Suppose $E^1$ and $E^2$ are two encryption methods. Let $K_1$ and $K_2$

be keys and consider the double encryption

$E_{K_1,K_2}(m)=E_{K_1}^1(E_{K_2}^2(m)).$

Suppose you know a plaintext$-$ciphertext pair. Show

how to perform a meet$-$in$-$the$-$middle attack on this

double encryption.

An affine encryption given by $x|x+(mod26)$

can be regarded as a double encryption, where one

encryption is multiplying the plaintext by $$ and the

other is a shift by $.$ Assume that you have a plaintext

and ciphertext that are long enough that $$ and $$ are

unique. Show that the meet$-$in$-$the$-$middle attack from

part $($a$)$ takes at most $38$ steps $($not including the

comparisons between the lists$).$ Note that this is much

faster than a brute force search through all $312$ keys.