Wednesday, 10 February 2010

Application of Invertible Matrices: Coding

There are many ways to encrypt a message. And the use of coding has become particularly significant in recent years (due to the explosion of the internet for example). One way to encrypt or code a message uses matrices and their inverse. Indeed, consider a fixed invertible matrix A. Convert the message into a matrix B such that AB is possible to perform. Send the message generated by AB. At the other end, they will need to know A-1 in order to decrypt or decode the message sent. Indeed, we have
\begin{displaymath}A^{-1} \Big(AB\Big) = B\end{displaymath}

which is the original message. Keep in mind that whenever an undesired intruder finds A, we must be able to change it. So we should have a mechanical way of generating simple matrices A which are invertible and have simple inverse matrices. Note that, in general, the inverse of a matrix involves fractions which are not easy to send in an electronic form. The best is to have both A and its inverse with integers as their entries. In fact, we can use our previous knowledge to generate such class of matrices. Indeed, if A is a matrix such that its determinant is $\pm 1$ and all its entries are integers, then A-1 will have entries which are integers. So how do we generate such class of matrices? One practical way is to start with an upper triangular matrix with $\pm 1$ on the diagonal and integer-entries. Then we use the elementary row operations to change the matrix while keeping the determinant unchanged. Do not multiply rows with non-integers while doing elementary row operations. Let us illustrate this on an example.
Example. Consider the matrix

First we keep the first row and add it to the second as well as to the third rows. We obtain

Next we keep the first row again, we add the second to the third, and finally add the last one to the first multiplied by -2. We obtain

This is our matrix A. Easy calculations will give det(A) = -1, which we knew since the above elementary operations did not change the determinant from the original triangular matrix which obviously has -1 as its determinant. We leave the details of the calculations to the reader. The inverse of A is

Back to our original problem. Consider the message
\begin{displaymath}\mbox{I LOVE MONICA }\end{displaymath}

To every letter we will associate a number. The easiest way to do that is to associate 0 to a blank or space, 1 to A, 2 to B, etc... Another way is to associate 0 to a blank or space, 1 to A, -1 to B, 2 to C, -2 to D, etc... Let us use the second choice. So our message is given by the string
I& &L&O&V&E&&M&O&N&I&C&A\\

Now we rearrange these numbers into a matrix B. For example, we have
\begin{displaymath}B = \left(\begin{array}{rrrrrr}

Then we perform the product AB, where A is the matrix found above. We get
\begin{displaymath}AB = \left(\begin{array}{rrr}
1&-5&2\\ ...

The encrypted message to be sent is
\begin{displaymath}1,\; -52,\; -7,\; -66,\; \cdots,\; -1,\; -2,\; 1\end{displaymath}



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