1. Run the following Python code that implements the product cipher example given above:

import numpy as np

#confusion-diffusion example using product cipher

#plaintext message

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

print('Plaintext =',m)

print()

#XOR part of product cipher

key1 = np.matrix([1,0,1,0])

c1=np.mod(m+key1,2)

print('Encrypt 1=',c1)

print()

#permutation part of product cipher

key2 = np.matrix([[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]])

print('Key 2=',key2)

print()

#c2 is the ciphertext

c2=c1*key2

c2.astype(int)

c2=np.mod(c2,22)

print('Encrypt 2=',c2)

print()

#decrypt by by first inverting the pernutation

#and then inverting the exclusive XOR

Pinv=np.transpose(key2)

c1hat=c2*Pinv

c1hat.astype(int)

c1hat=np.mod(c1hat,2)

print()

print('P inverse=',Pinv)

print('Decrypt 2=',c1hat)

print()

m1hat=np.mod(c1hat+key1,2)

print('Decrypted ciphertext =',m1hat)

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?