% or 0.001.
b) The probability of falsely accepting an impostor using only iris recognition is given by the
false acceptance rate of the iris modality, which is 0.01% or 0.0001.
c) Since the system requires both modalities to match for authentication, the overall false ac-
ceptance rate of the system can be calculated by multiplying the false acceptance rates of each
modality:
Overall False Acceptance Rate =False Acceptance Rate of Fingerprint×False Acceptance Rate of Iris
= 0.001 ×0.0001 = 0.0000001
Therefore, the overall false acceptance rate of the system when using both fingerprint and iris
recognition modalities is 0.00001%.
8 8. AUTHENTICATION FAILURES DUE TO AGING OR CHANGING BIOMETRIC CHARAC-
TERISTICS
Problem 8. In a biometric security system, the false acceptance rate (FAR) is measured to
be 0.02 and the false rejection rate (FRR) is 0.03. Calculate the Equal Error Rate (EER) for this
system.
Solution 8. The Equal Error Rate (EER) is the point where the false acceptance rate (FAR)
and the false rejection rate (FRR) are equal. Mathematically, EER is given by the average of FAR
and FRR when they are equal:
EER = (FAR + FRR) / 2
Given: FAR = 0.02 FRR = 0.03
Substitute the values into the formula: EER = (0.02 + 0.03) / 2 EER = 0.05 / 2 EER = 0.025
Therefore, the Equal Error Rate (EER) for this biometric security system is 0.025.
9 9. PROBLEMS WITH BIOMETRIC MATCHING ALGORITHMS AND ACCURACY RATES
Problem 9.
A security system that uses fingerprint recognition has an accuracy rate of 99.5%. If a random
sample of 500 fingerprints are scanned, calculate the probability that:
a) Exactly 3 fingerprints are incorrectly recognized.
b) At most 5 fingerprints are incorrectly recognized.
c) More than 10 fingerprints are incorrectly recognized.
Solution 9.
a) To find the probability that exactly 3 fingerprints are incorrectly recognized, we can use the
binomial probability formula:
P(X=k) = n
k·pk·(1 −p)n−k
where n= 500 (number of fingerprint scans), k= 3 (number of incorrectly recognized finger-
prints), and p= 1 −0.995 = 0.005 (probability of incorrectly recognizing a fingerprint).
P(X= 3) = 500
3·0.0053·0.995497
Calculating this probability gives us the answer for part a.
b) To find the probability that at most 5 fingerprints are incorrectly recognized, we need to
calculate
P(X≤5) = P(X= 0) + P(X= 1) + P(X= 2) + P(X= 3) + P(X= 4) + P(X= 5)
Using the binomial probability formula for each value of k, we can add up the probabilities to
find the answer for part b.
c) To find the probability that more than 10 fingerprints are incorrectly recognized, we can cal-
culate the complement of the probability that at most 10 are incorrectly recognized:
P(X > 10) = 1 −P(X≤10)
Using the binomial probability formula as in part b, we can find P(X≤10) and subtract it from
1 to get the answer for part c.
10 10. PRIVACY CONCERNS AND DATA PROTECTION IN BIOMETRIC DATABASES
Problem 10. A biometric system has an FAR (False Acceptance Rate) of 0.1% and an FRR
(False Rejection Rate) of 5%. If an individual tries to authenticate using this system 50 times, what
is the probability that they will be falsely rejected at least once?
Solution 10. Given that the FRR (False Rejection Rate) is 5%, the probability of a successful
authentication (1 - FRR) for one attempt is 95% or 0.95. Let pbe the probability of being falsely
rejected at least once in 50 attempts.
The probability of being falsely rejected at least once in ntrials is given by 1−(1 −FRR)n.
Therefore, the probability of being falsely rejected at least once in 50 attempts is:
p= 1 −(1 −0.05)50 = 1 −0.9550 ≈0.9231
Therefore, the probability that the individual will be falsely rejected at least once when attempt-
ing to authenticate 50 times using this biometric system is approximately 92.31%.
I. Problem: Biometric System Performance Degradation Over Time
An iris recognition system has an initial false acceptance rate (FAR) of 0.01
Solution
Given: Initial FAR = 0.01Initial FRR = 0.05Degraded FAR = 0.03Degraded FRR = 0.08
EER is the point where FAR = FRR.
Initial system: EER = (Initial FAR + Initial FRR) / 2 EER = (0.0001 + 0.0005) / 2 EER = 0.0003
Degraded system: EER = (Degraded FAR + Degraded FRR) / 2 EER = (0.0003 + 0.0008) / 2
EER = 0.00055
Therefore, the EER for the initial system is 0.0003 and for the degraded system is 0.00055.
11 12. ISSUES WITH BIOMETRIC IDENTIFICATION IN LARGE-SCALE SYSTEMS
Problem 12. In a large-scale biometric identification system, the false acceptance rate (FAR) is
0.1% and the false rejection rate (FRR) is 5%. If 1000 individuals attempt to authenticate themselves
using the system, calculate:
a) The number of individuals falsely accepted by the system.
b) The number of individuals falsely rejected by the system.
c) The overall accuracy of the system.
Solution 12.
a) The false acceptance rate (FAR) is 0.1%, which means 0.001 or 1
1000 of attempts are falsely
accepted. For 1000 individuals, the number falsely accepted can be calculated as:
False Accepted = 1000 ×0.001 = 1 individual
Therefore, the system falsely accepts 1individual.
b) The false rejection rate (FRR) is 5%, which means 5% or 0.05 of attempts are falsely rejected.
For 1000 individuals, the number falsely rejected can be calculated as:
False Rejected = 1000 ×0.05 = 50 individuals
Therefore, the system falsely rejects 50 individuals.
c) The overall accuracy of the system can be calculated as:
Accuracy = (1 −FAR)×100% = (1 −0.001) ×100% = 99.9%
Thus, the overall accuracy of the system is 99.9%.
12 13. CHALLENGES IN SECURING BIOMETRIC DATA DURING TRANSMISSION AND
STORAGE
Problem 13. Biometric data of a user is being transmitted over a network for authentication
purposes. The biometric data consists of a fingerprint image of size 500 ×500 pixels, where each
pixel is represented by an 8-bit grayscale value. During transmission, encryption is applied using
a symmetric key algorithm that increases the size of the data by 20%. If the encryption key used
is 128 bits long, calculate the size of the encrypted biometric data in kilobytes.
Solution 13. a) The size of the original fingerprint image in bytes can be calculated as:
Image size =Width ×Height ×Bit depth
Since each pixel is represented by an 8-bit grayscale value, the bit depth is 8. Given the image
size is 500 ×500 pixels:
Image size = 500 ×500 ×8bits
b) To convert bits to bytes, we divide by 8:
Image size in bytes =500 ×500 ×8
8bytes
c) After encryption, the size of the data increases by 20%. The new size of the encrypted data
can be calculated as:
New encrypted data size = 1.2×Original data size (in bytes)
d) Given that the encryption key is 128 bits long, to convert it to bytes, we divide by 8:
Key size =128
8bytes
e) The total size of the encrypted biometric data is the sum of the encrypted data size and the
key size:
Total size =New encrypted data size +Key size
Now, substitute the values calculated in parts a, b, c, and d to find the total size of the encrypted
biometric data in kilobytes.
13 14. IMPLICATIONS OF BIOMETRIC DATA BREACHES AND IDENTITY THEFT
Problem 14. A company’s biometric security system has been breached, and the fingerprint
data of 1000 employees has been stolen. If each fingerprint template contains 500 bytes of data and
the company offers a reward of 500percompromisedfingerprinttemplate, calculatethetotalrewardamountthecompanywillneedtopaytoaf f ectedemployees.
Solution 14. Given that: - Number of compromised fingerprint templates = 1000 - Size of each
fingerprint template = 500 bytes - Reward offered per compromised template = 500
Step 1: Calculate the total amount of data compromised Total amount of data compromised
= Number of compromised templates ×Size of each template Total amount of data compromised
= 1000 ×500 bytes Total amount of data compromised = 500,000 bytes
Step 2: Calculate the total reward amount Total reward amount = Number of compromised
templates ×Reward offered per template Total reward amount = 1000 ×500T otalrewardamount =500,000
Therefore, the company will need to pay a total reward amount of 500,000totheaffectedemployees.
14 15. CULTURAL BIASES AND LIMITATIONS IN BIOMETRIC RECOGNITION SYSTEMS
Problem 15. A facial recognition system has been trained on a dataset composed mainly of
individuals of Asian descent. When tested on a diverse population that includes individuals of
African descent, the system shows a significantly higher error rate for this subgroup.
The error rates for the system on the test set are as follows:
•Overall error rate: 5
•Error rate for individuals of Asian descent: 3
•Error rate for individuals of African descent: 12
a) What is the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent?
b) If the system correctly identifies 90
c) Considering these results, discuss the potential impact of cultural biases on the effectiveness
of biometric recognition systems.
Solution 15.
a) The odds ratio of misidentification for individuals of African descent compared to individuals
of Asian descent can be calculated using the error rates:
Odds ratio =Error rate for African descent
Error rate for Asian descent =0.12
0.03 = 4
Therefore, the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent is 4.
b) Given that the system correctly identifies 90
Accuracy rate for African descent = 1 −Error rate for African descent = 1 −0.12 = 0.88 = 88%
Therefore, the accuracy rate for individuals of African descent is 88
c) The significantly higher error rate for individuals of African descent in the facial recognition
system compared to individuals of Asian descent highlights a bias that may have been introduced
during the training phase, where the dataset was skewed towards a specific ethnicity. This cultural
bias can lead to inequitable outcomes, with certain demographic groups facing higher risks of
misidentification and potential discrimination. Addressing cultural biases in biometric recognition
systems is crucial to ensure fair and accurate identification across diverse populations.
15 16. LEGAL AND ETHICAL IMPLICATIONS OF BIOMETRIC DATA COLLECTION AND
USE
Problem 16. A company is implementing a biometric authentication system for its employees
using fingerprint recognition. The system is expected to process 1000 fingerprint scans per day. If
each fingerprint scan produces a file size of 500 KB, calculate the total amount of data collected
by the system in one month.
Solution 16. Given that the system processes 1000 fingerprint scans per day and each scan
produces a file size of 500 KB, we can calculate the total data collected by the system in one day
as follows:
Total data collected per day = Number of scans per day x File size per scan Total data collected
per day = 1000 x 500 KB Total data collected per day = 500,000 KB
To convert KB to GB, we use the conversion factor: 1 GB = 1024 KB
Total data collected per day in GB = 500,000 KB / 1024 = 488.28 GB
Since there are typically 30 days in a month, we can calculate the total amount of data collected
by the system in one month as follows:
Total data collected per month = Total data collected per day x Number of days in a month Total
data collected per month = 488.28 GB x 30 Total data collected per month = 14,648.4 GB
Therefore, the total amount of data collected by the system in one month is 14,648.4 GB.
16 17. CHALLENGES IN IMPLEMENTING BIOMETRIC AUTHENTICATION IN IOT DEVICES
Problem 17. A company is considering implementing biometric authentication for their IoT
devices to enhance security. They estimate that the false acceptance rate (FAR) of the biometric
system is 0.1% and the false rejection rate (FRR) is 1%. If an unauthorized user attempts to access
the device 50 times, calculate the probability that the system will grant access to this unauthorized
user at least once.
Solution 17. To calculate the probability of the system granting access to the unauthorized
user at least once, we need to consider the probability of false acceptance for each attempt. Let’s
denote the probability of a false acceptance as P(F A).
Given that FAR is 0.1%, we have:
P(F A) = 0.001
Therefore, the probability of rejecting a false attempt is:
P(¬F A) = 1 −P(F A) = 1 −0.001 = 0.999
The probability that the system will grant access to the unauthorized user at least once in 50
attempts is the complement of the probability that it will reject all 50 attempts. Thus, the probability
is:
P(Access granted at least once)=1−P(Access not granted at all)
Since each attempt is independent, the probability of the system not granting access to the
unauthorized user in a single attempt is P(¬F A). Therefore, the probability that the system will not
grant access in 50 attempts is:
P(Access not granted at all)=(P(¬F A))n= 0.99950
Calculating this probability:
P(Access not granted at all)=0.99950 ≈0.950 = 95%
So, the probability that the system will grant access to the unauthorized user at least once is
approximately:
P(Access granted at least once)=1−0.950 = 0.050 = 5%
Therefore, the system has a 5% chance of granting access to the unauthorized user at least
once out of 50 attempts.
17 18. USER ACCEPTANCE AND TRUST ISSUES WITH BIOMETRIC SECURITY MEASURES
Problem 18.
A study is conducted to analyze the user acceptance of a new biometric authentication system.
The study involved 100 participants, and the results showed that 65 participants were willing to use
the biometric system, while the remaining participants expressed concerns about privacy and did
not want to adopt it.
a) Calculate the user acceptance rate in percentage.
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same, how many participants would be willing to use the biometric system?
c) Discuss why user acceptance and trust are essential factors in the implementation of bio-
metric security measures.
Solution 18.
a) The user acceptance rate can be calculated as follows:
User acceptance rate =Number of participants willing to use the system
Total number of participants ×100%
In this case, the number of participants willing to use the system is 65, and the total number of
participants is 100. Therefore,
User acceptance rate =65
100 ×100% = 65%
So, the user acceptance rate is 65
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same at 65
Number of participants willing to use the system =Acceptance rate×Total number of participants
Number of participants willing to use the system = 0.65 ×150 = 97.5
Since we cannot have a fraction of a participant, we can estimate that approximately 98 partic-
ipants would be willing to use the biometric system.
c) User acceptance and trust are crucial factors in the implementation of biometric security
measures because:
1. If users do not trust the system, they might not use it properly or try to bypass it, leading to
security vulnerabilities. 2. Acceptance by users is essential for the success of the system, as low
adoption rates can render the biometric security measures ineffective. 3. Ensuring user acceptance
and trust can help improve the overall user experience and satisfaction with the security system. 4.
Addressing privacy concerns and building trust can help mitigate potential legal and ethical issues
related to biometric data collection and storage.
18 19. CHALLENGES IN SECURELY SHARING BIOMETRIC DATA ACROSS DIFFERENT
PLATFORMS
Problem 19. Consider a scenario where a company wants to securely share biometric data
(such as fingerprint scans) across different platforms for employee authentication. However, due to
security concerns, the company decides to implement a secure encryption scheme before sharing
the data.
The encryption scheme they choose involves using a symmetric key system where each pixel
value of the fingerprint image is replaced by the result of applying a modular operation. If the
original pixel value is p= 125 and the symmetric key is k= 7, determine the encrypted pixel value.
Solution 19.
To determine the encrypted pixel value, we need to apply the modular operation using the
symmetric key. The encryption process is represented by the formula:
Encrypted pixel value = (p+k) mod 256
Given p= 125 and k= 7, we can substitute these values into the formula:
Encrypted pixel value = (125 + 7) mod 256
Encrypted pixel value = 132 mod 256
Encrypted pixel value = 132 −256 ×132
256
Encrypted pixel value = 132 −256 ×0
Encrypted pixel value = 132
Therefore, the encrypted pixel value is 132.
19 20. BIOMETRIC IDENTIFICATION ERRORS AND THEIR IMPACT ON SECURITY MEA-
SURES
Problem 20. A biometric system is being evaluated for its performance in identifying individuals
based on facial recognition. In a test dataset of 500 individuals, the system produced the following
results:
- True positives: 450 individuals correctly identified - False positives: 20 individuals incorrectly
identified - False negatives: 10 individuals not identified
Calculate the following metrics for the system:
a) The accuracy of the system (in percentage).
b) The false acceptance rate (FAR) of the system (in percentage).
c) The false rejection rate (FRR) of the system (in percentage).
Solution 20.
a) The accuracy of the system is given by the formula:
Accuracy =True Positives +True Negatives
Total Population =450
500 = 0.9
Converting this to a percentage:
Accuracy = 0.9×100% = 90%
Therefore, the accuracy of the system is 90
b) The false acceptance rate (FAR) of the system is calculated as:
FAR =False Positives
False Positives + True Negatives =20
20 + 500 −450 =20
70 = 0.2857
Converting this to a percentage:
FAR = 0.2857 ×100% = 28.57%
Therefore, the false acceptance rate (FAR) of the system is 28.57
c) The false rejection rate (FRR) of the system is calculated as:
FRR =False Negatives
False Negatives + True Positives =10
10 + 450 =10
460 ≈0.0217
Converting this to a percentage:
FRR = 0.0217 ×100% ≈2.17%
Therefore, the false rejection rate (FRR) of the system is approximately 2.17
Solution 2. Let’s calculate the probability of none of the 4 attempts being successful, and then
subtract this from 1 to find the probability that at least one attempt is successful.
Let pbe the probability of success (tricking the system) on a single attempt, which is 10% or
0.1.
The probability of failure on a single attempt is 1−p= 1 −0.1=0.9.
The probability of none of the 4 attempts being successful is:
(0.9)4= 0.6561
Therefore, the probability of at least one attempt being successful is:
1−0.6561 = 0.3439
So, the probability that at least one of the 4 attempts will be successful is approximately 34.39%.
3 3. ISSUES WITH IRIS AUTHENTICATION IN LOW LIGHTING CONDITIONS
Problem 3. The iris recognition system at a high-security facility struggled to authenticate
individuals in low lighting conditions due to excessive noise in the captured images. One of the
authentication methods used is based on the comparison of feature vectors extracted from the iris
image.
Given two feature vectors X= [1.2,3.4,−2.1,0.9] and Y= [0.8,3.9,−2.3,1.1], calculate the
Euclidean distance between these feature vectors as a measure of similarity.
Solution 3.
1. Let’s calculate the Euclidean distance between the two feature vectors Xand Y, given by:
euclidean distance =v
u
u
t
n
X
i=1
(Xi−Yi)2
where Xiand Yiare the elements of the feature vectors Xand Yrespectively.
2. Plugging in the values for Xand Y:
euclidean distance =p(1.2−0.8)2+ (3.4−3.9)2+ (−2.1+2.3)2+ (0.9−1.1)2
euclidean distance =√0.16 + 0.25 + 0.04 + 0.04
euclidean distance =√0.49 = 0.7
Hence, the Euclidean distance between the feature vectors Xand Yis 0.7 units.
4 Biometric Security and Identity Management
Problem: Consider a biometric system that uses fingerprint recognition for authentication. The
system has an equal error rate of 0.05. If there are 1000 users attempting to authenticate using this
system, calculate the number of users that will be incorrectly authenticated due to false rejection
and false acceptance.
Solution: Given the equal error rate (EER) is 0.05, this means that the false rejection rate
(FRR) is equal to the false acceptance rate (FAR) at the threshold where they intersect.
Let Nbe the total number of users attempting to authenticate, which is 1000.
1. Finding the number of users incorrectly rejected:
The false rejection rate (FRR) is 0.05. This means that 5
Number of users incorrectly rejected = FRR ×N Number of users incorrectly rejected = 0.05 ×
1000 Number of users incorrectly rejected = 50 users
So, 50 users will be incorrectly rejected.
2. Finding the number of users incorrectly accepted:
The false acceptance rate (FAR) is also 0.05. This means that 5
Number of impostors incorrectly accepted = FAR ×N Number of impostors incorrectly accepted
= 0.05 ×1000 Number of impostors incorrectly accepted = 50 users
Therefore, 50 impostors will be incorrectly accepted during authentication process.
5 5. LACK OF STANDARDIZATION IN BIOMETRIC DATA STORAGE AND ENCRYPTION
Problem 5. Many biometric systems lack standardization in storing and encrypting biometric
data. As a result, a company is considering implementing a biometric access system for their
employees which uses fingerprint recognition technology. The system saves the fingerprint data in
an encrypted format using a unique encryption algorithm. The system has a false acceptance rate
(FAR) of 0.01 and a false rejection rate (FRR) of 0.05.
If the company has 1000 employees and each employee tries to access the system twice a day
on average, calculate the following:
a) The expected number of false acceptances per day.
b) The expected number of false rejections per day.
c) The total expected number of authentication errors per day.
Solution 5.
a) To calculate the expected number of false acceptances per day, we need to consider the
FAR and the total number of attempted accesses by the 1000 employees.
The FAR represents the probability of the system incorrectly accepting a false fingerprint as a
match. Since FAR = 0.01, the probability of a false acceptance is 0.01.
Let pF A be the probability of false acceptance, then:
pF A = 0.01
The company has 1000 employees trying to access the system twice a day on average, which
results in 2000 attempted accesses per day. Therefore, the expected number of false acceptances
per day can be calculated as:
Expected number of false acceptances per day =pF A ×2000
Expected number of false acceptances per day = 0.01 ×2000 = 20
Therefore, the expected number of false acceptances per day is 20.
b) To calculate the expected number of false rejections per day, we need to consider the FRR
and the total number of attempted accesses by the 1000 employees.
Similar to the FAR, the FRR represents the probability of the system incorrectly rejecting a valid
fingerprint as a non-match. Since FRR = 0.05, the probability of a false rejection is 0.05.
Let pF R be the probability of false rejection, then:
pF R = 0.05
The expected number of false rejections per day can be calculated as:
Expected number of false rejections per day =pF R ×2000
Expected number of false rejections per day = 0.05 ×2000 = 100
Therefore, the expected number of false rejections per day is 100.
c) The total expected number of authentication errors per day is the sum of false acceptances
and false rejections. Therefore:
Total expected number of authentication errors per day =Number of false acceptances+Number of false rejections
Total expected number of authentication errors per day = 20 + 100 = 120
Thus, the total expected number of authentication errors per day is 120.
6 6. BIOMETRIC SPOOFING TECHNIQUES AND THE NEED FOR ANTI-SPOOFING MEA-
SURES
Problem 6. A biometric security system uses fingerprint recognition for authentication. In
testing the system, it is found that the False Acceptance Rate (FAR) is 0.1% and the False Rejection
Rate (FRR) is 1%. Calculate the Equal Error Rate (EER) for this system.
Solution 6. The Equal Error Rate (EER) is the point at which the False Acceptance Rate (FAR)
is equal to the False Rejection Rate (FRR).
Given FAR = 0.1% and FRR = 1%, we need to find the EER.
Let the threshold be denoted as T. The EER occurs when FAR = FRR.
From the definitions:
FAR =P(reject |genuine)
FRR =P(accept |imposter)
Therefore, the EER is the average of FAR and FRR.
EER =FAR +FRR
2
EER =0.1+1
2= 0.55%
So, the Equal Error Rate (EER) for this system is 0.55%.
By analyzing the EER, we can see how well the system balances between rejecting valid users
and accepting imposters.
7 Biometric Security and Identity Management
Problem: Biometric systems often use multiple modalities to enhance security. Consider a
biometric system that combines fingerprint and iris recognition technologies. The false acceptance
rates for fingerprint and iris recognition are 0.1% and 0.01%, respectively.
a) What is the probability of falsely accepting an impostor using only fingerprint recognition?
b) What is the probability of falsely accepting an impostor using only iris recognition?
c) If the system requires both modalities to match in order to authenticate, what is the overall
false acceptance rate of the system?
Solution:
a) The probability of falsely accepting an impostor using only fingerprint recognition is given by
the false acceptance rate of the fingerprint modality, which is 0.1% or 0.001.
b) The probability of falsely accepting an impostor using only iris recognition is given by the
false acceptance rate of the iris modality, which is 0.01% or 0.0001.
c) Since the system requires both modalities to match for authentication, the overall false ac-
ceptance rate of the system can be calculated by multiplying the false acceptance rates of each
modality:
Overall False Acceptance Rate =False Acceptance Rate of Fingerprint×False Acceptance Rate of Iris
= 0.001 ×0.0001 = 0.0000001
Therefore, the overall false acceptance rate of the system when using both fingerprint and iris
recognition modalities is 0.00001%.
8 8. AUTHENTICATION FAILURES DUE TO AGING OR CHANGING BIOMETRIC CHARAC-
TERISTICS
Problem 8. In a biometric security system, the false acceptance rate (FAR) is measured to
be 0.02 and the false rejection rate (FRR) is 0.03. Calculate the Equal Error Rate (EER) for this
system.
Solution 8. The Equal Error Rate (EER) is the point where the false acceptance rate (FAR)
and the false rejection rate (FRR) are equal. Mathematically, EER is given by the average of FAR
and FRR when they are equal:
EER = (FAR + FRR) / 2
Given: FAR = 0.02 FRR = 0.03
Substitute the values into the formula: EER = (0.02 + 0.03) / 2 EER = 0.05 / 2 EER = 0.025
Therefore, the Equal Error Rate (EER) for this biometric security system is 0.025.
9 9. PROBLEMS WITH BIOMETRIC MATCHING ALGORITHMS AND ACCURACY RATES
Problem 9.
A security system that uses fingerprint recognition has an accuracy rate of 99.5%. If a random
sample of 500 fingerprints are scanned, calculate the probability that:
a) Exactly 3 fingerprints are incorrectly recognized.
b) At most 5 fingerprints are incorrectly recognized.
c) More than 10 fingerprints are incorrectly recognized.
Solution 9.
a) To find the probability that exactly 3 fingerprints are incorrectly recognized, we can use the
binomial probability formula:
P(X=k) = n
k·pk·(1 −p)n−k
where n= 500 (number of fingerprint scans), k= 3 (number of incorrectly recognized finger-
prints), and p= 1 −0.995 = 0.005 (probability of incorrectly recognizing a fingerprint).
P(X= 3) = 500
3·0.0053·0.995497
Calculating this probability gives us the answer for part a.
b) To find the probability that at most 5 fingerprints are incorrectly recognized, we need to
calculate
P(X≤5) = P(X= 0) + P(X= 1) + P(X= 2) + P(X= 3) + P(X= 4) + P(X= 5)
Using the binomial probability formula for each value of k, we can add up the probabilities to
find the answer for part b.
c) To find the probability that more than 10 fingerprints are incorrectly recognized, we can cal-
culate the complement of the probability that at most 10 are incorrectly recognized:
P(X > 10) = 1 −P(X≤10)
Using the binomial probability formula as in part b, we can find P(X≤10) and subtract it from
1 to get the answer for part c.
10 10. PRIVACY CONCERNS AND DATA PROTECTION IN BIOMETRIC DATABASES
Problem 10. A biometric system has an FAR (False Acceptance Rate) of 0.1% and an FRR
(False Rejection Rate) of 5%. If an individual tries to authenticate using this system 50 times, what
is the probability that they will be falsely rejected at least once?
Solution 10. Given that the FRR (False Rejection Rate) is 5%, the probability of a successful
authentication (1 - FRR) for one attempt is 95% or 0.95. Let pbe the probability of being falsely
rejected at least once in 50 attempts.
The probability of being falsely rejected at least once in ntrials is given by 1−(1 −FRR)n.
Therefore, the probability of being falsely rejected at least once in 50 attempts is:
p= 1 −(1 −0.05)50 = 1 −0.9550 ≈0.9231
Therefore, the probability that the individual will be falsely rejected at least once when attempt-
ing to authenticate 50 times using this biometric system is approximately 92.31%.
I. Problem: Biometric System Performance Degradation Over Time
An iris recognition system has an initial false acceptance rate (FAR) of 0.01
Solution
Given: Initial FAR = 0.01Initial FRR = 0.05Degraded FAR = 0.03Degraded FRR = 0.08
EER is the point where FAR = FRR.
Initial system: EER = (Initial FAR + Initial FRR) / 2 EER = (0.0001 + 0.0005) / 2 EER = 0.0003
Degraded system: EER = (Degraded FAR + Degraded FRR) / 2 EER = (0.0003 + 0.0008) / 2
EER = 0.00055
Therefore, the EER for the initial system is 0.0003 and for the degraded system is 0.00055.
11 12. ISSUES WITH BIOMETRIC IDENTIFICATION IN LARGE-SCALE SYSTEMS
Problem 12. In a large-scale biometric identification system, the false acceptance rate (FAR) is
0.1% and the false rejection rate (FRR) is 5%. If 1000 individuals attempt to authenticate themselves
using the system, calculate:
a) The number of individuals falsely accepted by the system.
b) The number of individuals falsely rejected by the system.
c) The overall accuracy of the system.
Solution 12.
a) The false acceptance rate (FAR) is 0.1%, which means 0.001 or 1
1000 of attempts are falsely
accepted. For 1000 individuals, the number falsely accepted can be calculated as:
False Accepted = 1000 ×0.001 = 1 individual
Therefore, the system falsely accepts 1individual.
b) The false rejection rate (FRR) is 5%, which means 5% or 0.05 of attempts are falsely rejected.
For 1000 individuals, the number falsely rejected can be calculated as:
False Rejected = 1000 ×0.05 = 50 individuals
Therefore, the system falsely rejects 50 individuals.
c) The overall accuracy of the system can be calculated as:
Accuracy = (1 −FAR)×100% = (1 −0.001) ×100% = 99.9%
Thus, the overall accuracy of the system is 99.9%.
12 13. CHALLENGES IN SECURING BIOMETRIC DATA DURING TRANSMISSION AND
STORAGE
Problem 13. Biometric data of a user is being transmitted over a network for authentication
purposes. The biometric data consists of a fingerprint image of size 500 ×500 pixels, where each
pixel is represented by an 8-bit grayscale value. During transmission, encryption is applied using
a symmetric key algorithm that increases the size of the data by 20%. If the encryption key used
is 128 bits long, calculate the size of the encrypted biometric data in kilobytes.
Solution 13. a) The size of the original fingerprint image in bytes can be calculated as:
Image size =Width ×Height ×Bit depth
Since each pixel is represented by an 8-bit grayscale value, the bit depth is 8. Given the image
size is 500 ×500 pixels:
Image size = 500 ×500 ×8bits
b) To convert bits to bytes, we divide by 8:
Image size in bytes =500 ×500 ×8
8bytes
c) After encryption, the size of the data increases by 20%. The new size of the encrypted data
can be calculated as:
New encrypted data size = 1.2×Original data size (in bytes)
d) Given that the encryption key is 128 bits long, to convert it to bytes, we divide by 8:
Key size =128
8bytes
e) The total size of the encrypted biometric data is the sum of the encrypted data size and the
key size:
Total size =New encrypted data size +Key size
Now, substitute the values calculated in parts a, b, c, and d to find the total size of the encrypted
biometric data in kilobytes.
13 14. IMPLICATIONS OF BIOMETRIC DATA BREACHES AND IDENTITY THEFT
Problem 14. A company’s biometric security system has been breached, and the fingerprint
data of 1000 employees has been stolen. If each fingerprint template contains 500 bytes of data and
the company offers a reward of 500percompromisedfingerprinttemplate, calculatethetotalrewardamountthecompanywillneedtopaytoaf f ectedemployees.
Solution 14. Given that: - Number of compromised fingerprint templates = 1000 - Size of each
fingerprint template = 500 bytes - Reward offered per compromised template = 500
Step 1: Calculate the total amount of data compromised Total amount of data compromised
= Number of compromised templates ×Size of each template Total amount of data compromised
= 1000 ×500 bytes Total amount of data compromised = 500,000 bytes
Step 2: Calculate the total reward amount Total reward amount = Number of compromised
templates ×Reward offered per template Total reward amount = 1000 ×500T otalrewardamount =500,000
Therefore, the company will need to pay a total reward amount of 500,000totheaffectedemployees.
14 15. CULTURAL BIASES AND LIMITATIONS IN BIOMETRIC RECOGNITION SYSTEMS
Problem 15. A facial recognition system has been trained on a dataset composed mainly of
individuals of Asian descent. When tested on a diverse population that includes individuals of
African descent, the system shows a significantly higher error rate for this subgroup.
The error rates for the system on the test set are as follows:
•Overall error rate: 5
•Error rate for individuals of Asian descent: 3
•Error rate for individuals of African descent: 12
a) What is the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent?
b) If the system correctly identifies 90
c) Considering these results, discuss the potential impact of cultural biases on the effectiveness
of biometric recognition systems.
Solution 15.
a) The odds ratio of misidentification for individuals of African descent compared to individuals
of Asian descent can be calculated using the error rates:
Odds ratio =Error rate for African descent
Error rate for Asian descent =0.12
0.03 = 4
Therefore, the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent is 4.
b) Given that the system correctly identifies 90
Accuracy rate for African descent = 1 −Error rate for African descent = 1 −0.12 = 0.88 = 88%
Therefore, the accuracy rate for individuals of African descent is 88
c) The significantly higher error rate for individuals of African descent in the facial recognition
system compared to individuals of Asian descent highlights a bias that may have been introduced
during the training phase, where the dataset was skewed towards a specific ethnicity. This cultural
bias can lead to inequitable outcomes, with certain demographic groups facing higher risks of
misidentification and potential discrimination. Addressing cultural biases in biometric recognition
systems is crucial to ensure fair and accurate identification across diverse populations.
15 16. LEGAL AND ETHICAL IMPLICATIONS OF BIOMETRIC DATA COLLECTION AND
USE
Problem 16. A company is implementing a biometric authentication system for its employees
using fingerprint recognition. The system is expected to process 1000 fingerprint scans per day. If
each fingerprint scan produces a file size of 500 KB, calculate the total amount of data collected
by the system in one month.
Solution 16. Given that the system processes 1000 fingerprint scans per day and each scan
produces a file size of 500 KB, we can calculate the total data collected by the system in one day
as follows:
Total data collected per day = Number of scans per day x File size per scan Total data collected
per day = 1000 x 500 KB Total data collected per day = 500,000 KB
To convert KB to GB, we use the conversion factor: 1 GB = 1024 KB
Total data collected per day in GB = 500,000 KB / 1024 = 488.28 GB
Since there are typically 30 days in a month, we can calculate the total amount of data collected
by the system in one month as follows:
Total data collected per month = Total data collected per day x Number of days in a month Total
data collected per month = 488.28 GB x 30 Total data collected per month = 14,648.4 GB
Therefore, the total amount of data collected by the system in one month is 14,648.4 GB.
16 17. CHALLENGES IN IMPLEMENTING BIOMETRIC AUTHENTICATION IN IOT DEVICES
Problem 17. A company is considering implementing biometric authentication for their IoT
devices to enhance security. They estimate that the false acceptance rate (FAR) of the biometric
system is 0.1% and the false rejection rate (FRR) is 1%. If an unauthorized user attempts to access
the device 50 times, calculate the probability that the system will grant access to this unauthorized
user at least once.
Solution 17. To calculate the probability of the system granting access to the unauthorized
user at least once, we need to consider the probability of false acceptance for each attempt. Let’s
denote the probability of a false acceptance as P(F A).
Given that FAR is 0.1%, we have:
P(F A) = 0.001
Therefore, the probability of rejecting a false attempt is:
P(¬F A) = 1 −P(F A) = 1 −0.001 = 0.999
The probability that the system will grant access to the unauthorized user at least once in 50
attempts is the complement of the probability that it will reject all 50 attempts. Thus, the probability
is:
P(Access granted at least once)=1−P(Access not granted at all)
Since each attempt is independent, the probability of the system not granting access to the
unauthorized user in a single attempt is P(¬F A). Therefore, the probability that the system will not
grant access in 50 attempts is:
P(Access not granted at all)=(P(¬F A))n= 0.99950
Calculating this probability:
P(Access not granted at all)=0.99950 ≈0.950 = 95%
So, the probability that the system will grant access to the unauthorized user at least once is
approximately:
P(Access granted at least once)=1−0.950 = 0.050 = 5%
Therefore, the system has a 5% chance of granting access to the unauthorized user at least
once out of 50 attempts.
17 18. USER ACCEPTANCE AND TRUST ISSUES WITH BIOMETRIC SECURITY MEASURES
Problem 18.
A study is conducted to analyze the user acceptance of a new biometric authentication system.
The study involved 100 participants, and the results showed that 65 participants were willing to use
the biometric system, while the remaining participants expressed concerns about privacy and did
not want to adopt it.
a) Calculate the user acceptance rate in percentage.
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same, how many participants would be willing to use the biometric system?
c) Discuss why user acceptance and trust are essential factors in the implementation of bio-
metric security measures.
Solution 18.
a) The user acceptance rate can be calculated as follows:
User acceptance rate =Number of participants willing to use the system
Total number of participants ×100%
In this case, the number of participants willing to use the system is 65, and the total number of
participants is 100. Therefore,
User acceptance rate =65
100 ×100% = 65%
So, the user acceptance rate is 65
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same at 65
Number of participants willing to use the system =Acceptance rate×Total number of participants
Number of participants willing to use the system = 0.65 ×150 = 97.5
Since we cannot have a fraction of a participant, we can estimate that approximately 98 partic-
ipants would be willing to use the biometric system.
c) User acceptance and trust are crucial factors in the implementation of biometric security
measures because:
1. If users do not trust the system, they might not use it properly or try to bypass it, leading to
security vulnerabilities. 2. Acceptance by users is essential for the success of the system, as low
adoption rates can render the biometric security measures ineffective. 3. Ensuring user acceptance
and trust can help improve the overall user experience and satisfaction with the security system. 4.
Addressing privacy concerns and building trust can help mitigate potential legal and ethical issues
related to biometric data collection and storage.
18 19. CHALLENGES IN SECURELY SHARING BIOMETRIC DATA ACROSS DIFFERENT
PLATFORMS
Problem 19. Consider a scenario where a company wants to securely share biometric data
(such as fingerprint scans) across different platforms for employee authentication. However, due to
security concerns, the company decides to implement a secure encryption scheme before sharing
the data.
The encryption scheme they choose involves using a symmetric key system where each pixel
value of the fingerprint image is replaced by the result of applying a modular operation. If the
original pixel value is p= 125 and the symmetric key is k= 7, determine the encrypted pixel value.
Solution 19.
To determine the encrypted pixel value, we need to apply the modular operation using the
symmetric key. The encryption process is represented by the formula:
Encrypted pixel value = (p+k) mod 256
Given p= 125 and k= 7, we can substitute these values into the formula:
Encrypted pixel value = (125 + 7) mod 256
Encrypted pixel value = 132 mod 256
Encrypted pixel value = 132 −256 ×132
256
Encrypted pixel value = 132 −256 ×0
Encrypted pixel value = 132
Therefore, the encrypted pixel value is 132.
19 20. BIOMETRIC IDENTIFICATION ERRORS AND THEIR IMPACT ON SECURITY MEA-
SURES
Problem 20. A biometric system is being evaluated for its performance in identifying individuals
based on facial recognition. In a test dataset of 500 individuals, the system produced the following
results:
- True positives: 450 individuals correctly identified - False positives: 20 individuals incorrectly
identified - False negatives: 10 individuals not identified
Calculate the following metrics for the system:
a) The accuracy of the system (in percentage).
b) The false acceptance rate (FAR) of the system (in percentage).
c) The false rejection rate (FRR) of the system (in percentage).
Solution 20.
a) The accuracy of the system is given by the formula:
Accuracy =True Positives +True Negatives
Total Population =450
500 = 0.9
Converting this to a percentage:
Accuracy = 0.9×100% = 90%
Therefore, the accuracy of the system is 90
b) The false acceptance rate (FAR) of the system is calculated as:
FAR =False Positives
False Positives + True Negatives =20
20 + 500 −450 =20
70 = 0.2857
Converting this to a percentage:
FAR = 0.2857 ×100% = 28.57%
Therefore, the false acceptance rate (FAR) of the system is 28.57
c) The false rejection rate (FRR) of the system is calculated as:
FRR =False Negatives
False Negatives + True Positives =10
10 + 450 =10
460 ≈0.0217
Converting this to a percentage:
FRR = 0.0217 ×100% ≈2.17%
Therefore, the false rejection rate (FRR) of the system is approximately 2.17
Solution 2. Let’s calculate the probability of none of the 4 attempts being successful, and then
subtract this from 1 to find the probability that at least one attempt is successful.
Let pbe the probability of success (tricking the system) on a single attempt, which is 10% or
0.1.
The probability of failure on a single attempt is 1−p= 1 −0.1=0.9.
The probability of none of the 4 attempts being successful is:
(0.9)4= 0.6561
Therefore, the probability of at least one attempt being successful is:
1−0.6561 = 0.3439
So, the probability that at least one of the 4 attempts will be successful is approximately 34.39%.
3 3. ISSUES WITH IRIS AUTHENTICATION IN LOW LIGHTING CONDITIONS
Problem 3. The iris recognition system at a high-security facility struggled to authenticate
individuals in low lighting conditions due to excessive noise in the captured images. One of the
authentication methods used is based on the comparison of feature vectors extracted from the iris
image.
Given two feature vectors X= [1.2,3.4,−2.1,0.9] and Y= [0.8,3.9,−2.3,1.1], calculate the
Euclidean distance between these feature vectors as a measure of similarity.
Solution 3.
1. Let’s calculate the Euclidean distance between the two feature vectors Xand Y, given by:
euclidean distance =v
u
u
t
n
X
i=1
(Xi−Yi)2
where Xiand Yiare the elements of the feature vectors Xand Yrespectively.
2. Plugging in the values for Xand Y:
euclidean distance =p(1.2−0.8)2+ (3.4−3.9)2+ (−2.1+2.3)2+ (0.9−1.1)2
euclidean distance =√0.16 + 0.25 + 0.04 + 0.04
euclidean distance =√0.49 = 0.7
Hence, the Euclidean distance between the feature vectors Xand Yis 0.7 units.
4 Biometric Security and Identity Management
Problem: Consider a biometric system that uses fingerprint recognition for authentication. The
system has an equal error rate of 0.05. If there are 1000 users attempting to authenticate using this
system, calculate the number of users that will be incorrectly authenticated due to false rejection
and false acceptance.
Solution: Given the equal error rate (EER) is 0.05, this means that the false rejection rate
(FRR) is equal to the false acceptance rate (FAR) at the threshold where they intersect.
Let Nbe the total number of users attempting to authenticate, which is 1000.
1. Finding the number of users incorrectly rejected:
The false rejection rate (FRR) is 0.05. This means that 5
Number of users incorrectly rejected = FRR ×N Number of users incorrectly rejected = 0.05 ×
1000 Number of users incorrectly rejected = 50 users
So, 50 users will be incorrectly rejected.
2. Finding the number of users incorrectly accepted:
The false acceptance rate (FAR) is also 0.05. This means that 5
Number of impostors incorrectly accepted = FAR ×N Number of impostors incorrectly accepted
= 0.05 ×1000 Number of impostors incorrectly accepted = 50 users
Therefore, 50 impostors will be incorrectly accepted during authentication process.
5 5. LACK OF STANDARDIZATION IN BIOMETRIC DATA STORAGE AND ENCRYPTION
Problem 5. Many biometric systems lack standardization in storing and encrypting biometric
data. As a result, a company is considering implementing a biometric access system for their
employees which uses fingerprint recognition technology. The system saves the fingerprint data in
an encrypted format using a unique encryption algorithm. The system has a false acceptance rate
(FAR) of 0.01 and a false rejection rate (FRR) of 0.05.
If the company has 1000 employees and each employee tries to access the system twice a day
on average, calculate the following:
a) The expected number of false acceptances per day.
b) The expected number of false rejections per day.
c) The total expected number of authentication errors per day.
Solution 5.
a) To calculate the expected number of false acceptances per day, we need to consider the
FAR and the total number of attempted accesses by the 1000 employees.
The FAR represents the probability of the system incorrectly accepting a false fingerprint as a
match. Since FAR = 0.01, the probability of a false acceptance is 0.01.
Let pF A be the probability of false acceptance, then:
pF A = 0.01
The company has 1000 employees trying to access the system twice a day on average, which
results in 2000 attempted accesses per day. Therefore, the expected number of false acceptances
per day can be calculated as:
Expected number of false acceptances per day =pF A ×2000
Expected number of false acceptances per day = 0.01 ×2000 = 20
Therefore, the expected number of false acceptances per day is 20.
b) To calculate the expected number of false rejections per day, we need to consider the FRR
and the total number of attempted accesses by the 1000 employees.
Similar to the FAR, the FRR represents the probability of the system incorrectly rejecting a valid
fingerprint as a non-match. Since FRR = 0.05, the probability of a false rejection is 0.05.
Let pF R be the probability of false rejection, then:
pF R = 0.05
The expected number of false rejections per day can be calculated as:
Expected number of false rejections per day =pF R ×2000
Expected number of false rejections per day = 0.05 ×2000 = 100
Therefore, the expected number of false rejections per day is 100.
c) The total expected number of authentication errors per day is the sum of false acceptances
and false rejections. Therefore:
Total expected number of authentication errors per day =Number of false acceptances+Number of false rejections
Total expected number of authentication errors per day = 20 + 100 = 120
Thus, the total expected number of authentication errors per day is 120.
6 6. BIOMETRIC SPOOFING TECHNIQUES AND THE NEED FOR ANTI-SPOOFING MEA-
SURES
Problem 6. A biometric security system uses fingerprint recognition for authentication. In
testing the system, it is found that the False Acceptance Rate (FAR) is 0.1% and the False Rejection
Rate (FRR) is 1%. Calculate the Equal Error Rate (EER) for this system.
Solution 6. The Equal Error Rate (EER) is the point at which the False Acceptance Rate (FAR)
is equal to the False Rejection Rate (FRR).
Given FAR = 0.1% and FRR = 1%, we need to find the EER.
Let the threshold be denoted as T. The EER occurs when FAR = FRR.
From the definitions:
FAR =P(reject |genuine)
FRR =P(accept |imposter)
Therefore, the EER is the average of FAR and FRR.
EER =FAR +FRR
2
EER =0.1+1
2= 0.55%
So, the Equal Error Rate (EER) for this system is 0.55%.
By analyzing the EER, we can see how well the system balances between rejecting valid users
and accepting imposters.
7 Biometric Security and Identity Management
Problem: Biometric systems often use multiple modalities to enhance security. Consider a
biometric system that combines fingerprint and iris recognition technologies. The false acceptance
rates for fingerprint and iris recognition are 0.1% and 0.01%, respectively.
a) What is the probability of falsely accepting an impostor using only fingerprint recognition?
b) What is the probability of falsely accepting an impostor using only iris recognition?
c) If the system requires both modalities to match in order to authenticate, what is the overall
false acceptance rate of the system?
Solution:
a) The probability of falsely accepting an impostor using only fingerprint recognition is given by
the false acceptance rate of the fingerprint modality, which is 0.1% or 0.001.
b) The probability of falsely accepting an impostor using only iris recognition is given by the
false acceptance rate of the iris modality, which is 0.01% or 0.0001.
c) Since the system requires both modalities to match for authentication, the overall false ac-
ceptance rate of the system can be calculated by multiplying the false acceptance rates of each
modality:
Overall False Acceptance Rate =False Acceptance Rate of Fingerprint×False Acceptance Rate of Iris
= 0.001 ×0.0001 = 0.0000001
Therefore, the overall false acceptance rate of the system when using both fingerprint and iris
recognition modalities is 0.00001%.
8 8. AUTHENTICATION FAILURES DUE TO AGING OR CHANGING BIOMETRIC CHARAC-
TERISTICS
Problem 8. In a biometric security system, the false acceptance rate (FAR) is measured to
be 0.02 and the false rejection rate (FRR) is 0.03. Calculate the Equal Error Rate (EER) for this
system.
Solution 8. The Equal Error Rate (EER) is the point where the false acceptance rate (FAR)
and the false rejection rate (FRR) are equal. Mathematically, EER is given by the average of FAR
and FRR when they are equal:
EER = (FAR + FRR) / 2
Given: FAR = 0.02 FRR = 0.03
Substitute the values into the formula: EER = (0.02 + 0.03) / 2 EER = 0.05 / 2 EER = 0.025
Therefore, the Equal Error Rate (EER) for this biometric security system is 0.025.
9 9. PROBLEMS WITH BIOMETRIC MATCHING ALGORITHMS AND ACCURACY RATES
Problem 9.
A security system that uses fingerprint recognition has an accuracy rate of 99.5%. If a random
sample of 500 fingerprints are scanned, calculate the probability that:
a) Exactly 3 fingerprints are incorrectly recognized.
b) At most 5 fingerprints are incorrectly recognized.
c) More than 10 fingerprints are incorrectly recognized.
Solution 9.
a) To find the probability that exactly 3 fingerprints are incorrectly recognized, we can use the
binomial probability formula:
P(X=k) = n
k·pk·(1 −p)n−k
where n= 500 (number of fingerprint scans), k= 3 (number of incorrectly recognized finger-
prints), and p= 1 −0.995 = 0.005 (probability of incorrectly recognizing a fingerprint).
P(X= 3) = 500
3·0.0053·0.995497
Calculating this probability gives us the answer for part a.
b) To find the probability that at most 5 fingerprints are incorrectly recognized, we need to
calculate
P(X≤5) = P(X= 0) + P(X= 1) + P(X= 2) + P(X= 3) + P(X= 4) + P(X= 5)
Using the binomial probability formula for each value of k, we can add up the probabilities to
find the answer for part b.
c) To find the probability that more than 10 fingerprints are incorrectly recognized, we can cal-
culate the complement of the probability that at most 10 are incorrectly recognized:
P(X > 10) = 1 −P(X≤10)
Using the binomial probability formula as in part b, we can find P(X≤10) and subtract it from
1 to get the answer for part c.
10 10. PRIVACY CONCERNS AND DATA PROTECTION IN BIOMETRIC DATABASES
Problem 10. A biometric system has an FAR (False Acceptance Rate) of 0.1% and an FRR
(False Rejection Rate) of 5%. If an individual tries to authenticate using this system 50 times, what
is the probability that they will be falsely rejected at least once?
Solution 10. Given that the FRR (False Rejection Rate) is 5%, the probability of a successful
authentication (1 - FRR) for one attempt is 95% or 0.95. Let pbe the probability of being falsely
rejected at least once in 50 attempts.
The probability of being falsely rejected at least once in ntrials is given by 1−(1 −FRR)n.
Therefore, the probability of being falsely rejected at least once in 50 attempts is:
p= 1 −(1 −0.05)50 = 1 −0.9550 ≈0.9231
Therefore, the probability that the individual will be falsely rejected at least once when attempt-
ing to authenticate 50 times using this biometric system is approximately 92.31%.
I. Problem: Biometric System Performance Degradation Over Time
An iris recognition system has an initial false acceptance rate (FAR) of 0.01
Solution
Given: Initial FAR = 0.01Initial FRR = 0.05Degraded FAR = 0.03Degraded FRR = 0.08
EER is the point where FAR = FRR.
Initial system: EER = (Initial FAR + Initial FRR) / 2 EER = (0.0001 + 0.0005) / 2 EER = 0.0003
Degraded system: EER = (Degraded FAR + Degraded FRR) / 2 EER = (0.0003 + 0.0008) / 2
EER = 0.00055
Therefore, the EER for the initial system is 0.0003 and for the degraded system is 0.00055.
11 12. ISSUES WITH BIOMETRIC IDENTIFICATION IN LARGE-SCALE SYSTEMS
Problem 12. In a large-scale biometric identification system, the false acceptance rate (FAR) is
0.1% and the false rejection rate (FRR) is 5%. If 1000 individuals attempt to authenticate themselves
using the system, calculate:
a) The number of individuals falsely accepted by the system.
b) The number of individuals falsely rejected by the system.
c) The overall accuracy of the system.
Solution 12.
a) The false acceptance rate (FAR) is 0.1%, which means 0.001 or 1
1000 of attempts are falsely
accepted. For 1000 individuals, the number falsely accepted can be calculated as:
False Accepted = 1000 ×0.001 = 1 individual
Therefore, the system falsely accepts 1individual.
b) The false rejection rate (FRR) is 5%, which means 5% or 0.05 of attempts are falsely rejected.
For 1000 individuals, the number falsely rejected can be calculated as:
False Rejected = 1000 ×0.05 = 50 individuals
Therefore, the system falsely rejects 50 individuals.
c) The overall accuracy of the system can be calculated as:
Accuracy = (1 −FAR)×100% = (1 −0.001) ×100% = 99.9%
Thus, the overall accuracy of the system is 99.9%.
12 13. CHALLENGES IN SECURING BIOMETRIC DATA DURING TRANSMISSION AND
STORAGE
Problem 13. Biometric data of a user is being transmitted over a network for authentication
purposes. The biometric data consists of a fingerprint image of size 500 ×500 pixels, where each
pixel is represented by an 8-bit grayscale value. During transmission, encryption is applied using
a symmetric key algorithm that increases the size of the data by 20%. If the encryption key used
is 128 bits long, calculate the size of the encrypted biometric data in kilobytes.
Solution 13. a) The size of the original fingerprint image in bytes can be calculated as:
Image size =Width ×Height ×Bit depth
Since each pixel is represented by an 8-bit grayscale value, the bit depth is 8. Given the image
size is 500 ×500 pixels:
Image size = 500 ×500 ×8bits
b) To convert bits to bytes, we divide by 8:
Image size in bytes =500 ×500 ×8
8bytes
c) After encryption, the size of the data increases by 20%. The new size of the encrypted data
can be calculated as:
New encrypted data size = 1.2×Original data size (in bytes)
d) Given that the encryption key is 128 bits long, to convert it to bytes, we divide by 8:
Key size =128
8bytes
e) The total size of the encrypted biometric data is the sum of the encrypted data size and the
key size:
Total size =New encrypted data size +Key size
Now, substitute the values calculated in parts a, b, c, and d to find the total size of the encrypted
biometric data in kilobytes.
13 14. IMPLICATIONS OF BIOMETRIC DATA BREACHES AND IDENTITY THEFT
Problem 14. A company’s biometric security system has been breached, and the fingerprint
data of 1000 employees has been stolen. If each fingerprint template contains 500 bytes of data and
the company offers a reward of 500percompromisedfingerprinttemplate, calculatethetotalrewardamountthecompanywillneedtopaytoaf f ectedemployees.
Solution 14. Given that: - Number of compromised fingerprint templates = 1000 - Size of each
fingerprint template = 500 bytes - Reward offered per compromised template = 500
Step 1: Calculate the total amount of data compromised Total amount of data compromised
= Number of compromised templates ×Size of each template Total amount of data compromised
= 1000 ×500 bytes Total amount of data compromised = 500,000 bytes
Step 2: Calculate the total reward amount Total reward amount = Number of compromised
templates ×Reward offered per template Total reward amount = 1000 ×500T otalrewardamount =500,000
Therefore, the company will need to pay a total reward amount of 500,000totheaffectedemployees.
14 15. CULTURAL BIASES AND LIMITATIONS IN BIOMETRIC RECOGNITION SYSTEMS
Problem 15. A facial recognition system has been trained on a dataset composed mainly of
individuals of Asian descent. When tested on a diverse population that includes individuals of
African descent, the system shows a significantly higher error rate for this subgroup.
The error rates for the system on the test set are as follows:
•Overall error rate: 5
•Error rate for individuals of Asian descent: 3
•Error rate for individuals of African descent: 12
a) What is the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent?
b) If the system correctly identifies 90
c) Considering these results, discuss the potential impact of cultural biases on the effectiveness
of biometric recognition systems.
Solution 15.
a) The odds ratio of misidentification for individuals of African descent compared to individuals
of Asian descent can be calculated using the error rates:
Odds ratio =Error rate for African descent
Error rate for Asian descent =0.12
0.03 = 4
Therefore, the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent is 4.
b) Given that the system correctly identifies 90
Accuracy rate for African descent = 1 −Error rate for African descent = 1 −0.12 = 0.88 = 88%
Therefore, the accuracy rate for individuals of African descent is 88
c) The significantly higher error rate for individuals of African descent in the facial recognition
system compared to individuals of Asian descent highlights a bias that may have been introduced
during the training phase, where the dataset was skewed towards a specific ethnicity. This cultural
bias can lead to inequitable outcomes, with certain demographic groups facing higher risks of
misidentification and potential discrimination. Addressing cultural biases in biometric recognition
systems is crucial to ensure fair and accurate identification across diverse populations.
15 16. LEGAL AND ETHICAL IMPLICATIONS OF BIOMETRIC DATA COLLECTION AND
USE
Problem 16. A company is implementing a biometric authentication system for its employees
using fingerprint recognition. The system is expected to process 1000 fingerprint scans per day. If
each fingerprint scan produces a file size of 500 KB, calculate the total amount of data collected
by the system in one month.
Solution 16. Given that the system processes 1000 fingerprint scans per day and each scan
produces a file size of 500 KB, we can calculate the total data collected by the system in one day
as follows:
Total data collected per day = Number of scans per day x File size per scan Total data collected
per day = 1000 x 500 KB Total data collected per day = 500,000 KB
To convert KB to GB, we use the conversion factor: 1 GB = 1024 KB
Total data collected per day in GB = 500,000 KB / 1024 = 488.28 GB
Since there are typically 30 days in a month, we can calculate the total amount of data collected
by the system in one month as follows:
Total data collected per month = Total data collected per day x Number of days in a month Total
data collected per month = 488.28 GB x 30 Total data collected per month = 14,648.4 GB
Therefore, the total amount of data collected by the system in one month is 14,648.4 GB.
16 17. CHALLENGES IN IMPLEMENTING BIOMETRIC AUTHENTICATION IN IOT DEVICES
Problem 17. A company is considering implementing biometric authentication for their IoT
devices to enhance security. They estimate that the false acceptance rate (FAR) of the biometric
system is 0.1% and the false rejection rate (FRR) is 1%. If an unauthorized user attempts to access
the device 50 times, calculate the probability that the system will grant access to this unauthorized
user at least once.
Solution 17. To calculate the probability of the system granting access to the unauthorized
user at least once, we need to consider the probability of false acceptance for each attempt. Let’s
denote the probability of a false acceptance as P(F A).
Given that FAR is 0.1%, we have:
P(F A) = 0.001
Therefore, the probability of rejecting a false attempt is:
P(¬F A) = 1 −P(F A) = 1 −0.001 = 0.999
The probability that the system will grant access to the unauthorized user at least once in 50
attempts is the complement of the probability that it will reject all 50 attempts. Thus, the probability
is:
P(Access granted at least once)=1−P(Access not granted at all)
Since each attempt is independent, the probability of the system not granting access to the
unauthorized user in a single attempt is P(¬F A). Therefore, the probability that the system will not
grant access in 50 attempts is:
P(Access not granted at all)=(P(¬F A))n= 0.99950
Calculating this probability:
P(Access not granted at all)=0.99950 ≈0.950 = 95%
So, the probability that the system will grant access to the unauthorized user at least once is
approximately:
P(Access granted at least once)=1−0.950 = 0.050 = 5%
Therefore, the system has a 5% chance of granting access to the unauthorized user at least
once out of 50 attempts.
17 18. USER ACCEPTANCE AND TRUST ISSUES WITH BIOMETRIC SECURITY MEASURES
Problem 18.
A study is conducted to analyze the user acceptance of a new biometric authentication system.
The study involved 100 participants, and the results showed that 65 participants were willing to use
the biometric system, while the remaining participants expressed concerns about privacy and did
not want to adopt it.
a) Calculate the user acceptance rate in percentage.
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same, how many participants would be willing to use the biometric system?
c) Discuss why user acceptance and trust are essential factors in the implementation of bio-
metric security measures.
Solution 18.
a) The user acceptance rate can be calculated as follows:
User acceptance rate =Number of participants willing to use the system
Total number of participants ×100%
In this case, the number of participants willing to use the system is 65, and the total number of
participants is 100. Therefore,
User acceptance rate =65
100 ×100% = 65%
So, the user acceptance rate is 65
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same at 65
Number of participants willing to use the system =Acceptance rate×Total number of participants
Number of participants willing to use the system = 0.65 ×150 = 97.5
Since we cannot have a fraction of a participant, we can estimate that approximately 98 partic-
ipants would be willing to use the biometric system.
c) User acceptance and trust are crucial factors in the implementation of biometric security
measures because:
1. If users do not trust the system, they might not use it properly or try to bypass it, leading to
security vulnerabilities. 2. Acceptance by users is essential for the success of the system, as low
adoption rates can render the biometric security measures ineffective. 3. Ensuring user acceptance
and trust can help improve the overall user experience and satisfaction with the security system. 4.
Addressing privacy concerns and building trust can help mitigate potential legal and ethical issues
related to biometric data collection and storage.
18 19. CHALLENGES IN SECURELY SHARING BIOMETRIC DATA ACROSS DIFFERENT
PLATFORMS
Problem 19. Consider a scenario where a company wants to securely share biometric data
(such as fingerprint scans) across different platforms for employee authentication. However, due to
security concerns, the company decides to implement a secure encryption scheme before sharing
the data.
The encryption scheme they choose involves using a symmetric key system where each pixel
value of the fingerprint image is replaced by the result of applying a modular operation. If the
original pixel value is p= 125 and the symmetric key is k= 7, determine the encrypted pixel value.
Solution 19.
To determine the encrypted pixel value, we need to apply the modular operation using the
symmetric key. The encryption process is represented by the formula:
Encrypted pixel value = (p+k) mod 256
Given p= 125 and k= 7, we can substitute these values into the formula:
Encrypted pixel value = (125 + 7) mod 256
Encrypted pixel value = 132 mod 256
Encrypted pixel value = 132 −256 ×132
256
Encrypted pixel value = 132 −256 ×0
Encrypted pixel value = 132
Therefore, the encrypted pixel value is 132.
19 20. BIOMETRIC IDENTIFICATION ERRORS AND THEIR IMPACT ON SECURITY MEA-
SURES
Problem 20. A biometric system is being evaluated for its performance in identifying individuals
based on facial recognition. In a test dataset of 500 individuals, the system produced the following
results:
- True positives: 450 individuals correctly identified - False positives: 20 individuals incorrectly
identified - False negatives: 10 individuals not identified
Calculate the following metrics for the system:
a) The accuracy of the system (in percentage).
b) The false acceptance rate (FAR) of the system (in percentage).
c) The false rejection rate (FRR) of the system (in percentage).
Solution 20.
a) The accuracy of the system is given by the formula:
Accuracy =True Positives +True Negatives
Total Population =450
500 = 0.9
Converting this to a percentage:
Accuracy = 0.9×100% = 90%
Therefore, the accuracy of the system is 90
b) The false acceptance rate (FAR) of the system is calculated as:
FAR =False Positives
False Positives + True Negatives =20
20 + 500 −450 =20
70 = 0.2857
Converting this to a percentage:
FAR = 0.2857 ×100% = 28.57%
Therefore, the false acceptance rate (FAR) of the system is 28.57
c) The false rejection rate (FRR) of the system is calculated as:
FRR =False Negatives
False Negatives + True Positives =10
10 + 450 =10
460 ≈0.0217
Converting this to a percentage:
FRR = 0.0217 ×100% ≈2.17%
Therefore, the false rejection rate (FRR) of the system is approximately 2.17
Solution 2. Let’s calculate the probability of none of the 4 attempts being successful, and then
subtract this from 1 to find the probability that at least one attempt is successful.
Let pbe the probability of success (tricking the system) on a single attempt, which is 10% or
0.1.
The probability of failure on a single attempt is 1−p= 1 −0.1=0.9.
The probability of none of the 4 attempts being successful is:
(0.9)4= 0.6561
Therefore, the probability of at least one attempt being successful is:
1−0.6561 = 0.3439
So, the probability that at least one of the 4 attempts will be successful is approximately 34.39%.
3 3. ISSUES WITH IRIS AUTHENTICATION IN LOW LIGHTING CONDITIONS
Problem 3. The iris recognition system at a high-security facility struggled to authenticate
individuals in low lighting conditions due to excessive noise in the captured images. One of the
authentication methods used is based on the comparison of feature vectors extracted from the iris
image.
Given two feature vectors X= [1.2,3.4,−2.1,0.9] and Y= [0.8,3.9,−2.3,1.1], calculate the
Euclidean distance between these feature vectors as a measure of similarity.
Solution 3.
1. Let’s calculate the Euclidean distance between the two feature vectors Xand Y, given by:
euclidean distance =v
u
u
t
n
X
i=1
(Xi−Yi)2
where Xiand Yiare the elements of the feature vectors Xand Yrespectively.
2. Plugging in the values for Xand Y:
euclidean distance =p(1.2−0.8)2+ (3.4−3.9)2+ (−2.1+2.3)2+ (0.9−1.1)2
euclidean distance =√0.16 + 0.25 + 0.04 + 0.04
euclidean distance =√0.49 = 0.7
Hence, the Euclidean distance between the feature vectors Xand Yis 0.7 units.
4 Biometric Security and Identity Management
Problem: Consider a biometric system that uses fingerprint recognition for authentication. The
system has an equal error rate of 0.05. If there are 1000 users attempting to authenticate using this
system, calculate the number of users that will be incorrectly authenticated due to false rejection
and false acceptance.
Solution: Given the equal error rate (EER) is 0.05, this means that the false rejection rate
(FRR) is equal to the false acceptance rate (FAR) at the threshold where they intersect.
Let Nbe the total number of users attempting to authenticate, which is 1000.
1. Finding the number of users incorrectly rejected:
The false rejection rate (FRR) is 0.05. This means that 5
Number of users incorrectly rejected = FRR ×N Number of users incorrectly rejected = 0.05 ×
1000 Number of users incorrectly rejected = 50 users
So, 50 users will be incorrectly rejected.
2. Finding the number of users incorrectly accepted:
The false acceptance rate (FAR) is also 0.05. This means that 5
Number of impostors incorrectly accepted = FAR ×N Number of impostors incorrectly accepted
= 0.05 ×1000 Number of impostors incorrectly accepted = 50 users
Therefore, 50 impostors will be incorrectly accepted during authentication process.
5 5. LACK OF STANDARDIZATION IN BIOMETRIC DATA STORAGE AND ENCRYPTION
Problem 5. Many biometric systems lack standardization in storing and encrypting biometric
data. As a result, a company is considering implementing a biometric access system for their
employees which uses fingerprint recognition technology. The system saves the fingerprint data in
an encrypted format using a unique encryption algorithm. The system has a false acceptance rate
(FAR) of 0.01 and a false rejection rate (FRR) of 0.05.
If the company has 1000 employees and each employee tries to access the system twice a day
on average, calculate the following:
a) The expected number of false acceptances per day.
b) The expected number of false rejections per day.
c) The total expected number of authentication errors per day.
Solution 5.
a) To calculate the expected number of false acceptances per day, we need to consider the
FAR and the total number of attempted accesses by the 1000 employees.
The FAR represents the probability of the system incorrectly accepting a false fingerprint as a
match. Since FAR = 0.01, the probability of a false acceptance is 0.01.
Let pF A be the probability of false acceptance, then:
pF A = 0.01
The company has 1000 employees trying to access the system twice a day on average, which
results in 2000 attempted accesses per day. Therefore, the expected number of false acceptances
per day can be calculated as:
Expected number of false acceptances per day =pF A ×2000
Expected number of false acceptances per day = 0.01 ×2000 = 20
Therefore, the expected number of false acceptances per day is 20.
b) To calculate the expected number of false rejections per day, we need to consider the FRR
and the total number of attempted accesses by the 1000 employees.
Similar to the FAR, the FRR represents the probability of the system incorrectly rejecting a valid
fingerprint as a non-match. Since FRR = 0.05, the probability of a false rejection is 0.05.
Let pF R be the probability of false rejection, then:
pF R = 0.05
The expected number of false rejections per day can be calculated as:
Expected number of false rejections per day =pF R ×2000
Expected number of false rejections per day = 0.05 ×2000 = 100
Therefore, the expected number of false rejections per day is 100.
c) The total expected number of authentication errors per day is the sum of false acceptances
and false rejections. Therefore:
Total expected number of authentication errors per day =Number of false acceptances+Number of false rejections
Total expected number of authentication errors per day = 20 + 100 = 120
Thus, the total expected number of authentication errors per day is 120.
6 6. BIOMETRIC SPOOFING TECHNIQUES AND THE NEED FOR ANTI-SPOOFING MEA-
SURES
Problem 6. A biometric security system uses fingerprint recognition for authentication. In
testing the system, it is found that the False Acceptance Rate (FAR) is 0.1% and the False Rejection
Rate (FRR) is 1%. Calculate the Equal Error Rate (EER) for this system.
Solution 6. The Equal Error Rate (EER) is the point at which the False Acceptance Rate (FAR)
is equal to the False Rejection Rate (FRR).
Given FAR = 0.1% and FRR = 1%, we need to find the EER.
Let the threshold be denoted as T. The EER occurs when FAR = FRR.
From the definitions:
FAR =P(reject |genuine)
FRR =P(accept |imposter)
Therefore, the EER is the average of FAR and FRR.
EER =FAR +FRR
2
EER =0.1+1
2= 0.55%
So, the Equal Error Rate (EER) for this system is 0.55%.
By analyzing the EER, we can see how well the system balances between rejecting valid users
and accepting imposters.
7 Biometric Security and Identity Management
Problem: Biometric systems often use multiple modalities to enhance security. Consider a
biometric system that combines fingerprint and iris recognition technologies. The false acceptance
rates for fingerprint and iris recognition are 0.1% and 0.01%, respectively.
a) What is the probability of falsely accepting an impostor using only fingerprint recognition?
b) What is the probability of falsely accepting an impostor using only iris recognition?
c) If the system requires both modalities to match in order to authenticate, what is the overall
false acceptance rate of the system?
Solution:
a) The probability of falsely accepting an impostor using only fingerprint recognition is given by
the false acceptance rate of the fingerprint modality, which is 0.1% or 0.001.
b) The probability of falsely accepting an impostor using only iris recognition is given by the
false acceptance rate of the iris modality, which is 0.01% or 0.0001.
c) Since the system requires both modalities to match for authentication, the overall false ac-
ceptance rate of the system can be calculated by multiplying the false acceptance rates of each
modality:
Overall False Acceptance Rate =False Acceptance Rate of Fingerprint×False Acceptance Rate of Iris
= 0.001 ×0.0001 = 0.0000001
Therefore, the overall false acceptance rate of the system when using both fingerprint and iris
recognition modalities is 0.00001%.
8 8. AUTHENTICATION FAILURES DUE TO AGING OR CHANGING BIOMETRIC CHARAC-
TERISTICS
Problem 8. In a biometric security system, the false acceptance rate (FAR) is measured to
be 0.02 and the false rejection rate (FRR) is 0.03. Calculate the Equal Error Rate (EER) for this
system.
Solution 8. The Equal Error Rate (EER) is the point where the false acceptance rate (FAR)
and the false rejection rate (FRR) are equal. Mathematically, EER is given by the average of FAR
and FRR when they are equal:
EER = (FAR + FRR) / 2
Given: FAR = 0.02 FRR = 0.03
Substitute the values into the formula: EER = (0.02 + 0.03) / 2 EER = 0.05 / 2 EER = 0.025
Therefore, the Equal Error Rate (EER) for this biometric security system is 0.025.
9 9. PROBLEMS WITH BIOMETRIC MATCHING ALGORITHMS AND ACCURACY RATES
Problem 9.
A security system that uses fingerprint recognition has an accuracy rate of 99.5%. If a random
sample of 500 fingerprints are scanned, calculate the probability that:
a) Exactly 3 fingerprints are incorrectly recognized.
b) At most 5 fingerprints are incorrectly recognized.
c) More than 10 fingerprints are incorrectly recognized.
Solution 9.
a) To find the probability that exactly 3 fingerprints are incorrectly recognized, we can use the
binomial probability formula:
P(X=k) = n
k·pk·(1 −p)n−k
where n= 500 (number of fingerprint scans), k= 3 (number of incorrectly recognized finger-
prints), and p= 1 −0.995 = 0.005 (probability of incorrectly recognizing a fingerprint).
P(X= 3) = 500
3·0.0053·0.995497
Calculating this probability gives us the answer for part a.
b) To find the probability that at most 5 fingerprints are incorrectly recognized, we need to
calculate
P(X≤5) = P(X= 0) + P(X= 1) + P(X= 2) + P(X= 3) + P(X= 4) + P(X= 5)
Using the binomial probability formula for each value of k, we can add up the probabilities to
find the answer for part b.
c) To find the probability that more than 10 fingerprints are incorrectly recognized, we can cal-
culate the complement of the probability that at most 10 are incorrectly recognized:
P(X > 10) = 1 −P(X≤10)
Using the binomial probability formula as in part b, we can find P(X≤10) and subtract it from
1 to get the answer for part c.
10 10. PRIVACY CONCERNS AND DATA PROTECTION IN BIOMETRIC DATABASES
Problem 10. A biometric system has an FAR (False Acceptance Rate) of 0.1% and an FRR
(False Rejection Rate) of 5%. If an individual tries to authenticate using this system 50 times, what
is the probability that they will be falsely rejected at least once?
Solution 10. Given that the FRR (False Rejection Rate) is 5%, the probability of a successful
authentication (1 - FRR) for one attempt is 95% or 0.95. Let pbe the probability of being falsely
rejected at least once in 50 attempts.
The probability of being falsely rejected at least once in ntrials is given by 1−(1 −FRR)n.
Therefore, the probability of being falsely rejected at least once in 50 attempts is:
p= 1 −(1 −0.05)50 = 1 −0.9550 ≈0.9231
Therefore, the probability that the individual will be falsely rejected at least once when attempt-
ing to authenticate 50 times using this biometric system is approximately 92.31%.
I. Problem: Biometric System Performance Degradation Over Time
An iris recognition system has an initial false acceptance rate (FAR) of 0.01
Solution
Given: Initial FAR = 0.01Initial FRR = 0.05Degraded FAR = 0.03Degraded FRR = 0.08
EER is the point where FAR = FRR.
Initial system: EER = (Initial FAR + Initial FRR) / 2 EER = (0.0001 + 0.0005) / 2 EER = 0.0003
Degraded system: EER = (Degraded FAR + Degraded FRR) / 2 EER = (0.0003 + 0.0008) / 2
EER = 0.00055
Therefore, the EER for the initial system is 0.0003 and for the degraded system is 0.00055.
11 12. ISSUES WITH BIOMETRIC IDENTIFICATION IN LARGE-SCALE SYSTEMS
Problem 12. In a large-scale biometric identification system, the false acceptance rate (FAR) is
0.1% and the false rejection rate (FRR) is 5%. If 1000 individuals attempt to authenticate themselves
using the system, calculate:
a) The number of individuals falsely accepted by the system.
b) The number of individuals falsely rejected by the system.
c) The overall accuracy of the system.
Solution 12.
a) The false acceptance rate (FAR) is 0.1%, which means 0.001 or 1
1000 of attempts are falsely
accepted. For 1000 individuals, the number falsely accepted can be calculated as:
False Accepted = 1000 ×0.001 = 1 individual
Therefore, the system falsely accepts 1individual.
b) The false rejection rate (FRR) is 5%, which means 5% or 0.05 of attempts are falsely rejected.
For 1000 individuals, the number falsely rejected can be calculated as:
False Rejected = 1000 ×0.05 = 50 individuals
Therefore, the system falsely rejects 50 individuals.
c) The overall accuracy of the system can be calculated as:
Accuracy = (1 −FAR)×100% = (1 −0.001) ×100% = 99.9%
Thus, the overall accuracy of the system is 99.9%.
12 13. CHALLENGES IN SECURING BIOMETRIC DATA DURING TRANSMISSION AND
STORAGE
Problem 13. Biometric data of a user is being transmitted over a network for authentication
purposes. The biometric data consists of a fingerprint image of size 500 ×500 pixels, where each
pixel is represented by an 8-bit grayscale value. During transmission, encryption is applied using
a symmetric key algorithm that increases the size of the data by 20%. If the encryption key used
is 128 bits long, calculate the size of the encrypted biometric data in kilobytes.
Solution 13. a) The size of the original fingerprint image in bytes can be calculated as:
Image size =Width ×Height ×Bit depth
Since each pixel is represented by an 8-bit grayscale value, the bit depth is 8. Given the image
size is 500 ×500 pixels:
Image size = 500 ×500 ×8bits
b) To convert bits to bytes, we divide by 8:
Image size in bytes =500 ×500 ×8
8bytes
c) After encryption, the size of the data increases by 20%. The new size of the encrypted data
can be calculated as:
New encrypted data size = 1.2×Original data size (in bytes)
d) Given that the encryption key is 128 bits long, to convert it to bytes, we divide by 8:
Key size =128
8bytes
e) The total size of the encrypted biometric data is the sum of the encrypted data size and the
key size:
Total size =New encrypted data size +Key size
Now, substitute the values calculated in parts a, b, c, and d to find the total size of the encrypted
biometric data in kilobytes.
13 14. IMPLICATIONS OF BIOMETRIC DATA BREACHES AND IDENTITY THEFT
Problem 14. A company’s biometric security system has been breached, and the fingerprint
data of 1000 employees has been stolen. If each fingerprint template contains 500 bytes of data and
the company offers a reward of 500percompromisedfingerprinttemplate, calculatethetotalrewardamountthecompanywillneedtopaytoaf f ectedemployees.
Solution 14. Given that: - Number of compromised fingerprint templates = 1000 - Size of each
fingerprint template = 500 bytes - Reward offered per compromised template = 500
Step 1: Calculate the total amount of data compromised Total amount of data compromised
= Number of compromised templates ×Size of each template Total amount of data compromised
= 1000 ×500 bytes Total amount of data compromised = 500,000 bytes
Step 2: Calculate the total reward amount Total reward amount = Number of compromised
templates ×Reward offered per template Total reward amount = 1000 ×500T otalrewardamount =500,000
Therefore, the company will need to pay a total reward amount of 500,000totheaffectedemployees.
14 15. CULTURAL BIASES AND LIMITATIONS IN BIOMETRIC RECOGNITION SYSTEMS
Problem 15. A facial recognition system has been trained on a dataset composed mainly of
individuals of Asian descent. When tested on a diverse population that includes individuals of
African descent, the system shows a significantly higher error rate for this subgroup.
The error rates for the system on the test set are as follows:
•Overall error rate: 5
•Error rate for individuals of Asian descent: 3
•Error rate for individuals of African descent: 12
a) What is the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent?
b) If the system correctly identifies 90
c) Considering these results, discuss the potential impact of cultural biases on the effectiveness
of biometric recognition systems.
Solution 15.
a) The odds ratio of misidentification for individuals of African descent compared to individuals
of Asian descent can be calculated using the error rates:
Odds ratio =Error rate for African descent
Error rate for Asian descent =0.12
0.03 = 4
Therefore, the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent is 4.
b) Given that the system correctly identifies 90
Accuracy rate for African descent = 1 −Error rate for African descent = 1 −0.12 = 0.88 = 88%
Therefore, the accuracy rate for individuals of African descent is 88
c) The significantly higher error rate for individuals of African descent in the facial recognition
system compared to individuals of Asian descent highlights a bias that may have been introduced
during the training phase, where the dataset was skewed towards a specific ethnicity. This cultural
bias can lead to inequitable outcomes, with certain demographic groups facing higher risks of
misidentification and potential discrimination. Addressing cultural biases in biometric recognition
systems is crucial to ensure fair and accurate identification across diverse populations.
15 16. LEGAL AND ETHICAL IMPLICATIONS OF BIOMETRIC DATA COLLECTION AND
USE
Problem 16. A company is implementing a biometric authentication system for its employees
using fingerprint recognition. The system is expected to process 1000 fingerprint scans per day. If
each fingerprint scan produces a file size of 500 KB, calculate the total amount of data collected
by the system in one month.
Solution 16. Given that the system processes 1000 fingerprint scans per day and each scan
produces a file size of 500 KB, we can calculate the total data collected by the system in one day
as follows:
Total data collected per day = Number of scans per day x File size per scan Total data collected
per day = 1000 x 500 KB Total data collected per day = 500,000 KB
To convert KB to GB, we use the conversion factor: 1 GB = 1024 KB
Total data collected per day in GB = 500,000 KB / 1024 = 488.28 GB
Since there are typically 30 days in a month, we can calculate the total amount of data collected
by the system in one month as follows:
Total data collected per month = Total data collected per day x Number of days in a month Total
data collected per month = 488.28 GB x 30 Total data collected per month = 14,648.4 GB
Therefore, the total amount of data collected by the system in one month is 14,648.4 GB.
16 17. CHALLENGES IN IMPLEMENTING BIOMETRIC AUTHENTICATION IN IOT DEVICES
Problem 17. A company is considering implementing biometric authentication for their IoT
devices to enhance security. They estimate that the false acceptance rate (FAR) of the biometric
system is 0.1% and the false rejection rate (FRR) is 1%. If an unauthorized user attempts to access
the device 50 times, calculate the probability that the system will grant access to this unauthorized
user at least once.
Solution 17. To calculate the probability of the system granting access to the unauthorized
user at least once, we need to consider the probability of false acceptance for each attempt. Let’s
denote the probability of a false acceptance as P(F A).
Given that FAR is 0.1%, we have:
P(F A) = 0.001
Therefore, the probability of rejecting a false attempt is:
P(¬F A) = 1 −P(F A) = 1 −0.001 = 0.999
The probability that the system will grant access to the unauthorized user at least once in 50
attempts is the complement of the probability that it will reject all 50 attempts. Thus, the probability
is:
P(Access granted at least once)=1−P(Access not granted at all)
Since each attempt is independent, the probability of the system not granting access to the
unauthorized user in a single attempt is P(¬F A). Therefore, the probability that the system will not
grant access in 50 attempts is:
P(Access not granted at all)=(P(¬F A))n= 0.99950
Calculating this probability:
P(Access not granted at all)=0.99950 ≈0.950 = 95%
So, the probability that the system will grant access to the unauthorized user at least once is
approximately:
P(Access granted at least once)=1−0.950 = 0.050 = 5%
Therefore, the system has a 5% chance of granting access to the unauthorized user at least
once out of 50 attempts.
17 18. USER ACCEPTANCE AND TRUST ISSUES WITH BIOMETRIC SECURITY MEASURES
Problem 18.
A study is conducted to analyze the user acceptance of a new biometric authentication system.
The study involved 100 participants, and the results showed that 65 participants were willing to use
the biometric system, while the remaining participants expressed concerns about privacy and did
not want to adopt it.
a) Calculate the user acceptance rate in percentage.
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same, how many participants would be willing to use the biometric system?
c) Discuss why user acceptance and trust are essential factors in the implementation of bio-
metric security measures.
Solution 18.
a) The user acceptance rate can be calculated as follows:
User acceptance rate =Number of participants willing to use the system
Total number of participants ×100%
In this case, the number of participants willing to use the system is 65, and the total number of
participants is 100. Therefore,
User acceptance rate =65
100 ×100% = 65%
So, the user acceptance rate is 65
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same at 65
Number of participants willing to use the system =Acceptance rate×Total number of participants
Number of participants willing to use the system = 0.65 ×150 = 97.5
Since we cannot have a fraction of a participant, we can estimate that approximately 98 partic-
ipants would be willing to use the biometric system.
c) User acceptance and trust are crucial factors in the implementation of biometric security
measures because:
1. If users do not trust the system, they might not use it properly or try to bypass it, leading to
security vulnerabilities. 2. Acceptance by users is essential for the success of the system, as low
adoption rates can render the biometric security measures ineffective. 3. Ensuring user acceptance
and trust can help improve the overall user experience and satisfaction with the security system. 4.
Addressing privacy concerns and building trust can help mitigate potential legal and ethical issues
related to biometric data collection and storage.
18 19. CHALLENGES IN SECURELY SHARING BIOMETRIC DATA ACROSS DIFFERENT
PLATFORMS
Problem 19. Consider a scenario where a company wants to securely share biometric data
(such as fingerprint scans) across different platforms for employee authentication. However, due to
security concerns, the company decides to implement a secure encryption scheme before sharing
the data.
The encryption scheme they choose involves using a symmetric key system where each pixel
value of the fingerprint image is replaced by the result of applying a modular operation. If the
original pixel value is p= 125 and the symmetric key is k= 7, determine the encrypted pixel value.
Solution 19.
To determine the encrypted pixel value, we need to apply the modular operation using the
symmetric key. The encryption process is represented by the formula:
Encrypted pixel value = (p+k) mod 256
Given p= 125 and k= 7, we can substitute these values into the formula:
Encrypted pixel value = (125 + 7) mod 256
Encrypted pixel value = 132 mod 256
Encrypted pixel value = 132 −256 ×132
256
Encrypted pixel value = 132 −256 ×0
Encrypted pixel value = 132
Therefore, the encrypted pixel value is 132.
19 20. BIOMETRIC IDENTIFICATION ERRORS AND THEIR IMPACT ON SECURITY MEA-
SURES
Problem 20. A biometric system is being evaluated for its performance in identifying individuals
based on facial recognition. In a test dataset of 500 individuals, the system produced the following
results:
- True positives: 450 individuals correctly identified - False positives: 20 individuals incorrectly
identified - False negatives: 10 individuals not identified
Calculate the following metrics for the system:
a) The accuracy of the system (in percentage).
b) The false acceptance rate (FAR) of the system (in percentage).
c) The false rejection rate (FRR) of the system (in percentage).
Solution 20.
a) The accuracy of the system is given by the formula:
Accuracy =True Positives +True Negatives
Total Population =450
500 = 0.9
Converting this to a percentage:
Accuracy = 0.9×100% = 90%
Therefore, the accuracy of the system is 90
b) The false acceptance rate (FAR) of the system is calculated as:
FAR =False Positives
False Positives + True Negatives =20
20 + 500 −450 =20
70 = 0.2857
Converting this to a percentage:
FAR = 0.2857 ×100% = 28.57%
Therefore, the false acceptance rate (FAR) of the system is 28.57
c) The false rejection rate (FRR) of the system is calculated as:
FRR =False Negatives
False Negatives + True Positives =10
10 + 450 =10
460 ≈0.0217
Converting this to a percentage:
FRR = 0.0217 ×100% ≈2.17%
Therefore, the false rejection rate (FRR) of the system is approximately 2.17
Solution 2. Let’s calculate the probability of none of the 4 attempts being successful, and then
subtract this from 1 to find the probability that at least one attempt is successful.
Let pbe the probability of success (tricking the system) on a single attempt, which is 10% or
0.1.
The probability of failure on a single attempt is 1−p= 1 −0.1=0.9.
The probability of none of the 4 attempts being successful is:
(0.9)4= 0.6561
Therefore, the probability of at least one attempt being successful is:
1−0.6561 = 0.3439
So, the probability that at least one of the 4 attempts will be successful is approximately 34.39%.
3 3. ISSUES WITH IRIS AUTHENTICATION IN LOW LIGHTING CONDITIONS
Problem 3. The iris recognition system at a high-security facility struggled to authenticate
individuals in low lighting conditions due to excessive noise in the captured images. One of the
authentication methods used is based on the comparison of feature vectors extracted from the iris
image.
Given two feature vectors X= [1.2,3.4,−2.1,0.9] and Y= [0.8,3.9,−2.3,1.1], calculate the
Euclidean distance between these feature vectors as a measure of similarity.
Solution 3.
1. Let’s calculate the Euclidean distance between the two feature vectors Xand Y, given by:
euclidean distance =v
u
u
t
n
X
i=1
(Xi−Yi)2
where Xiand Yiare the elements of the feature vectors Xand Yrespectively.
2. Plugging in the values for Xand Y:
euclidean distance =p(1.2−0.8)2+ (3.4−3.9)2+ (−2.1+2.3)2+ (0.9−1.1)2
euclidean distance =√0.16 + 0.25 + 0.04 + 0.04
euclidean distance =√0.49 = 0.7
Hence, the Euclidean distance between the feature vectors Xand Yis 0.7 units.
4 Biometric Security and Identity Management
Problem: Consider a biometric system that uses fingerprint recognition for authentication. The
system has an equal error rate of 0.05. If there are 1000 users attempting to authenticate using this
system, calculate the number of users that will be incorrectly authenticated due to false rejection
and false acceptance.
Solution: Given the equal error rate (EER) is 0.05, this means that the false rejection rate
(FRR) is equal to the false acceptance rate (FAR) at the threshold where they intersect.
Let Nbe the total number of users attempting to authenticate, which is 1000.
1. Finding the number of users incorrectly rejected:
The false rejection rate (FRR) is 0.05. This means that 5
Number of users incorrectly rejected = FRR ×N Number of users incorrectly rejected = 0.05 ×
1000 Number of users incorrectly rejected = 50 users
So, 50 users will be incorrectly rejected.
2. Finding the number of users incorrectly accepted:
The false acceptance rate (FAR) is also 0.05. This means that 5
Number of impostors incorrectly accepted = FAR ×N Number of impostors incorrectly accepted
= 0.05 ×1000 Number of impostors incorrectly accepted = 50 users
Therefore, 50 impostors will be incorrectly accepted during authentication process.
5 5. LACK OF STANDARDIZATION IN BIOMETRIC DATA STORAGE AND ENCRYPTION
Problem 5. Many biometric systems lack standardization in storing and encrypting biometric
data. As a result, a company is considering implementing a biometric access system for their
employees which uses fingerprint recognition technology. The system saves the fingerprint data in
an encrypted format using a unique encryption algorithm. The system has a false acceptance rate
(FAR) of 0.01 and a false rejection rate (FRR) of 0.05.
If the company has 1000 employees and each employee tries to access the system twice a day
on average, calculate the following:
a) The expected number of false acceptances per day.
b) The expected number of false rejections per day.
c) The total expected number of authentication errors per day.
Solution 5.
a) To calculate the expected number of false acceptances per day, we need to consider the
FAR and the total number of attempted accesses by the 1000 employees.
The FAR represents the probability of the system incorrectly accepting a false fingerprint as a
match. Since FAR = 0.01, the probability of a false acceptance is 0.01.
Let pF A be the probability of false acceptance, then:
pF A = 0.01
The company has 1000 employees trying to access the system twice a day on average, which
results in 2000 attempted accesses per day. Therefore, the expected number of false acceptances
per day can be calculated as:
Expected number of false acceptances per day =pF A ×2000
Expected number of false acceptances per day = 0.01 ×2000 = 20
Therefore, the expected number of false acceptances per day is 20.
b) To calculate the expected number of false rejections per day, we need to consider the FRR
and the total number of attempted accesses by the 1000 employees.
Similar to the FAR, the FRR represents the probability of the system incorrectly rejecting a valid
fingerprint as a non-match. Since FRR = 0.05, the probability of a false rejection is 0.05.
Let pF R be the probability of false rejection, then:
pF R = 0.05
The expected number of false rejections per day can be calculated as:
Expected number of false rejections per day =pF R ×2000
Expected number of false rejections per day = 0.05 ×2000 = 100
Therefore, the expected number of false rejections per day is 100.
c) The total expected number of authentication errors per day is the sum of false acceptances
and false rejections. Therefore:
Total expected number of authentication errors per day =Number of false acceptances+Number of false rejections
Total expected number of authentication errors per day = 20 + 100 = 120
Thus, the total expected number of authentication errors per day is 120.
6 6. BIOMETRIC SPOOFING TECHNIQUES AND THE NEED FOR ANTI-SPOOFING MEA-
SURES
Problem 6. A biometric security system uses fingerprint recognition for authentication. In
testing the system, it is found that the False Acceptance Rate (FAR) is 0.1% and the False Rejection
Rate (FRR) is 1%. Calculate the Equal Error Rate (EER) for this system.
Solution 6. The Equal Error Rate (EER) is the point at which the False Acceptance Rate (FAR)
is equal to the False Rejection Rate (FRR).
Given FAR = 0.1% and FRR = 1%, we need to find the EER.
Let the threshold be denoted as T. The EER occurs when FAR = FRR.
From the definitions:
FAR =P(reject |genuine)
FRR =P(accept |imposter)
Therefore, the EER is the average of FAR and FRR.
EER =FAR +FRR
2
EER =0.1+1
2= 0.55%
So, the Equal Error Rate (EER) for this system is 0.55%.
By analyzing the EER, we can see how well the system balances between rejecting valid users
and accepting imposters.
7 Biometric Security and Identity Management
Problem: Biometric systems often use multiple modalities to enhance security. Consider a
biometric system that combines fingerprint and iris recognition technologies. The false acceptance
rates for fingerprint and iris recognition are 0.1% and 0.01%, respectively.
a) What is the probability of falsely accepting an impostor using only fingerprint recognition?
b) What is the probability of falsely accepting an impostor using only iris recognition?
c) If the system requires both modalities to match in order to authenticate, what is the overall
false acceptance rate of the system?
Solution:
a) The probability of falsely accepting an impostor using only fingerprint recognition is given by
the false acceptance rate of the fingerprint modality, which is 0.1% or 0.001.
b) The probability of falsely accepting an impostor using only iris recognition is given by the
false acceptance rate of the iris modality, which is 0.01% or 0.0001.
c) Since the system requires both modalities to match for authentication, the overall false ac-
ceptance rate of the system can be calculated by multiplying the false acceptance rates of each
modality:
Overall False Acceptance Rate =False Acceptance Rate of Fingerprint×False Acceptance Rate of Iris
= 0.001 ×0.0001 = 0.0000001
Therefore, the overall false acceptance rate of the system when using both fingerprint and iris
recognition modalities is 0.00001%.
8 8. AUTHENTICATION FAILURES DUE TO AGING OR CHANGING BIOMETRIC CHARAC-
TERISTICS
Problem 8. In a biometric security system, the false acceptance rate (FAR) is measured to
be 0.02 and the false rejection rate (FRR) is 0.03. Calculate the Equal Error Rate (EER) for this
system.
Solution 8. The Equal Error Rate (EER) is the point where the false acceptance rate (FAR)
and the false rejection rate (FRR) are equal. Mathematically, EER is given by the average of FAR
and FRR when they are equal:
EER = (FAR + FRR) / 2
Given: FAR = 0.02 FRR = 0.03
Substitute the values into the formula: EER = (0.02 + 0.03) / 2 EER = 0.05 / 2 EER = 0.025
Therefore, the Equal Error Rate (EER) for this biometric security system is 0.025.
9 9. PROBLEMS WITH BIOMETRIC MATCHING ALGORITHMS AND ACCURACY RATES
Problem 9.
A security system that uses fingerprint recognition has an accuracy rate of 99.5%. If a random
sample of 500 fingerprints are scanned, calculate the probability that:
a) Exactly 3 fingerprints are incorrectly recognized.
b) At most 5 fingerprints are incorrectly recognized.
c) More than 10 fingerprints are incorrectly recognized.
Solution 9.
a) To find the probability that exactly 3 fingerprints are incorrectly recognized, we can use the
binomial probability formula:
P(X=k) = n
k·pk·(1 −p)n−k
where n= 500 (number of fingerprint scans), k= 3 (number of incorrectly recognized finger-
prints), and p= 1 −0.995 = 0.005 (probability of incorrectly recognizing a fingerprint).
P(X= 3) = 500
3·0.0053·0.995497
Calculating this probability gives us the answer for part a.
b) To find the probability that at most 5 fingerprints are incorrectly recognized, we need to
calculate
P(X≤5) = P(X= 0) + P(X= 1) + P(X= 2) + P(X= 3) + P(X= 4) + P(X= 5)
Using the binomial probability formula for each value of k, we can add up the probabilities to
find the answer for part b.
c) To find the probability that more than 10 fingerprints are incorrectly recognized, we can cal-
culate the complement of the probability that at most 10 are incorrectly recognized:
P(X > 10) = 1 −P(X≤10)
Using the binomial probability formula as in part b, we can find P(X≤10) and subtract it from
1 to get the answer for part c.
10 10. PRIVACY CONCERNS AND DATA PROTECTION IN BIOMETRIC DATABASES
Problem 10. A biometric system has an FAR (False Acceptance Rate) of 0.1% and an FRR
(False Rejection Rate) of 5%. If an individual tries to authenticate using this system 50 times, what
is the probability that they will be falsely rejected at least once?
Solution 10. Given that the FRR (False Rejection Rate) is 5%, the probability of a successful
authentication (1 - FRR) for one attempt is 95% or 0.95. Let pbe the probability of being falsely
rejected at least once in 50 attempts.
The probability of being falsely rejected at least once in ntrials is given by 1−(1 −FRR)n.
Therefore, the probability of being falsely rejected at least once in 50 attempts is:
p= 1 −(1 −0.05)50 = 1 −0.9550 ≈0.9231
Therefore, the probability that the individual will be falsely rejected at least once when attempt-
ing to authenticate 50 times using this biometric system is approximately 92.31%.
I. Problem: Biometric System Performance Degradation Over Time
An iris recognition system has an initial false acceptance rate (FAR) of 0.01
Solution
Given: Initial FAR = 0.01Initial FRR = 0.05Degraded FAR = 0.03Degraded FRR = 0.08
EER is the point where FAR = FRR.
Initial system: EER = (Initial FAR + Initial FRR) / 2 EER = (0.0001 + 0.0005) / 2 EER = 0.0003
Degraded system: EER = (Degraded FAR + Degraded FRR) / 2 EER = (0.0003 + 0.0008) / 2
EER = 0.00055
Therefore, the EER for the initial system is 0.0003 and for the degraded system is 0.00055.
11 12. ISSUES WITH BIOMETRIC IDENTIFICATION IN LARGE-SCALE SYSTEMS
Problem 12. In a large-scale biometric identification system, the false acceptance rate (FAR) is
0.1% and the false rejection rate (FRR) is 5%. If 1000 individuals attempt to authenticate themselves
using the system, calculate:
a) The number of individuals falsely accepted by the system.
b) The number of individuals falsely rejected by the system.
c) The overall accuracy of the system.
Solution 12.
a) The false acceptance rate (FAR) is 0.1%, which means 0.001 or 1
1000 of attempts are falsely
accepted. For 1000 individuals, the number falsely accepted can be calculated as:
False Accepted = 1000 ×0.001 = 1 individual
Therefore, the system falsely accepts 1individual.
b) The false rejection rate (FRR) is 5%, which means 5% or 0.05 of attempts are falsely rejected.
For 1000 individuals, the number falsely rejected can be calculated as:
False Rejected = 1000 ×0.05 = 50 individuals
Therefore, the system falsely rejects 50 individuals.
c) The overall accuracy of the system can be calculated as:
Accuracy = (1 −FAR)×100% = (1 −0.001) ×100% = 99.9%
Thus, the overall accuracy of the system is 99.9%.
12 13. CHALLENGES IN SECURING BIOMETRIC DATA DURING TRANSMISSION AND
STORAGE
Problem 13. Biometric data of a user is being transmitted over a network for authentication
purposes. The biometric data consists of a fingerprint image of size 500 ×500 pixels, where each
pixel is represented by an 8-bit grayscale value. During transmission, encryption is applied using
a symmetric key algorithm that increases the size of the data by 20%. If the encryption key used
is 128 bits long, calculate the size of the encrypted biometric data in kilobytes.
Solution 13. a) The size of the original fingerprint image in bytes can be calculated as:
Image size =Width ×Height ×Bit depth
Since each pixel is represented by an 8-bit grayscale value, the bit depth is 8. Given the image
size is 500 ×500 pixels:
Image size = 500 ×500 ×8bits
b) To convert bits to bytes, we divide by 8:
Image size in bytes =500 ×500 ×8
8bytes
c) After encryption, the size of the data increases by 20%. The new size of the encrypted data
can be calculated as:
New encrypted data size = 1.2×Original data size (in bytes)
d) Given that the encryption key is 128 bits long, to convert it to bytes, we divide by 8:
Key size =128
8bytes
e) The total size of the encrypted biometric data is the sum of the encrypted data size and the
key size:
Total size =New encrypted data size +Key size
Now, substitute the values calculated in parts a, b, c, and d to find the total size of the encrypted
biometric data in kilobytes.
13 14. IMPLICATIONS OF BIOMETRIC DATA BREACHES AND IDENTITY THEFT
Problem 14. A company’s biometric security system has been breached, and the fingerprint
data of 1000 employees has been stolen. If each fingerprint template contains 500 bytes of data and
the company offers a reward of 500percompromisedfingerprinttemplate, calculatethetotalrewardamountthecompanywillneedtopaytoaf f ectedemployees.
Solution 14. Given that: - Number of compromised fingerprint templates = 1000 - Size of each
fingerprint template = 500 bytes - Reward offered per compromised template = 500
Step 1: Calculate the total amount of data compromised Total amount of data compromised
= Number of compromised templates ×Size of each template Total amount of data compromised
= 1000 ×500 bytes Total amount of data compromised = 500,000 bytes
Step 2: Calculate the total reward amount Total reward amount = Number of compromised
templates ×Reward offered per template Total reward amount = 1000 ×500T otalrewardamount =500,000
Therefore, the company will need to pay a total reward amount of 500,000totheaffectedemployees.
14 15. CULTURAL BIASES AND LIMITATIONS IN BIOMETRIC RECOGNITION SYSTEMS
Problem 15. A facial recognition system has been trained on a dataset composed mainly of
individuals of Asian descent. When tested on a diverse population that includes individuals of
African descent, the system shows a significantly higher error rate for this subgroup.
The error rates for the system on the test set are as follows:
•Overall error rate: 5
•Error rate for individuals of Asian descent: 3
•Error rate for individuals of African descent: 12
a) What is the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent?
b) If the system correctly identifies 90
c) Considering these results, discuss the potential impact of cultural biases on the effectiveness
of biometric recognition systems.
Solution 15.
a) The odds ratio of misidentification for individuals of African descent compared to individuals
of Asian descent can be calculated using the error rates:
Odds ratio =Error rate for African descent
Error rate for Asian descent =0.12
0.03 = 4
Therefore, the odds ratio of misidentification for individuals of African descent compared to
individuals of Asian descent is 4.
b) Given that the system correctly identifies 90
Accuracy rate for African descent = 1 −Error rate for African descent = 1 −0.12 = 0.88 = 88%
Therefore, the accuracy rate for individuals of African descent is 88
c) The significantly higher error rate for individuals of African descent in the facial recognition
system compared to individuals of Asian descent highlights a bias that may have been introduced
during the training phase, where the dataset was skewed towards a specific ethnicity. This cultural
bias can lead to inequitable outcomes, with certain demographic groups facing higher risks of
misidentification and potential discrimination. Addressing cultural biases in biometric recognition
systems is crucial to ensure fair and accurate identification across diverse populations.
15 16. LEGAL AND ETHICAL IMPLICATIONS OF BIOMETRIC DATA COLLECTION AND
USE
Problem 16. A company is implementing a biometric authentication system for its employees
using fingerprint recognition. The system is expected to process 1000 fingerprint scans per day. If
each fingerprint scan produces a file size of 500 KB, calculate the total amount of data collected
by the system in one month.
Solution 16. Given that the system processes 1000 fingerprint scans per day and each scan
produces a file size of 500 KB, we can calculate the total data collected by the system in one day
as follows:
Total data collected per day = Number of scans per day x File size per scan Total data collected
per day = 1000 x 500 KB Total data collected per day = 500,000 KB
To convert KB to GB, we use the conversion factor: 1 GB = 1024 KB
Total data collected per day in GB = 500,000 KB / 1024 = 488.28 GB
Since there are typically 30 days in a month, we can calculate the total amount of data collected
by the system in one month as follows:
Total data collected per month = Total data collected per day x Number of days in a month Total
data collected per month = 488.28 GB x 30 Total data collected per month = 14,648.4 GB
Therefore, the total amount of data collected by the system in one month is 14,648.4 GB.
16 17. CHALLENGES IN IMPLEMENTING BIOMETRIC AUTHENTICATION IN IOT DEVICES
Problem 17. A company is considering implementing biometric authentication for their IoT
devices to enhance security. They estimate that the false acceptance rate (FAR) of the biometric
system is 0.1% and the false rejection rate (FRR) is 1%. If an unauthorized user attempts to access
the device 50 times, calculate the probability that the system will grant access to this unauthorized
user at least once.
Solution 17. To calculate the probability of the system granting access to the unauthorized
user at least once, we need to consider the probability of false acceptance for each attempt. Let’s
denote the probability of a false acceptance as P(F A).
Given that FAR is 0.1%, we have:
P(F A) = 0.001
Therefore, the probability of rejecting a false attempt is:
P(¬F A) = 1 −P(F A) = 1 −0.001 = 0.999
The probability that the system will grant access to the unauthorized user at least once in 50
attempts is the complement of the probability that it will reject all 50 attempts. Thus, the probability
is:
P(Access granted at least once)=1−P(Access not granted at all)
Since each attempt is independent, the probability of the system not granting access to the
unauthorized user in a single attempt is P(¬F A). Therefore, the probability that the system will not
grant access in 50 attempts is:
P(Access not granted at all)=(P(¬F A))n= 0.99950
Calculating this probability:
P(Access not granted at all)=0.99950 ≈0.950 = 95%
So, the probability that the system will grant access to the unauthorized user at least once is
approximately:
P(Access granted at least once)=1−0.950 = 0.050 = 5%
Therefore, the system has a 5% chance of granting access to the unauthorized user at least
once out of 50 attempts.
17 18. USER ACCEPTANCE AND TRUST ISSUES WITH BIOMETRIC SECURITY MEASURES
Problem 18.
A study is conducted to analyze the user acceptance of a new biometric authentication system.
The study involved 100 participants, and the results showed that 65 participants were willing to use
the biometric system, while the remaining participants expressed concerns about privacy and did
not want to adopt it.
a) Calculate the user acceptance rate in percentage.
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same, how many participants would be willing to use the biometric system?
c) Discuss why user acceptance and trust are essential factors in the implementation of bio-
metric security measures.
Solution 18.
a) The user acceptance rate can be calculated as follows:
User acceptance rate =Number of participants willing to use the system
Total number of participants ×100%
In this case, the number of participants willing to use the system is 65, and the total number of
participants is 100. Therefore,
User acceptance rate =65
100 ×100% = 65%
So, the user acceptance rate is 65
b) If the study was conducted again with 150 participants and the acceptance rate remained
the same at 65
Number of participants willing to use the system =Acceptance rate×Total number of participants
Number of participants willing to use the system = 0.65 ×150 = 97.5
Since we cannot have a fraction of a participant, we can estimate that approximately 98 partic-
ipants would be willing to use the biometric system.
c) User acceptance and trust are crucial factors in the implementation of biometric security
measures because:
1. If users do not trust the system, they might not use it properly or try to bypass it, leading to
security vulnerabilities. 2. Acceptance by users is essential for the success of the system, as low
adoption rates can render the biometric security measures ineffective. 3. Ensuring user acceptance
and trust can help improve the overall user experience and satisfaction with the security system. 4.
Addressing privacy concerns and building trust can help mitigate potential legal and ethical issues
related to biometric data collection and storage.
18 19. CHALLENGES IN SECURELY SHARING BIOMETRIC DATA ACROSS DIFFERENT
PLATFORMS
Problem 19. Consider a scenario where a company wants to securely share biometric data
(such as fingerprint scans) across different platforms for employee authentication. However, due to
security concerns, the company decides to implement a secure encryption scheme before sharing
the data.
The encryption scheme they choose involves using a symmetric key system where each pixel
value of the fingerprint image is replaced by the result of applying a modular operation. If the
original pixel value is p= 125 and the symmetric key is k= 7, determine the encrypted pixel value.
Solution 19.
To determine the encrypted pixel value, we need to apply the modular operation using the
symmetric key. The encryption process is represented by the formula:
Encrypted pixel value = (p+k) mod 256
Given p= 125 and k= 7, we can substitute these values into the formula:
Encrypted pixel value = (125 + 7) mod 256
Encrypted pixel value = 132 mod 256
Encrypted pixel value = 132 −256 ×132
256
Encrypted pixel value = 132 −256 ×0
Encrypted pixel value = 132
Therefore, the encrypted pixel value is 132.
19 20. BIOMETRIC IDENTIFICATION ERRORS AND THEIR IMPACT ON SECURITY MEA-
SURES
Problem 20. A biometric system is being evaluated for its performance in identifying individuals
based on facial recognition. In a test dataset of 500 individuals, the system produced the following
results:
- True positives: 450 individuals correctly identified - False positives: 20 individuals incorrectly
identified - False negatives: 10 individuals not identified
Calculate the following metrics for the system:
a) The accuracy of the system (in percentage).
b) The false acceptance rate (FAR) of the system (in percentage).
c) The false rejection rate (FRR) of the system (in percentage).
Solution 20.
a) The accuracy of the system is given by the formula:
Accuracy =True Positives +True Negatives
Total Population =450
500 = 0.9
Converting this to a percentage:
Accuracy = 0.9×100% = 90%
Therefore, the accuracy of the system is 90
b) The false acceptance rate (FAR) of the system is calculated as:
FAR =False Positives
False Positives + True Negatives =20
20 + 500 −450 =20
70 = 0.2857
Converting this to a percentage:
FAR = 0.2857 ×100% = 28.57%
Therefore, the false acceptance rate (FAR) of the system is 28.57
c) The false rejection rate (FRR) of the system is calculated as:
FRR =False Negatives
False Negatives + True Positives =10
10 + 450 =10
460 ≈0.0217
Converting this to a percentage:
FRR = 0.0217 ×100% ≈2.17%
Therefore, the false rejection rate (FRR) of the system is approximately 2.17
