Notes - Affine Ciphers

The following are notes from Introduction to Cryptography with Coding Theory.

• This is a strengthened version of a shift cipher where x -> ax+B
• example a = 9, B = 2
• affine -> CVVWPM
• Decryption involves reversing the cipher
• y = 9 x + 2
• In modular arithmetic we need to find the inverse differently than we would in algebra, more clearly described in Chapter 3
• Since we are using the alphabet, we need to work with mod 26
• therefore we need to find gcd(9,26) = 1
• What we can do is use the extended euclidean algorithm to solve for a multiplier to 9 that would result 1 mod 26 aka
• 9 x ≡ 1 (mod 26)
• we can also see that multiplying 9 by 3 will also result in a remainder of 1 which satisfies this congruence
• Therefore 3 is the desired inverse
• x ≡ 3(y-2) ≡ 3y -6 ≡ 3y +20 (mod 26)
• Example
• map the letter V
• V = 21
• 3*21 +20
• 81
• 81 ≡ 5 mod(26)
• the letter f
• This decryption will allow us to return the plaintext of affine from CVVWPM
• We must be able to find the inverse in modular arithmetic in order to have a usable key so arbitrary keys may result in multiple returns for a singular input
• Possible Attacks
• Ciphertext only
• 312 keys only means its easy for a computer to break, however difficult by hand
• Known Plaintext
• knowing even some of the plaintext reduces the possibilities and can potentially result in breaking the code even without a computer
• Chosen Plaintext
•  if ab is chosen as plaintext we can immediately break the code
• Chosen Ciphertext
• again if AB is chosen as ciphertext we find the encryption key but we already have the decryption key in this case so no additional work is needed

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