Polygraphic System

Another way of encoding is to divide plain text into sets of n-letters, and replace them with n code letters. In this case invertible matrices can be used to provide a better coding, than substitution.

First associate a different number with every letter of alphabet.

For example we may use the following conversion table:

$$\begin{array}{ccccccccccccc} A & B & C & D& E & F & G & H & I... ...& 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \end{array}$$

$$\begin{array}{ccccccccccccc} N & O & P & Q & R & S & T & U & ... ... &15 &16& 17& 18& 19 &20 &21 &22 &23& 24 &25 &26\\ \end{array}$$

Suppose $n=3,$ choose a 3 x 3 invertible matrix A,

$$A = \left[ \begin{array}{rrr} 0&2&-1\\ 1&-2&1\\ -1&-1&1\\ \end{array}\right]$$

Then

$$A^{-1} = \left[ \begin{array}{rrr} 1&1&0\\ 2&1&1\\ 3&2&2\\ \end{array}\right]$$

Both communicating parties should have knowledge of the table which list association of letters to numbers, the matrices A and $A^{-1}$. The party that want to send a message, needs to covert it to a string of numbers. As a second step that string of numbers needs to be divided into groups of 3, then multiple each group by the matrix $A$ to form new groups and build a new string of numbers. Then, send the resulted string as a string of numbers or letters. The party who is receiving the coded string should divide the string into groups of 3, then multiple by $A^{-1}$ and finally convert it to letters.