**Matrix-Based Data Encryption Tutorial**Matrices are commonly used to encrypt data. Here is a straightforward example of such an encryption technique. First, we represent each letter in the alphabet by a number, so let us take `` as 0, A = 1, B = 2, and so on. Thus, for example, "ABORT MISSION" becomes:\[ [1\ 2\ 15\ 18\ 20\ 0\ 13\ 9\ 19\ 19\ 9\ 15\ 14] \]To encrypt this coded phrase, we use an invertible matrix of any size with integer entries. For instance, let us take A to be the $2 \times 2$ matrix:\[ \begin{bmatrix}4 & 2 \\3 & 1 \\\end{bmatrix}\]We can first arrange the coded sequence of numbers in the form of a matrix with two rows (using zero in the last place if we have an odd number of characters) and then multiply on the left by A.Encrypted Matrix = \[ \begin{bmatrix}4 & 2 \\3 & 1 \\\end{bmatrix}\begin{bmatrix}1 & 15 & 20 & 13 & 9 & 19 & 15 \\2 & 18 & 0 & 9 & 19 & 9 & 14 \\\end{bmatrix}=\begin{bmatrix}8 & 96 & 80 & 70 & 48 & 114 & 66 & 42 & 42 \\\end{bmatrix}\]which we can also write as:\[ [8\ 5\ 96\ 63\ 80\ 60\ 70\ 48\ 114\ 76\ 66\ 42\ 56\ 42] \]To decipher the encoded message, multiply the encrypted matrix by $A^{-1}$ (the inverse of matrix A). The following exercise uses the above matrix A for encoding and decoding:Use the matrix A to encode the phrase "GO TO PLAN B".\[ \boxed{} \boxed{} \boxed{} \boxed{} \boxed{} \boxed{} \boxed{} \boxed{} \boxed{} \]**Need Help?**- [Read It]- [

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Math

Advanced Math

Use the matrix A to encode the phrase "GO TO PLAN B".

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Matrix-Based Data Encryption Tutorial**

Matrices are commonly used to encrypt data. Here is a straightforward example of such an encryption technique. First, we represent each letter in the alphabet by a number, so let us take `<space>` as 0, A = 1, B = 2, and so on. Thus, for example, "ABORT MISSION" becomes:

\[ [1\ 2\ 15\ 18\ 20\ 0\ 13\ 9\ 19\ 19\ 9\ 15\ 14] \]

To encrypt this coded phrase, we use an invertible matrix of any size with integer entries. For instance, let us take A to be the $2 \times 2$ matrix:

\[
\begin{bmatrix}
4 & 2 \\
3 & 1 \\
\end{bmatrix}
\]

We can first arrange the coded sequence of numbers in the form of a matrix with two rows (using zero in the last place if we have an odd number of characters) and then multiply on the left by A.

Encrypted Matrix =
\[
\begin{bmatrix}
4 & 2 \\
3 & 1 \\
\end{bmatrix}
\begin{bmatrix}
1 & 15 & 20 & 13 & 9 & 19 & 15 \\
2 & 18 & 0 & 9 & 19 & 9 & 14 \\
\end{bmatrix}
=
\begin{bmatrix}
8 & 96 & 80 & 70 & 48 & 114 & 66 & 42 & 42 \\
\end{bmatrix}
\]

which we can also write as:

\[ [8\ 5\ 96\ 63\ 80\ 60\ 70\ 48\ 114\ 76\ 66\ 42\ 56\ 42] \]

To decipher the encoded message, multiply the encrypted matrix by $A^{-1}$ (the inverse of matrix A). The following exercise uses the above matrix A for encoding and decoding:

Use the matrix A to encode the phrase "GO TO PLAN B".

\[ \boxed{} \boxed{} \boxed{} \boxed{} \boxed{} \boxed{} \boxed{} \boxed{} \boxed{} \]

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