Run the following Python code that implements the product cipher example explained in class:

productcipher.py

import numpy as np

#confusion$-$diffusion example using product cipher

#plaintext message

m $=$ np$.$matrix$([1,0,0,1])$

print$(\text{'}$Plaintext $=\text{'},$m$)$

print$()$

#XOR part of product cipher

key$1=$ np$.$matrix$([1,0,1,0])$

c$1=$np$.$mod$($m$+$key$1,2)$

print$(\text{'}$Encrypt $1=\text{'},$c$1)$

print$()$

#permutation part of product cipher

key$2=$ np$.$matrix$([[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]])$

print$(\text{'}$Key $2=\text{'},$key$2)$

print$()$

#c$2$ is the ciphertext

c$2=$c$1*$key$2$

c$2.$astype$($int$)$

c$2=$np$.$mod$($c$2,2)$

print$(\text{'}$Encrypt $2=\text{'},$c$2)$

print$()$

#decrypt by by first inverting the pernutation

#and then inverting the exclusive XOR

Pinv$=$np$.$transpose$($key$2)$

c$1$hat$=$c$2*$Pinv

c$1$hat.astype$($int$)$

c$1$hat$=$np$.$mod$($c$1$hat,$2)$

print$()$

print$(\text{'}$P inverse$=\text{'},$Pinv$)$

print$(\text{'}$Decrypt $2=\text{'},$c$1$hat$)$

print$()$

m$1$hat$=$np$.$mod$($c$1$hat$+$key$1,2)$

print$(\text{'}$Decrypted ciphertext $=\text{'},$m$1$hat$)$

print$()$

a$.$ Explain what the permutation matrix is doing to the bits in the plaintext.

b$.$ Use the Python script productcipher.py to construct a table relating the ciphertext to every possible $4-$bit plaintext message. Is it the case that all plaintext messages are scrambled? Is there any flaw in the system would allow an attacker to determine a key?

c$.$ Choose a single $4-$bit plaintext message and, by hand, work out each step in the

encryption$/$decryption process to show how the product cipher works.

d$.$ Why is the product cipher a symmetric cipher?

e$.$ How is the product cipher similar to the Vernam cipher discussed in Unit $1?$ How does it differ from the Vernam cipher?