For any block cipher, the fact that it is a nonlinear function is crucial to its security. To see

this, suppose that we have a linear block cipher EnLinear that encrypts $256-$bit blocks of

plaintext into $256-$bit blocks of ciphertext. Let EnLinear$(k,m)$ denote the encryption of a

$256-$bit message $m$ under a key $k($the actual bit length of $k$ is irrelevant$).$ Thus,

EnLinear$(k,[m1o+m2])=$EnLinear$(k,m1)o+$EnLinear$(k,m2)$ for all $128-$bit patterns $m1,$

$m2.$

Describe how, with $256$ chosen ciphertexts, an adversary can decrypt any ciphertext

without knowledge of the secret key $k.($A "chosen ciphertext" means that an adversary

has the ability to choose a ciphertext and then obtain its decryption. Here, you have $256$

plaintext$/$ciphertext pairs to work with, and you have the ability to choose the value of the

ciphertexts. I answered the question below. Is my work correct or not?

$m=[m_1,m_2,$dots,$m_{256}]$

$c=[c_1,c_2,$dots,$c_{256}]$

$k=[k_1,k_2,$dots,$k_{256}]$

$m,c,$ and $k$ are represented as vectors

Given that EnLinear $(k,m_{1}o+m_2)=$EnLinear$(k,m_1)o+$EnLinear$(k,m_2)$ for all $m_1$ and $m_2,$

let $C$ be the ciphertext corresponding to the plaintext $P$ and EnLinear as the encryption

function using the key $k.$

We can also set $m_1=P^{\text{'}},m_2=P_1,m_1=P^{\text{'}\text{'}},m_2=P_2.$ This can represented as the

following:

EnLinear$(k,P^{\text{'}}o+P_1)=$EnLinear$(k,P^{\text{'}})o+$EnLinear$(k,P_1)$

EnLinear$(k,P^{\text{'}\text{'}}o+P_2)=$EnLinear$(k,P^{\text{'}\text{'}})o+$EnLinear$(k,P_2)$

Both equations can now be rearranged to isolate $E_k(k,P^{\text{'}})$ and $E_k(k,P^{\text{'}\text{'}})$ as the following:

EnLinear$(k,P^{\text{'}})=$EnLinear$(k,P^{\text{'}}o+P_1)o+$EnLinear$(k,P_1)$

EnLinear$($k$,P$

The adversary can then choose two plaintexts, $P_1$ and $P_2,$ that correspond to

ciphertexts, $C_1$ and $C_2$ as the following:

EnLinear $(k,P^{\text{'}}o+P_1)=$EnLinear$(k,P^{\text{'}})o+$EnLinear$(k,P_1)$

EnLinear $(k,P^{\text{'}\text{'}}o+P_2)=$EnLinear$(k,P^{\text{'}\text{'}})o+$EnLinear$(k,P_2)$

After XORing both of the above equations, the adversary would get the following

equation:

EnLinear $(k,P_1)o+$EnLinear$(k,P_2)=C_{1}o+C_2$