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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 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 × 2 matrix [ 1 2 3 4 ] . 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 = [ 1 2 3 4 ] [ 1 15 20 13 19 9 14 2 18 0 9 19 15 0 ] = [ 5 51 20 31 57 39 14 11 117 60 75 133 87 42 ] Which we can also write as [ 5 11 51 117 20 60 31 75 57 133 39 87 14 42 ] To decipher the encoded message, multiply the encrypted matrix by A − 1 . Exercises 63–66 use the above matrix A for encoding and decoding. Decode the following message, which was encrypted using the matrix A . [ 33 69 54 126 11 27 20 60 29 59 65 149 41 87 ]

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Finite Mathematics

7th Edition
Stefan Waner + 1 other
Publisher: Cengage Learning
ISBN: 9781337280426
Chapter 4.3, Problem 65E
Textbook Problem
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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 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 × 2 matrix [ 1 2 3 4 ] . 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 = [ 1 2 3 4 ] [ 1 15 20 13 19 9 14 2 18 0 9 19 15 0 ]

= [ 5 51 20 31 57 39 14 11 117 60 75 133 87 42 ]

Which we can also write as

[ 5 11 51 117 20 60 31 75 57 133 39 87 14 42 ]

To decipher the encoded message, multiply the encrypted matrix by A 1 . Exercises 63–66 use the above matrix A for encoding and decoding.

Decode the following message, which was encrypted using the matrix A.

[ 33 69 54 126 11 27 20 60 29 59 65 149 41 87 ]

To determine

To calculate: The decoded form of the message [336954126112720602959651494187] where each letter in the alphabet is represented by a number, and the sequence of numbers is arranged in the form of a matrix with two rows and the sequence is multiplied on the left by A where A=[1234].

Explanation of Solution

Given Information:

Each letter in the alphabet is represented by a number, and the sequence of numbers is arranged in the form of a matrix with two rows and the sequence is multiplied on the left by A where A=[1234].

Formula used:

A matrix with m rows and n columns is of dimension m×n, where m and n are positive integers.

Calculation:

Consider the encoded message,

[336954126112720602959651494187]

Write as a matrix with two rows as below,

[335411202965416912627605914987]

To calculate the decoded message, multiply the inverse of the matrix A by the encoded message.

Thus, decoded message is [1234]1[335411202965416912627605914987].

Since, number of rows in [1234]1 is 2 and number of columns is 2. Thus, dimension of [1234]1 is 2×2.

Since, number of rows in [335411202965416912627605914987] is 2 and number of columns is 7. Thus, dimension of [335411202965416912627605914987] is 2×7.

Since, number of columns in [1234]1 is equal to the number of rows in [335411202965416912627605914987], the product [1234]1[335411202965416912627605914987] is defined

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Chapter 4 Solutions

Finite Mathematics
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In Exercises 1720, write the given system of...Ch. 4 - In Exercises 1720, write the given system of...Ch. 4 - In Exercises 2124, solve the game with the given...Ch. 4 - In Exercises 2124, solve the game with the given...Ch. 4 - In Exercises 2124, solve the game with the given...Ch. 4 - In Exercises 2124, solve the game with the given...Ch. 4 - In Exercises 2528, find the production vector X...Ch. 4 - In Exercises 2528, find the production vector X...Ch. 4 - In Exercises 2528, find the production vector X...Ch. 4 - In Exercises 2528, find the production vector X...Ch. 4 - It is now July 1, and online sales of romance,...Ch. 4 - It is now July 1, and online sales of romance,...Ch. 4 - It is now July 1, and online sales of romance,...Ch. 4 - It is now July 1, and online sales of romance,...Ch. 4 - It is now July 1, and online sales of romance,...Ch. 4 - It is now July 1, and online sales of romance,...Ch. 4 - Acting on a tip from Marjory Duffin, John OHagan...Ch. 4 - Acting on a tip from Marjory Duffin, John OHagan...Ch. 4 - Acting on a tip from Marjory Duffin, John OHagan...Ch. 4 - Acting on a tip from Marjory Duffin, John OHagan...Ch. 4 - OHaganBooks.com has two main...Ch. 4 - OHaganBooks.com has two main...Ch. 4 - OHaganBooks.com has two main...Ch. 4 - OHaganBooks.com has two main...Ch. 4 - Publisher Marjory Duffin reveals that...Ch. 4 - Publisher Marjory Duffin reveals that...Ch. 4 - Publisher Marjory Duffin reveals that...Ch. 4 - Publisher Marjory Duffin reveals that...Ch. 4 - Publisher Marjory Duffin reveals that...Ch. 4 - Publisher Marjory Duffin reveals that...Ch. 4 - Some of the books sold by OHaganBooks.com are...Ch. 4 - Some of the books sold by OHaganBooks.com are...Ch. 4 - Some of the books sold by OHaganBooks.com are...Ch. 4 - Some of the books sold by OHaganBooks.com are...

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The graph of x = 2 + 3t, y = 4 − t is a: circle ellipse line parabola

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th