Affine Cipher

Summary: Each letter is transformed using the formula: E(x) = (ax + b) mod 26, where 'a' and 'b' are the keys. The value 'a' must be coprime with 26 (valid values: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25).

Letter to Number: A=0, B=1, C=2, ..., Z=25

Example:

• Given: a = 5, b = 8
• Encryption formula: E(x) = (5x + 8) mod 26

Encryption:

• Plaintext: HELP
• H (x=7): (5*7 + 8) mod 26 = 43 mod 26 = 17 -> R
• E (x=4): (5*4 + 8) mod 26 = 28 mod 26 = 2 -> C
• L (x=11): (5*11 + 8) mod 26 = 63 mod 26 = 11 -> L
• P (x=15): (5*15 + 8) mod 26 = 83 mod 26 = 5 -> F
• Ciphertext: RCLF

Decryption (given a=5, b=8):

• Find modular multiplicative inverse of a=5: 5^(-1) = 21 (mod 26)
  • Because 5 * 21 = 105 = 4*26 + 1 = 1 (mod 26)
• Decryption formula: D(y) = 21(y - 8) mod 26
• Examples:
  • R (y=17): 21(17-8) mod 26 = 21*9 mod 26 = 189 mod 26 = 7 -> H
  • C (y=2): 21(2-8) mod 26 = 21*(-6) mod 26 = -126 mod 26 = 4 -> E
  • L (y=11): 21(11-8) mod 26 = 21*3 mod 26 = 63 mod 26 = 11 -> L
  • F (y=5): 21(5-8) mod 26 = 21*(-3) mod 26 = -63 mod 26 = 15 -> P
• Plaintext: HELP

Additional Example:

• Given: a = 7, b = 3
• Encryption formula: E(x) = (7x + 3) mod 26

Encrypting "CODE":

• C (x=2): (7*2 + 3) mod 26 = 17 -> R
• O (x=14): (7*14 + 3) mod 26 = 101 mod 26 = 23 -> X
• D (x=3): (7*3 + 3) mod 26 = 24 -> Y
• E (x=4): (7*4 + 3) mod 26 = 31 mod 26 = 5 -> F
• Ciphertext: RXYF

Decrypting RXYF with a=7, b=3:

• Find modular inverse of 7: 7^(-1) = 15 (mod 26)
  • Because 7 * 15 = 105 = 4*26 + 1 = 1 (mod 26)
• Decryption formula: D(y) = 15(y - 3) mod 26
• R (y=17): 15(17-3) mod 26 = 15*14 mod 26 = 210 mod 26 = 2 -> C
• X (y=23): 15(23-3) mod 26 = 15*20 mod 26 = 300 mod 26 = 14 -> O
• Y (y=24): 15(24-3) mod 26 = 15*21 mod 26 = 315 mod 26 = 3 -> D
• F (y=5): 15(5-3) mod 26 = 15*2 mod 26 = 30 mod 26 = 4 -> E
• Plaintext: CODE

Finding Modular Inverses (for common 'a' values):

• a=3: inverse=9 (3*9=27=1 mod 26)
• a=5: inverse=21 (5*21=105=1 mod 26)
• a=7: inverse=15 (7*15=105=1 mod 26)
• a=9: inverse=3 (9*3=27=1 mod 26)
• a=11: inverse=19 (11*19=209=1 mod 26)
• a=15: inverse=7 (15*7=105=1 mod 26)
• a=17: inverse=23 (17*23=391=1 mod 26)
• a=19: inverse=11 (19*11=209=1 mod 26)
• a=21: inverse=5 (21*5=105=1 mod 26)
• a=23: inverse=17 (23*17=391=1 mod 26)
• a=25: inverse=25 (25*25=625=1 mod 26)