2.5.1: Application of Matrices in Cryptography (Exercises)

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SECTION 2.5 PROBLEM SET: APPLICATION OF MATRICES IN CRYPTOGRAPHY

In problems 1 - 8, the letters A to Z correspond to the numbers 1 to 26, as shown below, and a space is represented by the number 27.

\[\begin{array}{lllllllllllll}
\mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} & \mathrm{F} & \mathrm{G} & \mathrm{H} & \mathrm{I} & \mathrm{J} & \mathrm{K} & \mathrm{L} & \mathrm{M} \\
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\
\hline \mathrm{N} & \mathrm{O} & \mathrm{P} & \mathrm{Q} & \mathrm{R} & \mathrm{S} & \mathrm{T} & \mathrm{U} & \mathrm{V} & \mathrm{W} & \mathrm{X} & \mathrm{Y} & \mathrm{Z} \\
14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26
\end{array} \nonumber \]

In problems 1 - 2, use the matrix \(A\), given below, to encode the given messages.

\[A=\left[\begin{array}{ll}
3 & 2 \\
1 & 1
\end{array}\right] \nonumber \]

In problems 3 - 4, decode the messages that were encoded using matrix \(A\).
Make sure to consider the spaces between words, but ignore all punctuation. Add a final space if necessary.

Encode the message: WATCH OUT!

Encode the message: HELP IS ON THE WAY.

Decode the following message:

64 23 102 41 82 32 97 35 71 28 69 32

Decode the following message:

105 40 117 48 39 19 69 32 72 27 37 15 114 47

SECTION 2.5 PROBLEM SET: APPLICATION OF MATRICES IN CRYPTOGRAPHY

In problems 5 - 6, use the matrix \(B\), given below, to encode the given messages.

\[\mathrm{B}=\left[\begin{array}{lll}
1 & 0 & 0 \\
2 & 1 & 2 \\
1 & 0 & -1
\end{array}\right] \nonumber \]

In problems 7 - 8, decode the messages that were encoded using matrix \(B\).

Make sure to consider the spaces between words, but ignore all punctuation. Add a final space if necessary.

Encode the message using matrix \(B\):

LUCK IS ON YOUR SIDE.

Encode the message using matrix \(B\):

MAY THE FORCE BE WITH YOU.

Decode the following message that was encoded using matrix \(B\):

8 23 7 4 47 - 2 15 102 -12 20 58 15 27 80 18 12 74 -7

Decode the following message that was encoded using matrix \(B\):

12 69 - 3 11 53 9 5 46 -10 18 95 - 9 25 107 4 27 76 22 1 72 -26