One of the important applications of inverse of a non-singular square matrix is in cryptography.

**Application of matrices to Cryptography**

One of the important applications of inverse of a non-singular
square matrix is in cryptography. Cryptography is an art of communication
between two people by keeping the information not known to others. It is based
upon two factors, namely encryption and decryption. **Encryption **means the process of
transformation of an information (plain form) into an unreadable form (coded
form). On the other hand, **Decryption **means the
transformation of the coded message back into
original form. Encryption and decryption require a secret technique which is
known only to the sender and the receiver.

This secret is called a **key**. One way of generating a key is by using a
non-singular matrix to encrypt a message by the sender. The receiver decodes
(decrypts) the message to retrieve the original message by using the inverse of
the matrix. The matrix used for encryption is called **encryption matrix **(**encoding matrix**) and that used for
decoding is called **decryption matrix ****(****decoding matrix****)**.

We explain the process of encryption and decryption by means of
an example.

Suppose that the sender and receiver consider messages in
alphabets *A *−
*Z *only, both assign the numbers 1-26 to the letters *A *− *Z* respectively,
and the number 0 to a blank space. For simplicity, the sender employs a key as
post-multiplication by a non-singular matrix of order 3 of his own choice. The
receiver uses post-multiplication by the inverse of the matrix which has been
chosen by the sender.

Let the encoding
matrix be

Let the message to be sent by the sender be “**WELCOME**”.

Since the key is taken as the operation of post-multiplication by
a square matrix of order 3, the message is cut into pieces **(WEL), (COM), (E)**, each of length 3, and
converted into a sequence of row matrices of numbers:

**[23 5 12],[3 15 13],[5 0 0].**

Note that, we have included two zeros in the last row matrix. The
reason is to get a row matrix with 5 as the first entry.

Next, we encode the
message by post-multiplying each row matrix as given below:

So the encoded message is [45 − 28
−23] [46 -18 3] [ 5 −5 5]

The receiver will decode the message
by the reverse key, post-multiplying by the inverse of A.

So the decoding matrix is

The receiver decodes the coded
message as follows:

So, the sequence of decoded row
matrices is [23 5 12] , [3 15 13], [5 0 0].

Thus, the receiver reads the message
as “**WELCOME**”.

Tags : Definition, Theorem, Solved Example Problems | Inverse of a Non-Singular Square Matrix , 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

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12th Mathematics : UNIT 1 : Applications of Matrices and Determinants : Application of matrices to Cryptography | Definition, Theorem, Solved Example Problems | Inverse of a Non-Singular Square Matrix

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