Encryption Matrices are commonly used to encrypt data. Here is a simple form such an encryption can take. First, we represent each letter in the alphabet by a number, so let us take = 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 [²3] 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. [1 15 20 13 19 9 14 Encrypted Matrix = 18 0 9 19 15 0 2 x 2 matrix which we can also write as 10 102 40 62 114 78 28] 7 69 20 40 76 54 14' [10 7 102 69 40 20 62 40 114 76 78 54 28 14]. To decipher the encoded message, multiply the encrypted matrix by A-1. The following exercise uses the above matrix A for encoding and decoding. Use the matrix A to encode the phrase "GO TO PLAN B".
Encryption Matrices are commonly used to encrypt data. Here is a simple form such an encryption can take. First, we represent each letter in the alphabet by a number, so let us take = 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 [²3] 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. [1 15 20 13 19 9 14 Encrypted Matrix = 18 0 9 19 15 0 2 x 2 matrix which we can also write as 10 102 40 62 114 78 28] 7 69 20 40 76 54 14' [10 7 102 69 40 20 62 40 114 76 78 54 28 14]. To decipher the encoded message, multiply the encrypted matrix by A-1. The following exercise uses the above matrix A for encoding and decoding. 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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![**Encryption**
Matrices are commonly used to encrypt data. Here is a simple form such an encryption can take. First, we represent each letter in the alphabet by a number, so let us take `<space> = 0, A = 1, B = 2,` and so on. Thus, for example, "ABORT MISSION" becomes
\[ [1 \ \ 2 \ \ 15 \ \ 18 \ \ 20 \ \ 0 \ \ 13 \ \ 9 \ \ 19 \ \ 19 \ \ 15 \ \ 14]. \]
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
\[ 2 \times 2 \text{ matrix }
\begin{bmatrix}
2 & 4 \\
1 & 3
\end{bmatrix}.
\]
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 =
\[
\begin{bmatrix}
2 & 4 \\
1 & 3
\end{bmatrix}
\begin{bmatrix}
1 & 15 & 20 & 13 & 19 & 9 & 14 \\
2 & 18 & 0 & 9 & 19 & 15 & 0
\end{bmatrix}
=
\begin{bmatrix}
10 & 102 & 40 & 62 & 40 & 76 & 28 \\
7 & 69 & 20 & 40 & 76 & 54 & 14
\end{bmatrix},
\]
which we can also write as
\[ [10 \ \ 7 \ \ 102 \ \ 69 \ \ 40 \ \ 20 \ \ 62 \ \ 40 \ \ 114 \ \ 76 \ \ 78 \ \ 54 \ \ 28 \ \ 14]. \]
To decipher the encoded message, multiply the encrypted matrix by \( A^{-1} \). The following exercise uses the above matrix \( A \) for encoding and decoding.
Use the matrix \( A \) to encode the phrase "GO TO PLAN B".
\[ [ \ \ \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb60d532a-9aac-45a2-a466-49da1bb79676%2Fdc9df8c9-347e-4068-b764-ed830f4fa2ef%2Fwbqa9u_processed.png&w=3840&q=75)
Transcribed Image Text:**Encryption**
Matrices are commonly used to encrypt data. Here is a simple form such an encryption can take. First, we represent each letter in the alphabet by a number, so let us take `<space> = 0, A = 1, B = 2,` and so on. Thus, for example, "ABORT MISSION" becomes
\[ [1 \ \ 2 \ \ 15 \ \ 18 \ \ 20 \ \ 0 \ \ 13 \ \ 9 \ \ 19 \ \ 19 \ \ 15 \ \ 14]. \]
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
\[ 2 \times 2 \text{ matrix }
\begin{bmatrix}
2 & 4 \\
1 & 3
\end{bmatrix}.
\]
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 =
\[
\begin{bmatrix}
2 & 4 \\
1 & 3
\end{bmatrix}
\begin{bmatrix}
1 & 15 & 20 & 13 & 19 & 9 & 14 \\
2 & 18 & 0 & 9 & 19 & 15 & 0
\end{bmatrix}
=
\begin{bmatrix}
10 & 102 & 40 & 62 & 40 & 76 & 28 \\
7 & 69 & 20 & 40 & 76 & 54 & 14
\end{bmatrix},
\]
which we can also write as
\[ [10 \ \ 7 \ \ 102 \ \ 69 \ \ 40 \ \ 20 \ \ 62 \ \ 40 \ \ 114 \ \ 76 \ \ 78 \ \ 54 \ \ 28 \ \ 14]. \]
To decipher the encoded message, multiply the encrypted matrix by \( A^{-1} \). The following exercise uses the above matrix \( A \) for encoding and decoding.
Use the matrix \( A \) to encode the phrase "GO TO PLAN B".
\[ [ \ \ \
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