Question

(8)Explain how the HILL cipher algorithm works. You may use the plaintext PLAY to and the key SECRET to show its encipherment procedure

Answer 1

1 Let f(a;b) & f(c;d) be two affine ciphers s/t
f(a;b)(x)  (a:x + b) mod 26
f(c;d)(x)  (c:x + d) mod 26
Is f(c;d)  f(a;b) a stronger encryption scheme than f(a;b)?
(10)
Using Affine Cipher Theory and given;a=9 and b=4, Encrpt the word "MAZERAS" and show the resulting cipher text.

Answer 2

What is the key-space of an affine cipher?

Answer 3

The HILL cipher algorithm is a polygraphic substitution cipher that operates on blocks of letters. It uses matrix multiplication to encrypt and decrypt the plaintext. Here's how it works:

1. Key Generation:
- The key, SECRET, is represented as a matrix using numerical equivalents of letters. For example, S=18, E=5, C=3, R=17, T=20.
- The matrix is created by arranging the numerical values of the letters in rows, with the number of columns determined by the length of the key. In this case, the key length is 6, so the matrix will have 2 rows and 3 columns:
```
S E C
R E T
```

2. Encryption:
- The plaintext, PLAY, is represented as a column vector using numerical equivalents of letters. P=16, L=12, A=1, Y=25:
```
P
L
A
Y
```
- The key matrix and plaintext vector are multiplied, resulting in a new column vector that represents the ciphertext:
```
S E C P (S*P + E*L + C*A)
R E T * L = (R*P + E*L + T*A)
(S*P + E*L + C*Y)
```

- Apply modulo 26 to the resulting values so that they fall within the range of the alphabet. For example, if any value is greater than or equal to 26, subtract 26 until it is in the range of 0-25.

- Convert the resulting numerical values back to their corresponding letters. In this case, the ciphertext will be:
```
A Y A
O W L
```

- Therefore, the encrypted ciphertext for the plaintext PLAY using the key SECRET is AYAWOL.

Note: Decryption is done in a similar manner but using the inverse of the key matrix in order to get back the original plaintext.

Answer 4

To determine if f(c;d) is a stronger encryption scheme than f(a;b), we need to consider the properties of both ciphers and their corresponding keys.

The affine cipher f(a;b)(x) encrypts a plaintext character x by multiplying it by a constant a, adding b, and taking the result modulo 26. The resulting ciphertext character is determined by the numerical value of the encrypted character.

Now let's encrypt the word "MAZERAS" using the given key values a=9 and b=4 with the affine cipher f(a;b):

1. Assign numerical values to the plaintext characters:
```
M = 12
A = 0
Z = 25
E = 4
R = 17
A = 0
S = 18
```

2. Apply the f(a;b) function to each character:
```
f(9;4)(12) = (9 * 12 + 4) mod 26 = 2
f(9;4)(0) = (9 * 0 + 4) mod 26 = 4
f(9;4)(25) = (9 * 25 + 4) mod 26 = 14
f(9;4)(4) = (9 * 4 + 4) mod 26 = 18
f(9;4)(17) = (9 * 17 + 4) mod 26 = 3
f(9;4)(0) = (9 * 0 + 4) mod 26 = 4
f(9;4)(18) = (9 * 18 + 4) mod 26 = 8
```

3. Convert the numerical values back to letters:
```
2 -> C
4 -> E
14 -> O
18 -> S
3 -> D
4 -> E
8 -> I
```

Therefore, the resulting ciphertext for the word "MAZERAS" encrypted with the key a=9 and b=4 using the affine cipher f(a;b) is "CEO SDEI".

Regarding the strength of the encryption scheme, f(c;d) is only stronger than f(a;b) if the values of c and d are carefully chosen to provide a higher level of security. Without knowing the values of c and d, it is not possible to determine if f(c;d) is stronger than f(a;b).

Answer 5

The key-space of an affine cipher refers to the number of possible combinations of values for the two key parameters, a and b. In the affine cipher, a is used as the multiplier and b as the additive constant.

In an affine cipher, both a and b need to be relatively prime with 26 (the number of characters in the English alphabet) in order to have a unique encryption and decryption process. This is because if a and 26 have any common factors, the encryption function may not be reversible, and multiple keys could lead to the same encryption or decryption.

The total number of possible values for a is the count of numbers that are relatively prime to 26, which is equal to the Euler's totient function of 26. For any given a, there are 12 possible values that are relatively prime to 26.

As for b, since it can take any value from 0 to 25 (inclusive), there are 26 possible values.

Therefore, the key-space of an affine cipher is the product of the number of possible values for a and the number of possible values for b, which is 12 * 26 = 312.

In conclusion, the key-space of an affine cipher is 312, meaning there are 312 unique combinations of values for the key parameters a and b that can be used in the encryption and decryption process.