Algebra applications in Cryptography

Today governments use sophisticated methods of coding and decoding messages. One type of code, which is extremely difficult to break, makes use of a large matrix to encode a message. The receiver of the message decodes it using the inverse of the matrix. This first matrix is called the encoding matrix and its inverse is called the decoding matrix.

Example Let the message be:

PREPARE TO NEGOTIATE

and the encoding matrix be

We assign a number for each letter of the alphabet. For simplicity, let us associate each letter with its position in the alphabet: A is 1, B is 2, and so on. Also, we assign the number 27 (remember we have only 26 letters in the alphabet) to a space between two words. Thus the message becomes:

Since we are using a 3 by 3 matrix, we break the enumerated message above into a sequence of 3 by 1 vectors:

Note that it was necessary to add a space at the end of the message to complete the last vector. We now encode the message by multiplying each of the above vectors by the encoding matrix. This can be done by writing the above vectors as columns of a matrix and perform the matrix multiplication of that matrix with the encoding matrix as follows:

which gives the matrix

The columns of this matrix give the encoded message. The message is transmitted in the following linear form

To decode the message, the receiver writes this string as a sequence of 3 by 1 column matrices and repeats the technique using the inverse of the encoding matrix. The inverse of this encoding matrix, the decoding matrix, is:

(make sure that you compute it yourself). Thus, to decode the message, perform the matrix multiplication

and get the matrix

The columns of this matrix, written in linear form, give the original message: