Application of Matrices to Cryptography

A striking use of the inverse of a non-singular square matrix is in cryptography -- the art of communicating so that the message stays hidden from everyone except the intended receiver. Two steps are involved: encryption, which turns the readable plain message into an unreadable coded form, and decryption, which recovers the original message from the code. Both sender and receiver share a secret called the key.

A convenient key is a non-singular matrix. The sender encodes the message with this encoding (encryption) matrix; the receiver decodes it using the matrix's inverse, the decoding (decryption) matrix. Because the chosen matrix is non-singular its inverse exists, so the original message is recovered exactly.

Example. Let the letters $A$--$Z$ be numbered $1$--$26$ and let a blank space be $0$. Take the encoding matrix (applied by post-multiplication)

To send the message WELCOME, cut it into blocks of three letters: $(W\,E\,L),\ (C\,O\,M),\ (E\,\_\,\_)$, padding the last block with two blanks. As row matrices of numbers these are

Encrypt by post-multiplying each row matrix by $A$:

So the transmitted (coded) sequence is $45,\,-28,\,23,\,46,\,-18,\,3,\,5,\,-5,\,5$. …