(8)Explain how the HILL cipher algorithm works. You may use the plaintext PLAY to and the key SECRET to show its encipherment procedure
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.
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.
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).
What is the key-space of an affine cipher?
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.
Step 1: Key Generation
To begin with, the key for the HILL cipher algorithm needs to be generated. The key is a matrix, typically a 2x2 or 3x3 matrix, depending on the desired level of security. For this example, we will use a 2x2 matrix key.
Given key: SECRET
To convert this key into a matrix, we can assign numeric values to each letter starting from 0. Let A = 0, B = 1, C = 2, and so on. Then, we arrange the letters of the key into the matrix as follows:
S E
C R
Step 2: Plaintext Conversion
The plaintext, in this case, is "PLAY". Similarly, we convert each letter into its corresponding numeric value using the same scheme mentioned above. Thus, "PLAY" becomes:
P = 15
L = 11
A = 0
Y = 24
Step 3: Encryption
Now that we have the key matrix and the plaintext values, we can proceed with the encryption process.
For the HILL cipher algorithm, each character of the plaintext is multiplied by the key matrix. The resulting matrix is then converted back to the corresponding letters.
To do the multiplication, we group the plaintext values into pairs since our key matrix is 2x2.
Pair 1: (15, 11)
Pair 2: (0, 24)
Let's encrypt Pair 1 first:
15 11 * S E = (15*S + 11*C) (15*E + 11*R)
Using the matrix multiplication formula, we calculate:
(15*S + 11*C) (15*E + 11*R) = (15*18 + 11*2) (15*4 + 11*17) = (444) (259)
Converting these resulting numeric values back into letters using the same scheme as before, we have:
444 = M
259 = C
So, Pair 1 encrypts to "MC".
Now, let's encrypt Pair 2 using the same procedure:
0 24 * S E = (0*S + 24*C) (0*E + 24*R) = (24*C) (24*R)
Converting these values back into letters, we have:
24*C = Y
24*R = A
Therefore, Pair 2 encrypts to "YA".
Putting these encrypted pairs together, we have the final ciphertext:
MCYA
Thus, the plaintext "PLAY" encrypted with the key "SECRET" using the HILL cipher algorithm results in the ciphertext "MCYA".
The HILL cipher algorithm is a cryptographic method that uses matrix algebra to encrypt and decrypt messages. It was developed by Lester S. Hill in 1929. To understand how the HILL cipher works, let's use the plaintext "PLAY" and the key "SECRET" to demonstrate the encipherment procedure.
1. Convert the plaintext and key into numbers:
First, we need to convert each letter in the plaintext "PLAY" and the key "SECRET" into their respective numerical equivalents. We can use a simple mapping such as A=0, B=1, C=2, and so on.
P = 15, L = 11, A = 0, Y = 24
S = 18, E = 4, C = 2, R = 17, E = 4, T = 19
2. Create a matrix from the key:
The key "SECRET" will be used to create a 2x2 matrix. We arrange the numerical equivalents of the letters in the key in a row-wise manner.
S E
C R
3. Calculate the matrix inverse:
To encrypt the plaintext, we need to calculate the inverse of the key matrix. In this case, the determinant of the key matrix is -42. Since -42 does not have a modular multiplicative inverse mod 26, we cannot proceed with this particular example. However, if the determinant had a modular multiplicative inverse, we would proceed to find the modular multiplicative inverse and then multiply it by the matrix adjugate.
4. Group the plaintext into pairs:
Since the key matrix should be a 2x2 matrix, we group the plaintext numbers into pairs. If the plaintext does not have an even number of characters, we can add padding letters to ensure it can be split evenly.
P L -> [15 11]
A Y -> [0 24]
5. Multiply matrices:
We multiply the key matrix with each pair of plaintext matrices. In matrix multiplication, each element of the resulting matrix is obtained by taking the dot product of corresponding row elements of the first matrix and column elements of the second matrix.
[15 11] * [S C] = [(15*S + 11*C) (15*R + 11*E)]
[ 0 24] [E R] [(0*S + 24*C) (0*R + 24*E)]
6. Take modulo 26:
After obtaining the resulting matrix from the matrix multiplication, we take modulo 26 for each element in the matrix to ensure the values remain within the range of alphabets (A=0 to Z=25).
7. Convert back to letters:
Finally, we convert the modulo 26 values back to letters using the inverse of the mapping we used earlier.
The resulting matrix will correspond to the encrypted message.
Note: In this particular example, we were unable to proceed with calculating the matrix inverse since the determinant did not have a modular multiplicative inverse mod 26. However, this is a general outline of how the HILL cipher algorithm works.