Question

Hello,

I need to think of a short message between 15 and 40 characters. I will need to encode it using a linear affine cipher with n=26. I am unsure of how to even start and am not understanding how to process this. Can you please help and show me how to do it step by step? Also, what is the best way I can decipher messages without the key?

Answer 1

linear affine cipher is v combination of additive and multiplicative ciphers which uses a pair of key.

first key - is used with multiplicative cipher (k₁ )

- belongs to Z₂₆* = > only some numbers in Z₂₆ will have multiplicative inverse, so only those have to be selected as k_1.

  k₁ = {1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25}

second key - is used with additive cipher (k₂ )

- belongs to Z₂₆

k₂ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 17, 18, 19,20, 21, 22, 23, 24, 25}

=> key domain = 12 * 26 = 312

Encryption : C = (P*k₁ +k₂)mod 26

Decryption: P = (C-k₂)*k₁^-1)mod 26

Plaintext - hello everyone

First the plaintext will be mapped with numbers ( 0 - a....25 - z)

Let k₁ and k₂ be 7, 2 respectively.

Both multiplication and addition will be done with respect to modulo n (=26)

Encryption:

h = 07 ------> (07 * 7 + 2)mod 26 => (49+2)mod 26 =  25 ---------> 25 = z

e = 04 ------> (04 * 7 + 2)mod 26 => (28+2)mod 26 = 04 ---------> 04 = e

l = 11 ------> (11 * 7 + 2)mod 26 => (77+2)mod 26 = 01 ---------> 01 = b

l = 11 ------> (11 * 7 + 2)mod 26 => (77+2)mod 26 = 01 ---------> 01 = b

o = 14 ------> (14 * 7 + 2)mod 26 => (98+2)mod 26 = 22 ---------> 22 = w

e = 04 ------> (04 * 7 + 2)mod 26 => (28+2)mod 26 = 04 ---------> 04 = e

v = 21 ------> (21 * 7 + 2)mod 26 => (28+2)mod 26 = 19 ---------> 19 = t

e = 04 ------> (04 * 7 + 2)mod 26 => (28+2)mod 26 = 04 ---------> 04 = e

r = 17 ------> (17 * 7 + 2)mod 26 => (28+2)mod 26 = 17 ---------> 17 = r

y = 24 ------> (24 * 7 + 2)mod 26 => (28+2)mod 26 = 14 ---------> 14 = o

o = 14 ------> (14 * 7 + 2)mod 26 => (98+2)mod 26 = 22 ---------> 22 = w

n = 13 ------> (13 * 7 + 2)mod 26 => (98+2)mod 26 = 14 ---------> 15 = p

e = 04 ------> (04 * 7 + 2)mod 26 => (28+2)mod 26 = 04 ---------> 04 = e

Ciphertext = zebbweterowpe

Decryption

Ciphertext = zebbweterowpe

Inverse can be found using Fermat's theorem or Euler's theorem or Euclidean method

Inverse of 7 is 15

z = 25 -----> ((25-2)*7^-1)mod 26 => (23*15)mod 26 = 07 ---------> 07 = h

e = 04 -----> ((04-2)*7^-1)mod 26 => (02*15)mod 26 = 04 ---------> 04 = e

b = 01 -----> ((01-2)*7^-1)mod 26 => (25*15)mod 26 = 11 ---------> 11 = l

b = 01 -----> ((01-2)*7^-1)mod 26 => (25*15)mod 26 = 11 ---------> 11 = l

w = 22 -----> ((22-2)*7^-1)mod 26 => (20*15)mod 26 = 14 ---------> 14 = o

e = 04 -----> ((04-2)*7^-1)mod 26 => (02*15)mod 26 = 04 ---------> 04 = e

t = 19 -----> ((19-2)*7^-1)mod 26 => (17*15)mod 26 = 21 ---------> 21 = v

e = 04 -----> ((04-2)*7^-1)mod 26 => (02*15)mod 26 = 04 ---------> 04 = e

r = 17 -----> ((17-2)*7^-1)mod 26 => (15*15)mod 26 = 17 ---------> 17 = r

o = 14 -----> ((14-2)*7^-1)mod 26 => (12*15)mod 26 = 24 ---------> 24 = y

w = 22 -----> ((22-2)*7^-1)mod 26 => (20*15)mod 26 = 14 ---------> 14 = o

p = 15 -----> ((15-2)*7^-1)mod 26 => (13*15)mod 26 = 13 ---------> 13 = n

e = 04 -----> ((04-2)*7^-1)mod 26 => (02*15)mod 26 = 04 ---------> 04 = e

Plaintext = helloeveryone

=> hello everyone

Brute-force and statistical methods can be used to identify the plaintext without knowing the key. But this will take time.

If suppose intruder has access to the sender's computer, they can go for chosen-plaintext attack. Here a text is given and corresponding ciphertext is found. Now, we have to map the letters from the text to ciphertext and have to find the key. In this way we'll be able to decode whatever sent between the sender and receiver.