Question $2[10$ marks$]$

What is the average time it would take to brute force the given ciphertext ciphers when using a machine specified below?

a$.$ Monoalphabetic substitution cipher using a machine ${10}^{12}$ operations $\frac{?}{sec}(1$ mark$)$

b$.$ Vigenere cipher with $10$ letter keywords using a machine ${10}^{15}o\frac{p}{sec}(1$ mark$)$

c$.$ Playfair cipher $|$Mono alphabetic substitution cipher using a machine ${10}^{10}o\frac{p}{sec}(1$ mark$)$

d$.$ Monoalphabetic substitution cipher using a machine mark$)$

e$.$ Vigen$$re cipher with $8$ letter keywords using a machine ${10}^{12}o\frac{p}{sec}(1$ mark$)$

Assume each cipher uses the $26$ letters from the English alphabet. A generalization of the Caesar cipher is called Affine Cipher

$C=E([a,b],p)=(ap+b)mod26$

For the Affine Caesar cipher to have a one$-$to$-$one mapping, the multiplicative inverse of a$,$ or $a^{-1},$ in mod $26$ must exist.

a$)$ Explain what is meant by a one$-$to$-$one mapping for a cipher. $(1$ Mark$)$

b$)$ For $b=4$ and $a>3,$ what is a value of a for which the Affine Caesar cipher has a one$-$to$-$one mapping? $(1$ Mark$)$

c$)$ For $b=4$ and $a>3,$ what is a value of a for which the Affine Caesar cipher does not have a one$-$to$-$one mapping? $(1$ Mark$)$

d$)$ Using the syntax a$-1$ for the multiplicative inverse of a$,$ write an equation for the decryption operation of the Affine Caesar cipher. $(1$ Mark$)$

e$)$ Assume the Affine Caesar cipher is extended for an n$-$character alphabet, i$.$e$.$ instead of mod $26$ it is mod n $.$ Write an expression that gives the number of values of a for which a one$-$to$-$one mapping exists. Explain your reasoning, i$.$e$.$ why the expression is valid. $(1$ Mark$)$