Question #271560

Q1: Use the matrix

9 1

7 2 

(a)  Obtain the Hill cipher for the plaintext message

                                              PAK ARMY

by letting  A=1,B=2,C=3,.......,Y=25 and Z=26

(b)  Decode Hill 2- cipher which was encrypted by this matrix.


1
Expert's answer
2021-11-29T08:50:41-0500

To encrypt a message, each block of n letters (considered as an n-component vector) is multiplied by an invertible n × n matrix, against modulus 26. To decrypt the message, each block is multiplied by the inverse of the matrix used for encryption.


a)

PAK ARMY(PA),(KA),(RM),(YX)(161),(111),(1813),(2524)PAK\ ARMY\to \begin{pmatrix} P \\ A \end{pmatrix}, \begin{pmatrix} K \\ A \end{pmatrix}, \begin{pmatrix} R \\ M \end{pmatrix}, \begin{pmatrix} Y \\ X \end{pmatrix}\to\begin{pmatrix} 16 \\ 1 \end{pmatrix}, \begin{pmatrix} 11 \\ 1 \end{pmatrix}, \begin{pmatrix} 18 \\ 13 \end{pmatrix}, \begin{pmatrix} 25 \\ 24 \end{pmatrix}


(9172)(161)=(145114)=(1510)(mod 26)\begin{pmatrix} 9 & 1 \\ 7 & 2 \end{pmatrix}\begin{pmatrix} 16 \\ 1 \end{pmatrix}=\begin{pmatrix} 145 \\ 114 \end{pmatrix}=\begin{pmatrix} 15 \\ 10 \end{pmatrix}(mod\ 26)


(9172)(111)=(10079)=(221)(mod 26)\begin{pmatrix} 9 & 1 \\ 7 & 2 \end{pmatrix}\begin{pmatrix} 11 \\ 1 \end{pmatrix}=\begin{pmatrix} 100 \\ 79 \end{pmatrix}=\begin{pmatrix} 22 \\ 1 \end{pmatrix}(mod\ 26)


(9172)(1813)=(175152)=(1922)(mod 26)\begin{pmatrix} 9 & 1 \\ 7 & 2 \end{pmatrix}\begin{pmatrix} 18 \\ 13 \end{pmatrix}=\begin{pmatrix} 175 \\ 152 \end{pmatrix}=\begin{pmatrix} 19 \\ 22 \end{pmatrix}(mod\ 26)


(9172)(2524)=(249199)=(1517)(mod 26)\begin{pmatrix} 9 & 1 \\ 7 & 2 \end{pmatrix}\begin{pmatrix} 25 \\ 24 \end{pmatrix}=\begin{pmatrix} 249 \\ 199 \end{pmatrix}=\begin{pmatrix} 15 \\ 17 \end{pmatrix}(mod\ 26)


(1510),(221),(1922),(1517)(OJ),(VA),(RM),(XY)\begin{pmatrix} 15 \\ 10 \end{pmatrix}, \begin{pmatrix} 22 \\ 1 \end{pmatrix}, \begin{pmatrix} 19 \\ 22 \end{pmatrix}, \begin{pmatrix} 15 \\ 17 \end{pmatrix}\to \begin{pmatrix} O \\ J \end{pmatrix}, \begin{pmatrix} V \\ A \end{pmatrix}, \begin{pmatrix} R \\ M \end{pmatrix}, \begin{pmatrix} X \\ Y \end{pmatrix}


b)

(9172)1=111(2179)19(2179)=(12251915)(mod 26)\begin{pmatrix} 9 & 1 \\ 7 & 2 \end{pmatrix}^{-1}=\frac{1}{11}\begin{pmatrix} 2 & -1 \\ -7 & 9 \end{pmatrix}\equiv 19\begin{pmatrix} 2 & -1 \\ -7 & 9 \end{pmatrix}=\begin{pmatrix} 12 & 25 \\ 19 & 15 \end{pmatrix}(mod\ 26)


(12251915)(1510)=(430435)=(161)(mod 26)\begin{pmatrix} 12 & 25 \\ 19 & 15 \end{pmatrix}\begin{pmatrix} 15 \\ 10 \end{pmatrix}=\begin{pmatrix} 430 \\ 435 \end{pmatrix}=\begin{pmatrix} 16 \\ 1 \end{pmatrix}(mod\ 26)


(12251915)(221)=(279433)=(111)(mod 26)\begin{pmatrix} 12 & 25 \\ 19 & 15 \end{pmatrix}\begin{pmatrix} 22 \\ 1 \end{pmatrix}=\begin{pmatrix} 279 \\ 433 \end{pmatrix}=\begin{pmatrix} 11 \\ 1 \end{pmatrix}(mod\ 26)


(12251915)(1922)=(778691)=(1813)(mod 26)\begin{pmatrix} 12 & 25 \\ 19 & 15 \end{pmatrix}\begin{pmatrix} 19 \\ 22 \end{pmatrix}=\begin{pmatrix} 778 \\ 691 \end{pmatrix}=\begin{pmatrix} 18 \\ 13 \end{pmatrix}(mod\ 26)


(12251915)(1517)=(605540)=(2524)(mod 26)\begin{pmatrix} 12 & 25 \\ 19 & 15 \end{pmatrix}\begin{pmatrix} 15 \\ 17 \end{pmatrix}=\begin{pmatrix} 605 \\ 540 \end{pmatrix}=\begin{pmatrix} 25 \\ 24 \end{pmatrix}(mod\ 26)


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