This problem concerns the five$-$bit block cipher introduced in our

Module $3$ lecture notes.

$($a$)$ The ciphertext

$y=1111010001$

was encrypted using the key $(P,k)=(R{S}_3,25).$ Decrypt it$,$ showing

each step clearly as in our lecture notes $($e$.$g$.$ show how you find $k^{-1},$

the different stages of decryption, etc.$).$

$($b$)$ Given the key $(w,P,k)=(10010,R{S}_1,13),$ encrypt the plaintext

$x=10110110$

showing each step clearly.

$($c$)$ The ciphertext

$y=0011111001$

was encrypted using the key $(w,P,k)=(10010,R{S}_1,13).$ Decrypt it$,$

showing each step clearly.

Our Module $3$ lecture notes state that for the five$-$bit block cipher

described there with key $k=23,$ a five$-$bit plaintext and its corre$-$

sponding ciphertext do not always have the same leftmost binary digit.

Verify this by doing the following:

$($a$)$ Find an example of a five$-$bit plaintext $x$ whose leftmost digit is

$1$ and whose corresponding ciphertext $y$ has leftmost digit $0.$ State $x$

and $y.$

$($b$)$ Find an example of a five$-$bit plaintext $x$ whose leftmost digit is

$0$ and whose corresponding ciphertext $y$ has leftmost digit $1.$ State $x$

and $y.$

Suppose you are using the ten$-$bit version of the block cipher intro$-$

duced in our Module $3$ lecture notes with a key of the form $(w,P,k).$

If Oscar has seven PCs$,$ each of which can check fifty thousand keys

per second, find on average how long $($in days$)$ a brute force key search

will take him, showing your reasoning.