1. Cryptopgraphy (a) We wish to encode a message using a simple cipher and matrix multiplication. For the cipher, we will assign to the letters of the alphabet A B C ... Z the numbers 1,..., 26 corresponding to their position in the alphabet and the number 27 represents a blank space. The idea is to write the message as a m × nmatrix and then use an invertible mxm matrix and matrix multiplication to encode and decode the message. For example, we may write "PEACE" as the 2 × 3 matrix M by using the cipher and then grouping the numbers into 2 × 1 columns, thus 16 1 5 M = 5 3 27 If we now take a 2 × 2 invertible matrix E as the encryption matrix, we may encrypt the message using matrix multiplication as EM. For example, if 4 E = 3 1 the message "PEACE", after the multiplication EM and 'unzipped' from the matrix becomes 54 53 26 34 42 The following message was encrypted using the matrix C where: 1 2 3 C = 2 5 3 1 0 8 107 193 133 108 183 156 76 91 177 136 236 187 95 172 118 Decode this message. You can use software or a site such as matrixcalc.org to perform the calcula- tions, but you must clearly show your procedure. 1 (b) Spy Headquarters sends a message to one of its agents using the encoding matrix The agent uses the matrix 2 -3 A = 3 3 5 B = 4 -1 to send messages to their informers. Find the encoding matrix that is consistent with this commu- nication pathway that enable Headquarters to send messages directly to the informers. Show all of your work.
1. Cryptopgraphy (a) We wish to encode a message using a simple cipher and matrix multiplication. For the cipher, we will assign to the letters of the alphabet A B C ... Z the numbers 1,..., 26 corresponding to their position in the alphabet and the number 27 represents a blank space. The idea is to write the message as a m × nmatrix and then use an invertible mxm matrix and matrix multiplication to encode and decode the message. For example, we may write "PEACE" as the 2 × 3 matrix M by using the cipher and then grouping the numbers into 2 × 1 columns, thus 16 1 5 M = 5 3 27 If we now take a 2 × 2 invertible matrix E as the encryption matrix, we may encrypt the message using matrix multiplication as EM. For example, if 4 E = 3 1 the message "PEACE", after the multiplication EM and 'unzipped' from the matrix becomes 54 53 26 34 42 The following message was encrypted using the matrix C where: 1 2 3 C = 2 5 3 1 0 8 107 193 133 108 183 156 76 91 177 136 236 187 95 172 118 Decode this message. You can use software or a site such as matrixcalc.org to perform the calcula- tions, but you must clearly show your procedure. 1 (b) Spy Headquarters sends a message to one of its agents using the encoding matrix The agent uses the matrix 2 -3 A = 3 3 5 B = 4 -1 to send messages to their informers. Find the encoding matrix that is consistent with this commu- nication pathway that enable Headquarters to send messages directly to the informers. Show all of your work.
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter12: Algebra Of Matrices
Section12.3: M X N Matrices
Problem 49PS
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Transcribed Image Text:1. Cryptopgraphy
(a) We wish to encode a message using a simple cipher and matrix multiplication. For the
cipher, we will assign to the letters of the alphabet A B C ... Z the numbers
1,..., 26
corresponding to their position in the alphabet and the number 27 represents a blank
space. The idea is to write the message as a m × nmatrix and then use an invertible
mxm matrix and matrix multiplication to encode and decode the message.
For example, we may write "PEACE" as the 2 × 3 matrix M by using the cipher
and then grouping the numbers into 2 × 1 columns, thus
16 1 5
M =
5 3 27
If we now take a 2 × 2 invertible matrix E as the encryption matrix, we may encrypt
the message using matrix multiplication as EM. For example, if
4
E =
3
1
the message "PEACE", after the multiplication EM and 'unzipped' from the matrix
becomes 54 53 26 34 42
The following message was encrypted using the matrix C where:
1 2 3
C =
2 5 3
1 0 8
107 193 133 108 183 156 76 91 177 136 236 187 95 172 118
Decode this message. You can use software or a site such as matrixcalc.org to perform the calcula-
tions, but you must clearly show your procedure.
1
(b) Spy Headquarters sends a message to one of its agents using the encoding matrix
The agent uses the matrix
2
-3
A =
3
3 5
B =
4 -1
to send messages to their informers. Find the encoding matrix that is consistent with this commu-
nication pathway that enable Headquarters to send messages directly to the informers. Show all of
your work.
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