ANALYSIS OF CRYPTOGRAPHIC ALGORITHMS’ TIME AND SPACE COMPLEXITY.

ANALYSIS OF CRYPTOGRAPHIC ALGORITHMS’ TIME AND

SPACE COMPLEXITY

1 SYMMETRIC ENCRYPTION ALGORITHMS

Problem 1. The Advanced Encryption Standard (AES) operates on 128-bit blocks. If the input

message is n bits long, what is the time complexity of encrypting this message using AES in

ECB mode? Express your answer in big O notation.

Solution 1. Step 1: Determine the number of blocks. Number of blocks = ⌈n/128⌉

Solution 2. Step 1: Calculate the total number of possible keys. Number of possible keys =

Step 2: In the worst case, we might need to try all possible keys. Worst-case number of

operations = 256

Therefore, the time complexity of an exhaustive key search for DES is O(256).

Problem 3. The RC4 stream cipher has an initialization phase where it creates a 256-byte

state array. What is the space complexity of RC4 in terms of the key size k? Express your

answer in big O notation.

Solution 3. Step 1: Analyze the space requirements. - The state array requires 256 bytes

(constant) - The key of size k bytes is stored

Solution 4. Step 1: Determine the range of numbers to check. For an n-bit number, the

maximum value is 2n− 1. We need to check up to √2n− 1, which is approximately 2n/2.

Step 2: Calculate the number of operations. Number of operations ≈ 2n/2

Therefore, the time complexity of the trial division primality test for an n-bit number is O(2n/2).

Problem 5. In the Diffie-Hellman key exchange protocol, both parties perform modular

exponentiation. If the modulus is an n-bit number, what is the time complexity of the modular

exponentiation operation using the square-and-multiply algorithm? Express your answer in big

O notation.

Solution 5. Step 1: Analyze the square-and-multiply algorithm. - The algorithm performs n

steps for an n-bit exponent - Each step involves at most two operations: squaring and

Therefore, the time complexity of modular exponentiation in Diffie-Hellman key exchange is

Problem 6. Elliptic Curve Cryptography (ECC) operates on points on an elliptic curve. If the

field size is q, what is the space complexity of storing a public key in ECC? Express your answer

in big O notation.

Solution 6. Step 1: Analyze the components of an ECC public key. An ECC public key consists

of: - The elliptic curve parameters (assumed to be fixed and not counted) - A point on the

curve, represented by two coordinates (x, y)

Step 2: Determine the size of each coordinate. Each coordinate is an element of the field of size

q, requiring log_2(q) bits to represent.

Solution 7. Step 1: Determine the number of blocks. Number of blocks = ⌈n/512⌉

Step 2: Analyze the time complexity. Each block processing takes constant time. Total time =

O(Number of blocks) = O(⌈n/512⌉) = O(n)

Therefore, the time complexity of computing the SHA-256 hash of an n-bit message is O(n).

Problem 8. In the Merkle-Damgård construction used in many hash functions, the message is

padded to a multiple of the block size. If the original message is n bits long and the block size is

b bits, what is the maximum length of the padded message? Express your answer in terms of n

and b.

Solution 8. Step 1: Analyze the padding scheme. - The message is padded to a multiple of

the block size - At least 1 bit is always added (the ’1’ bit) - The length of the original message

(n) is appended (usually 64 bits)

Solution 1. Step 1: Determine the number of blocks. Number of blocks = ⌈n/128⌉

Step 2: Analyze the time complexity. Each block encryption takes constant time. Total time =

O(Number of blocks) = O(⌈n/128⌉) = O(n)

Therefore, the time complexity of encrypting an n-bit message using AES in ECB mode is O(n).

Problem 2. The Data Encryption Standard (DES) uses a 56-bit key. How many operations

would be required for an exhaustive key search in the worst case? Express your answer in big O

notation.

Solution 2. Step 1: Calculate the total number of possible keys. Number of possible keys =

Step 2: In the worst case, we might need to try all possible keys. Worst-case number of

(constant) - The key of size k bytes is stored

Step 2: Express the total space in big O notation. Space complexity = O(256 + k) = O(k)

Therefore, the space complexity of RC4 is O(k), where k is the key size in bytes.

4 ASYMMETRIC ENCRYPTION ALGORITHMS

Problem 4. In RSA, key generation involves finding two large prime numbers. If we use the

trial division method to test primality up to √n for an n-bit number, what is the time complexity

of this primality test? Express your answer in big O notation.

Solution 4. Step 1: Determine the range of numbers to check. For an n-bit number, the

maximum value is 2n− 1. We need to check up to √2n− 1, which is approximately 2n/2.

Step 2: Calculate the number of operations. Number of operations ≈ 2n/2

Therefore, the time complexity of the trial division primality test for an n-bit number is O(2n/2).

multiplication

Step 2: Determine the complexity of each operation. - Squaring and multiplication of n-bit

numbers take O(n2) time using naive multiplication - Using more efficient algorithms like

Karatsuba, it can be reduced to O(nlog23) ≈ O(n1.585)

Step 3: Calculate the overall time complexity. Time complexity = n * O(nlog23) = O(n1+log23) ≈

Therefore, the time complexity of modular exponentiation in Diffie-Hellman key exchange is

Problem 6. Elliptic Curve Cryptography (ECC) operates on points on an elliptic curve. If the

field size is q, what is the space complexity of storing a public key in ECC? Express your answer

Step 3: Calculate the total space required. Space for public key = 2 * log_2(q) bits

Therefore, the space complexity of storing an ECC public key is O(log q).

5 HASH FUNCTIONS

Problem 7. The SHA-256 hash function processes messages in 512-bit blocks. If the input

message is n bits long, what is the time complexity of computing its SHA-256 hash? Express

your answer in big O notation.

Solution 7. Step 1: Determine the number of blocks. Number of blocks = ⌈n/512⌉

Step 2: Analyze the time complexity. Each block processing takes constant time. Total time =

O(Number of blocks) = O(⌈n/512⌉) = O(n)

Therefore, the time complexity of computing the SHA-256 hash of an n-bit message is O(n).

Problem 8. In the Merkle-Damgård construction used in many hash functions, the message is

padded to a multiple of the block size. If the original message is n bits long and the block size is

This can be simplified to: Maximum padded length = ⌈(n + 65)/b⌉ ∗ b

Therefore, the maximum length of the padded message is ⌈(n + 65)/b⌉ ∗ b bits.

The Advanced Encryption Standard (AES) operates on 128-bit blocks. If the input message is n

bits long, what is the time complexity of encrypting this message using AES in ECB mode?

Express your answer in big O notation.

Solution 1. Step 1: Determine the number of blocks. Number of blocks = ⌈n/128⌉

Step 2: Analyze the time complexity. Each block encryption takes constant time. Total time =

O(Number of blocks) = O(⌈n/128⌉) = O(n)

Therefore, the time complexity of encrypting an n-bit message using AES in ECB mode is O(n).

Problem 2. The Data Encryption Standard (DES) uses a 56-bit key. How many operations

Problem 3. The RC4 stream cipher has an initialization phase where it creates a 256-byte

state array. What is the space complexity of RC4 in terms of the key size k? Express your

in big O notation.

Solution 6. Step 1: Analyze the components of an ECC public key. An ECC public key consists

of: - The elliptic curve parameters (assumed to be fixed and not counted) - A point on the

Problem 2. The Data Encryption Standard (DES) uses a 56-bit key. How many operations

would be required for an exhaustive key search in the worst case? Express your answer in big O

Therefore, the time complexity of an exhaustive key search for DES is O(256).

Solution 7. Step 1: Determine the number of blocks. Number of blocks = ⌈n/512⌉

O(Number of blocks) = O(⌈n/512⌉) = O(n)

Therefore, the time complexity of computing the SHA-256 hash of an n-bit message is O(n).

Problem 8. In the Merkle-Damgård construction used in many hash functions, the message is

padded to a multiple of the block size. If the original message is n bits long and the block size is

b bits, what is the maximum length of the padded message? Express your answer in terms of n

and b.

(n) is appended (usually 64 bits)

Step 2: Calculate the maximum padded length. Maximum padded length = ⌈(n + 1 + 64)/b⌉ ∗ b

This can be simplified to: Maximum padded length = ⌈(n + 65)/b⌉ ∗ b

Therefore, the maximum length of the padded message is ⌈(n + 65)/b⌉ ∗ b bits.

Problem 7. The SHA-256 hash function processes messages in 512-bit blocks. If the input

message is n bits long, what is the time complexity of computing its SHA-256 hash? Express

your answer in big O notation.

Solution 7. Step 1: Determine the number of blocks. Number of blocks = ⌈n/512⌉

Step 2: Analyze the time complexity. Each block processing takes constant time. Total time =

O(Number of blocks) = O(⌈n/512⌉) = O(n)

Therefore, the time complexity of computing the SHA-256 hash of an n-bit message is O(n).

Problem 8. In the Merkle-Damgård construction used in many hash functions, the message is

padded to a multiple of the block size. If the original message is n bits long and the block size is

This can be simplified to: Maximum padded length = ⌈(n + 65)/b⌉ ∗ b

Therefore, the maximum length of the padded message is ⌈(n + 65)/b⌉ ∗ b bits.

The Advanced Encryption Standard (AES) operates on 128-bit blocks. If the input message is n

bits long, what is the time complexity of encrypting this message using AES in ECB mode?

Express your answer in big O notation.

Solution 1. Step 1: Determine the number of blocks. Number of blocks = ⌈n/128⌉

Step 2: Analyze the time complexity. Each block encryption takes constant time. Total time =

O(Number of blocks) = O(⌈n/128⌉) = O(n)

Therefore, the time complexity of encrypting an n-bit message using AES in ECB mode is O(n).

Problem 2. The Data Encryption Standard (DES) uses a 56-bit key. How many operations

Problem 3. The RC4 stream cipher has an initialization phase where it creates a 256-byte

state array. What is the space complexity of RC4 in terms of the key size k? Express your

in big O notation.

Solution 6. Step 1: Analyze the components of an ECC public key. An ECC public key consists

of: - The elliptic curve parameters (assumed to be fixed and not counted) - A point on the

Problem 2. The Data Encryption Standard (DES) uses a 56-bit key. How many operations

would be required for an exhaustive key search in the worst case? Express your answer in big O

Therefore, the time complexity of an exhaustive key search for DES is O(256).

Solution 7. Step 1: Determine the number of blocks. Number of blocks = ⌈n/512⌉

O(Number of blocks) = O(⌈n/512⌉) = O(n)

Therefore, the time complexity of computing the SHA-256 hash of an n-bit message is O(n).

Problem 8. In the Merkle-Damgård construction used in many hash functions, the message is

padded to a multiple of the block size. If the original message is n bits long and the block size is

b bits, what is the maximum length of the padded message? Express your answer in terms of n

and b.

(n) is appended (usually 64 bits)

Step 2: Calculate the maximum padded length. Maximum padded length = ⌈(n + 1 + 64)/b⌉ ∗ b

This can be simplified to: Maximum padded length = ⌈(n + 65)/b⌉ ∗ b

Therefore, the maximum length of the padded message is ⌈(n + 65)/b⌉ ∗ b bits.

The Advanced Encryption Standard (AES) operates on 128-bit blocks. If the input message is n

bits long, what is the time complexity of encrypting this message using AES in ECB mode?

Step 2: Analyze the time complexity. Each block encryption takes constant time. Total time =

O(Number of blocks) = O(⌈n/128⌉) = O(n)

Therefore, the time complexity of encrypting an n-bit message using AES in ECB mode is O(n).

Problem 3. The RC4 stream cipher has an initialization phase where it creates a 256-byte

state array. What is the space complexity of RC4 in terms of the key size k? Express your

answer in big O notation.

Solution 3. Step 1: Analyze the space requirements. - The state array requires 256 bytes

(constant) - The key of size k bytes is stored

Step 2: Express the total space in big O notation. Space complexity = O(256 + k) = O(k)

Therefore, the space complexity of RC4 is O(k), where k is the key size in bytes.

Problem 4. In RSA, key generation involves finding two large prime numbers. If we use the

trial division method to test primality up to √n for an n-bit number, what is the time complexity

of this primality test? Express your answer in big O notation.

Solution 4. Step 1: Determine the range of numbers to check. For an n-bit number, the

exponentiation operation using the square-and-multiply algorithm? Express your answer in big

O notation.

numbers take O(n2) time using naive multiplication - Using more efficient algorithms like

Karatsuba, it can be reduced to O(nlog23) ≈ O(n1.585)

Step 3: Calculate the overall time complexity. Time complexity = n * O(nlog23) = O(n1+log23) ≈

Therefore, the time complexity of modular exponentiation in Diffie-Hellman key exchange is

Problem 6. Elliptic Curve Cryptography (ECC) operates on points on an elliptic curve. If the

field size is q, what is the space complexity of storing a public key in ECC? Express your answer

in big O notation.

Solution 6. Step 1: Analyze the components of an ECC public key. An ECC public key consists

Problem 7. The SHA-256 hash function processes messages in 512-bit blocks. If the input

message is n bits long, what is the time complexity of computing its SHA-256 hash? Express

your answer in big O notation.

Solution 7. Step 1: Determine the number of blocks. Number of blocks = ⌈n/512⌉

Step 2: Analyze the time complexity. Each block processing takes constant time. Total time =

O(Number of blocks) = O(⌈n/512⌉) = O(n)

Therefore, the time complexity of computing the SHA-256 hash of an n-bit message is O(n).

Problem 8. In the Merkle-Damgård construction used in many hash functions, the message is

padded to a multiple of the block size. If the original message is n bits long and the block size is

Solution 8. Step 1: Analyze the padding scheme. - The message is padded to a multiple of

the block size - At least 1 bit is always added (the ’1’ bit) - The length of the original message

This can be simplified to: Maximum padded length = ⌈(n + 65)/b⌉ ∗ b

Solution 1. Step 1: Determine the number of blocks. Number of blocks = ⌈n/128⌉

Step 2: Analyze the time complexity. Each block encryption takes constant time. Total time =

O(Number of blocks) = O(⌈n/128⌉) = O(n)

Therefore, the time complexity of encrypting an n-bit message using AES in ECB mode is O(n).

Problem 2. The Data Encryption Standard (DES) uses a 56-bit key. How many operations

notation.

Solution 2. Step 1: Calculate the total number of possible keys. Number of possible keys =

Step 2: In the worst case, we might need to try all possible keys. Worst-case number of

operations = 256

Therefore, the time complexity of an exhaustive key search for DES is O(256).

Therefore, the space complexity of RC4 is O(k), where k is the key size in bytes.

Problem 4. In RSA, key generation involves finding two large prime numbers. If we use the

trial division method to test primality up to √n for an n-bit number, what is the time complexity

of this primality test? Express your answer in big O notation.

Solution 4. Step 1: Determine the range of numbers to check. For an n-bit number, the

maximum value is 2n− 1. We need to check up to √2n− 1, which is approximately 2n/2.

Step 2: Calculate the number of operations. Number of operations ≈ 2n/2

Therefore, the time complexity of the trial division primality test for an n-bit number is O(2n/2).

Problem 5. In the Diffie-Hellman key exchange protocol, both parties perform modular

exponentiation. If the modulus is an n-bit number, what is the time complexity of the modular

exponentiation operation using the square-and-multiply algorithm? Express your answer in big

O notation.

Solution 5. Step 1: Analyze the square-and-multiply algorithm. - The algorithm performs n

steps for an n-bit exponent - Each step involves at most two operations: squaring and

Therefore, the time complexity of modular exponentiation in Diffie-Hellman key exchange is

Problem 6. Elliptic Curve Cryptography (ECC) operates on points on an elliptic curve. If the

field size is q, what is the space complexity of storing a public key in ECC? Express your answer

in big O notation.

Solution 6. Step 1: Analyze the components of an ECC public key. An ECC public key consists

of: - The elliptic curve parameters (assumed to be fixed and not counted) - A point on the

curve, represented by two coordinates (x, y)

Step 2: Determine the size of each coordinate. Each coordinate is an element of the field of size

q, requiring log_2(q) bits to represent.

Solution 7. Step 1: Determine the number of blocks. Number of blocks = ⌈n/512⌉

Step 2: Analyze the time complexity. Each block processing takes constant time. Total time =

O(Number of blocks) = O(⌈n/512⌉) = O(n)

Therefore, the time complexity of computing the SHA-256 hash of an n-bit message is O(n).

Problem 8. In the Merkle-Damgård construction used in many hash functions, the message is

padded to a multiple of the block size. If the original message is n bits long and the block size is

b bits, what is the maximum length of the padded message? Express your answer in terms of n

and b.

Solution 8. Step 1: Analyze the padding scheme. - The message is padded to a multiple of

the block size - At least 1 bit is always added (the ’1’ bit) - The length of the original message

(n) is appended (usually 64 bits)

Solution 1. Step 1: Determine the number of blocks. Number of blocks = ⌈n/128⌉

Step 2: Analyze the time complexity. Each block encryption takes constant time. Total time =

O(Number of blocks) = O(⌈n/128⌉) = O(n)

Therefore, the time complexity of encrypting an n-bit message using AES in ECB mode is O(n).

Problem 2. The Data Encryption Standard (DES) uses a 56-bit key. How many operations

would be required for an exhaustive key search in the worst case? Express your answer in big O

notation.

Solution 2. Step 1: Calculate the total number of possible keys. Number of possible keys =

Step 2: In the worst case, we might need to try all possible keys. Worst-case number of

(constant) - The key of size k bytes is stored

Step 2: Express the total space in big O notation. Space complexity = O(256 + k) = O(k)

The Advanced Encryption Standard (AES) operates on 128-bit blocks. If the input message is n

bits long, what is the time complexity of encrypting this message using AES in ECB mode?

Step 2: Analyze the time complexity. Each block encryption takes constant time. Total time =

O(Number of blocks) = O(⌈n/128⌉) = O(n)

Therefore, the time complexity of encrypting an n-bit message using AES in ECB mode is O(n).

Problem 3. The RC4 stream cipher has an initialization phase where it creates a 256-byte

state array. What is the space complexity of RC4 in terms of the key size k? Express your

answer in big O notation.

Solution 3. Step 1: Analyze the space requirements. - The state array requires 256 bytes

(constant) - The key of size k bytes is stored

Step 2: Express the total space in big O notation. Space complexity = O(256 + k) = O(k)

Therefore, the space complexity of RC4 is O(k), where k is the key size in bytes.

Problem 4. In RSA, key generation involves finding two large prime numbers. If we use the

trial division method to test primality up to √n for an n-bit number, what is the time complexity

of this primality test? Express your answer in big O notation.

Solution 4. Step 1: Determine the range of numbers to check. For an n-bit number, the

exponentiation operation using the square-and-multiply algorithm? Express your answer in big

O notation.

numbers take O(n2) time using naive multiplication - Using more efficient algorithms like

Karatsuba, it can be reduced to O(nlog23) ≈ O(n1.585)

Step 3: Calculate the overall time complexity. Time complexity = n * O(nlog23) = O(n1+log23) ≈

Therefore, the time complexity of modular exponentiation in Diffie-Hellman key exchange is

Problem 6. Elliptic Curve Cryptography (ECC) operates on points on an elliptic curve. If the

field size is q, what is the space complexity of storing a public key in ECC? Express your answer

in big O notation.

Solution 6. Step 1: Analyze the components of an ECC public key. An ECC public key consists

Problem 7. The SHA-256 hash function processes messages in 512-bit blocks. If the input

message is n bits long, what is the time complexity of computing its SHA-256 hash? Express

your answer in big O notation.

Solution 7. Step 1: Determine the number of blocks. Number of blocks = ⌈n/512⌉

Step 2: Analyze the time complexity. Each block processing takes constant time. Total time =

O(Number of blocks) = O(⌈n/512⌉) = O(n)

Therefore, the time complexity of computing the SHA-256 hash of an n-bit message is O(n).

Problem 8. In the Merkle-Damgård construction used in many hash functions, the message is

padded to a multiple of the block size. If the original message is n bits long and the block size is

Solution 8. Step 1: Analyze the padding scheme. - The message is padded to a multiple of

the block size - At least 1 bit is always added (the ’1’ bit) - The length of the original message

This can be simplified to: Maximum padded length = ⌈(n + 65)/b⌉ ∗ b

Solution 1. Step 1: Determine the number of blocks. Number of blocks = ⌈n/128⌉

Step 2: Analyze the time complexity. Each block encryption takes constant time. Total time =

O(Number of blocks) = O(⌈n/128⌉) = O(n)

Therefore, the time complexity of encrypting an n-bit message using AES in ECB mode is O(n).

Problem 2. The Data Encryption Standard (DES) uses a 56-bit key. How many operations

notation.

Solution 2. Step 1: Calculate the total number of possible keys. Number of possible keys =

Step 2: In the worst case, we might need to try all possible keys. Worst-case number of