Suppose the matrix 1 3 2 4 is used as an encryption matrix F

Suppose the matrix (1 3 2 4) is used as an encryption matrix. Find two plaintexts that encrypt to the same ciphertext (so this is not a good matrix to use for encryption!).

Solution

Plaintext means a block of plaintext letters ( block ciphertext scheme)

Determinant of the given matrix: 1*4 - 2*3 = -2

-2 ---->coprime to MOD 26. As per ciphertext block

there are GCD (26, 2) = 2 plaintext blocks that map to it.

Equation : (Y1, Y2) (x1, x2)( 1 2 , 3 4 ) MOD26

Y1 = x1 +3x2 ; Y2 = 3x1 + 4x2    MOD26

Lets tak another pair of plaintext satiffying the equation:

(x1 x\' 1) + 3(x2 x\' 2) 0 (mod 26)

2(x1 x\'1) + 4(x2 x \'2) 0 (mod 26)

On solving the two equations we get :

2(x2 x\'2) 0 (mod 26) i.e. x2 = x\' 2 or x2 = x\'2 + 13.

From x2 = x\'2, then from the first equation, x1 = x\'1. (trivial sol.).

From x2 = x\'2 + 13, x1 = x\'1 + 13.

The pair plaintexts such that x1 = x\'1 + 13 and x2 = x\'2 + 13 encrypts to the same ciphertext block.