USE OF MATRICES IN CRYPTOGRAPHY

What is cryptography?

Cryptography, the science of encrypting and deciphering messages written in secret codes, has

played a vital role in securing information. With millions of financial transactions conducted over the

internet daily, cryptography has become more important than ever. Companies have begun to make

online transactions more secure by installing encryption software to prevent sensitive information

such as credit card numbers from falling into the wrong hands.

There are several cryptographic techniques and many extensive uses of mathematics to secure

information. One of the techniques uses for encoding and decoding messages is the concepts of

matrices where the algorithm for matrix multiplication and the concept of the multiplicative inverse

of a matrix are applied.

Using Matrices to encode and decipher messages

The matrix $A=\left[\begin{array}{ccc}-1& 5& -1\\ -2& 11& 7\\ 1& -5& 2\end{array}\right]$ is used for the following example.

Encoding a message

Step $1$: Set up a substitution cipher to encode the message "MATH IS SWEET"

For the cipher, we will associate a unique number to every letter in the alphabet.

This can be done randomly or using a systematic approach. Here we choose to

associate $0$ with a blank space, and assign $1$ to $A,-1$ to $B,2$ to $C,-2$ to $D,$

and so on$.$

Now encode the secret message as shown We next enter the coded message into a new matrix $B,$ by entering it letter by

letter into the columns of $B.$ Note that since the encrypting matrix $A$ is a $33,B$

must have a $3$ rows for multiplication to be possible.

$B=\left[\begin{array}{ccccc}M& H& S& W& T\\ A& E& ?& ?& ?\\ T& \text{I}& S& E& ?\end{array}\right]=\left[\begin{array}{ccccc}7& -4& 10& 12& -10\\ 1& 0& 0& 3& 0\\ -10& 5& 10& 3& 0\end{array}\right]$

If the message is too short to fill matrix $B,$ we use blank spaces to complete the final

column.

Step $2$: Use the matrix $A$ to encrypt it$.$

Computing the product $AB$ encrypts the message, and only someone with access to

$A^{-1}$ will be able to read it$.$

$AB=\left[\begin{array}{ccccccc}-1& 5& -1& 11& 7& -5& 2\end{array}\right]$

$1$

$1$

$-2\left[\begin{array}{ccccccccccccc}7& -4& 10& 12& -10& 0& 0& 3& 0& 5& 10& 3& 0\end{array}\right]$

$-10$

$1$

$=\left[\begin{array}{ccccccccccccc}8& -1& -20& 0& 10& 43& 50& 30& 20& 6& 30& 3& -10\end{array}\right]$

$-18$

$-73$

The coded message is $8,-73,-18,-1,43,6,-20,50,30,0,30,3,10,20$ and

$-10.$

Deciphering a message

Step $3$: Decipher the coded message from step $2.$

The received message is $8,-73,-18,-1,43,6,-20,50,30,0,30,3,10,20$ and

$-10.$ It is the result of the product $AB.$ To find matrix $B,$ we apply $A^{-1}$ since

$A^{-1}(AB)=(A^{-1}A)B=B.$ Writing the received message in matrix form, we

have

$AB=\left[\begin{array}{ccccc}8& -1& -20& 0& 10\\ -73& 43& 50& 30& 20\\ -18& 6& 30& 3& -10\end{array}\right]$ Step $4$: $,$ Next multiply $AB$ by $A^{-1}$ on the left to determine matrix $B$

$A^{-1}(AB)=\left[\begin{array}{ccccccc}-57& 5& -46& 1& -9& 0& 1\end{array}\right]$

$1$

$1$

$-11\left[\begin{array}{ccccccccccccc}8& -1& -20& 0& 10& 43& 50& 30& 20& 6& 30& 3& -10\end{array}\right]$

$-18$

$-73$

$=\left[\begin{array}{ccccccccccccc}7& -4& 10& 12& -10& 0& 0& 3& 0& 5& 10& 3& 0\end{array}\right]$

$-10$

$1=B$

Writing matrix $B$ in sentence form gives $7,1,-10,-4,0,5,10,0,10,12,3,3,-10,$

$0$&$0$ and using the substitution cipher to replace numbers with letters,

reveals the message MATH IS SWEET.

Activity $1($Unlock a coded message$)$

Doctor Strange is in deep trouble. He is being pursed by Thanos and is presently hiding in a secret

location to escape from being captured by him. He needs to send messages to two of his comrades,

Thor and Captain America, to tell them about his situation and whereabouts. As such, he composes

his first message by replacing each letter in the message by a number according to the code in

Table $1$ below.

Table $1$: Coding system

As Doctor Strange is afraid that his message will be intercepted and cracked by Thanos, therefore,

he decides to send the message in code together with a super hi$-$tech LOCK to Thor and Captain

America to unlock the coded message.

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

With the use of Table $1$ and LOCK, what is the message that Doctor Strange is trying to send to Thor

and Captain America? Please show your working clearly on how you unlock the code message. Activity $2($Design your own substitution cipher$)$