Matrix Ciphers - Introduction to Cryptography with Java Applets - page 142

CHAPTER 7

Matrix Ciphers

Matrices offer us an alternative way to implement a linear block cipher. We will call
such a matrix-based cryptosystem a matrix cipher. In the matrix ciphers, we use an
enciphering transformation

C AP + B (mod n )

but now A is a mm matrix (called the enciphering matrix), P is a column vector of num-
bers corresponding to a block of plaintext letters of length m , and B is a column vector of
length m . (When B is the zero vector, these are called Hill ciphers.) To decipher, we must
again solve for P :

AP + B C (mod n )
AP C B (mod n )
P IP AAP A ( C B ) (mod n ).
(Proposition 24 allows us to multiply both sides of a congruence by a matrix and preserve
the congruence.)

A represents an inverse of A modulo n ; that is, A must satisfy the con-

gruence

AA I (mod n )

where I represents the identity matrix. A must be chosen, of course, so that it has an inverse

modulo n .

E XAMPLE .

We use the ordinary alphabet, so n = 26. Let the enciphering matrix A be

517
415

let the shift vector B be

5
2