Problem$2.$ Recall that in a Feistel$-$network based block cipher, the round function $F(x,K_i)$ takes an input $x$ and a round key $K_i.$ Suppose that $x$ is $32-$bits and the Feistel network has $4$ rounds. Furthermore, suppose that all round keys are $32$ bits and the round function is defined as $F(x,K_i)=xo+K_i,$ where $o+$ denotes bit XOR. We assume that the key for the entire cipher is a concatenation of the $4$ round keys, i$.$e$.,$ the cipher key is $4*32=128$ bits long.

a$.$ Let the plaintext be $L_0||R_0,$ where $L_0$ and $R_0$ are $32-$bit blocks, and $||$ denotes concatenation. Let the key be $K_1||K_2||K_3||K_4.$ Let $L_i$ and $R_i$ be the output of the $i^{\text{'}}$ th round; then $L_4||R_4$ is the ciphertext. Write $L_4$ and $R_4$ in terms of $L_0,R_0,K_1,K_2,K_3,K_4.$

b$.$ Show that the resulting cipher is insecure again known$-$plaintext attack by describing an efficient algorithm that can decrypt any encrypted message give$$ one plaintext$/$ciphertext pair.

Hint: You do not have to recover the key completely to be able to decrypt encrypted messages.