Matrix Cryptography: Securing Messages with Invertible Matrices

1. Cryptography

2. Cryptography Today governments use sophisticated methods of coding and decoding messages. One type of code that is extremely difficult to break makes use of a large invertible matrix to encode a message. The receiver of the message decodes it using the inverse of the matrix. Cryptography is the process of coding and decoding messages. The word comes from the Greek “kryptos” meaning “hidden.” The technique can be traced back to the ancient Greeks.

3. The receiver of the message decodes it using the inverse of a matrix. This first matrix is called the encoded matrix. Its inverse is called the decoded matrix.

4. Let the message be BUY IBM STOCK and let the encoded matrix be

5. Assign a number to each letter. B U Y - I B M - S T O C K 2 21 25 27 9 2 13 27 19 20 15 3 11

6. Break the digital message up into a sequence of 3 x 1 column matrices as follows: Observe it was necessary to add two spaces at the end of the message in order to complete the last matrix.

7. We now put the message into code by multiplying each of the column matrices by the encoding matrix.

8. The columns of the matrix give the encoded message. Transmitted in linear form: -169, 46, 171, -116, 11, 143, -196, 46, 209, -117, 18, 137, -222, 54, 233

9. To Decode The receiver writes the string as a sequence of 3 x 1 column matrices and repeats the technique using the inverse of the encoding matrix. The decoding matrix is:

10. Decode the Message

11. The columns of the matrix, written in linear form, give the original message. 2 21 25 27 9 2 13 27 19 20 15 3 11 B U Y - I B M - S T O C K

12. For further reading Coding Theory and Cryptography edited by David Joyner, Springer-Verlag, 2000.