Problem One. Student received the following ciphertext which was encoded with a shift cipher:

GUCHQJVDLGSOHDVHGRKRPHZRUN

Perform an attack against the cipher based on a letter frequency count: How many letters do you

have to identify through a frequency count to recover the key? What is the cleartext?

Problem Two. Construct the addition and multiplication tables for Z$7.$

Problem Three. An obvious approach to increase the security of a symmetric algorithm is to

apply the same cipher twice, i$.$e$.$:

y $=$ ek$2($ek$1($x$))$

As is often the case in cryptography, things are very tricky and results are often different from

the expected and$/$ or desired ones. In this problem we show that a double encryption with the

affine cipher is only as secure as single encryption! Assume two affine ciphers ek$1=$ a$1$x $+$ b$1$ and

ek$2=$ a$2$x $+$ b$2.$

$1.$ Show that there is a single affine cipher ek$3=$ a$3$x $+$ b$3$ which performs exactly

$2.$ the same encryption $($and decryption$)$ as the combination ek$2($ek$1($x$)).$

$3.$ Find the values for a$3,$b$3$ when a$1=3,$ b$1=5$ and a$2=11,$ b$2=7.$

$4.$ For verification: $(1)$ encrypt the letter K first with ek$1$ and the result with ek$2,$ and $(2)$ encrypt

the letter K with ek$3.$

$5.$ Briefly describe what happens if an exhaustive key$-$search attack is applied to a doubleencrypted affine ciphertext. Is the effective key space increased?