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 encrypt the word "MAZERAS" using the affine cipher, we need to apply the encryption function f(a;b)(x) = (ax + b) mod 26.
Given a = 9 and b = 4, we can substitute these values into the encryption function.
For the letter 'M':
Using the numerical representation of the alphabet (A = 0, B = 1, ..., Z = 25), 'M' is represented by the number 12.
The cipher equation becomes: f(9;4)(12) = (9*12 + 4) mod 26 = 109 mod 26 = 3.
So, the encrypted letter for 'M' is 'C'.
Similarly, we can encrypt the remaining letters:
'A' -> f(9;4)(0) = (9*0 + 4) mod 26 = 4 mod 26 = 4 -> 'E'
'Z' -> f(9;4)(25) = (9*25 + 4) mod 26 = 229 mod 26 = 21 -> 'V'
'E' -> f(9;4)(4) = (9*4 + 4) mod 26 = 40 mod 26 = 14 -> 'O'
'R' -> f(9;4)(17) = (9*17 + 4) mod 26 = 157 mod 26 = 5 -> 'F'
'A' -> f(9;4)(0) = (9*0 + 4) mod 26 = 4 mod 26 = 4 -> 'E'
'S' -> f(9;4)(18) = (9*18 + 4) mod 26 = 166 mod 26 = 14 -> 'O'
So, the resulting ciphertext for the word "MAZERAS" is "CEVOFE".
To answer the second part of your question, whether f(c;d) is a stronger encryption scheme than f(a;b), we would need the values of c and d for a proper comparison. Without these values, it is not possible to determine the strength of one encryption scheme over the other.
To determine whether f(c;d) is a stronger encryption scheme than f(a;b), we need to compare the key space of both encryption schemes.
The key space of an affine cipher is determined by the possible values of the parameters 'a' and 'b' (or 'c' and 'd' in this case), and in this case, it is the set of all possible combinations of values of 'a' and 'b' (or 'c' and 'd') that satisfy the conditions for the parameters.
For f(a;b), the values of 'a' and 'b' must be relatively prime with the modulus 26 (which is the number of letters in the English alphabet) and 'a' must not be equal to 0. So, the key space for f(a;b) is limited to a subset of all possible combinations.
For f(c;d), the same conditions apply, and the key space is also limited to a subset of all possible combinations.
In general, a larger key space implies a stronger encryption scheme because it is more difficult for an attacker to guess the correct combination of parameters.
Without knowing the specific values of 'c' and 'd', we cannot determine whether f(c;d) is a stronger encryption scheme than f(a;b) in this case. However, we can move on to encrypting the word "MAZERAS" using the given values of 'a' and 'b'.
Using the encryption function f(a;b)(x) = (a*x + b) mod 26, and the given values of a=9 and b=4, we can encrypt the word "MAZERAS" as follows:
1. Convert each letter to its corresponding numerical value:
M -> 12
A -> 0
Z -> 25
E -> 4
R -> 17
A -> 0
S -> 18
2. Apply the encryption function to each numerical value:
f(9;4)(12) = (9*12 + 4) mod 26 = 118 mod 26 = 12 (which is the numerical value for M)
f(9;4)(0) = (9*0 + 4) mod 26 = 4 (which is the numerical value for E)
f(9;4)(25) = (9*25 + 4) mod 26 = 229 mod 26 = 11 (which is the numerical value for L)
f(9;4)(4) = (9*4 + 4) mod 26 = 40 mod 26 = 14 (which is the numerical value for O)
f(9;4)(17) = (9*17 + 4) mod 26 = 157 mod 26 = 25 (which is the numerical value for Z)
f(9;4)(0) = (9*0 + 4) mod 26 = 4 (which is the numerical value for E)
f(9;4)(18) = (9*18 + 4) mod 26 = 166 mod 26 = 18 (which is the numerical value for S)
3. Convert the numerical values back to letters:
12 -> M
4 -> E
11 -> L
14 -> O
25 -> Z
4 -> E
18 -> S
Therefore, the encrypted form of the word "MAZERAS" using the affine cipher with a=9 and b=4 is "MELOZES".
To determine if f(c;d) is a stronger encryption scheme than f(a;b), we need to compare their key values and properties.
In an affine cipher, the encryption function f(a;b)(x) is defined as (a * x + b) mod 26, where a and b are the key values, and x represents the plaintext. Similarly, f(c;d)(x) is (c * x + d) mod 26.
To compare the encryption schemes, we need to compare the key values (a, b) and (c, d). If the key values for f(c;d) provide better security than f(a;b), then it can be considered a stronger encryption scheme.
Now, let's encrypt the word "MAZERAS" using the key values a = 9 and b = 4.
1. Convert the letters of the word to their corresponding numerical values. A = 0, B = 1, C = 2, and so on.
MAZERAS becomes: [12, 0, 25, 4, 17, 0, 18]
2. Apply the encryption function f(a;b)(x) for each letter:
For the letter 'M' (x = 12): f(9;4)(12) = (9 * 12 + 4) mod 26 = 8
For letter 'A' (x = 0): f(9;4)(0) = (9 * 0 + 4) mod 26 = 4
For letter 'Z' (x = 25): f(9;4)(25) = (9 * 25 + 4) mod 26 = 17
For letter 'E' (x = 4): f(9;4)(4) = (9 * 4 + 4) mod 26 = 10
For letter 'R' (x = 17): f(9;4)(17) = (9 * 17 + 4) mod 26 = 23
For letter 'A' (x = 0): f(9;4)(0) = (9 * 0 + 4) mod 26 = 4
For letter 'S' (x = 18): f(9;4)(18) = (9 * 18 + 4) mod 26 = 2
3. Convert the resulting numerical values back to letters:
Ciphertext: [8, 4, 17, 10, 23, 4, 2]
Mapping these values back to alphabetical letters, we get: IETKXED
Therefore, the encrypted word "MAZERAS" using the key values a = 9 and b = 4 is "IETKXED".