$696229$In class, we defined the notion of perfect secrecy for secret$-$key encryption schemes. We then

showed that the one$-$time pad satisfies this notion by proving a lemma stating the the ciphertexts in

the one$-$time pad are uniformly distributed. Specifically, we showed the following:

Lemma. For every message m in $\{0,1\}$

n and ciphertext c in $\{0,1\}$

n$,$ it holds that

Pr

k$\{0,1\}$n

$[$Enc$($k$,$ m$)=$ c$]=1$

$2$

n

$.$

We then showed that any scheme satisfying the above lemma is perfectly secret. However, we did not

discuss the converse statement.

Prove or refute the following: An encryption scheme with ciphertext space C is perfectly secret if and

only if the ciphertexts are uniformly distributed on C$.$