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 min$\{0,1{\}}^n$ and ciphertext cin$\{0,1{\}}^n,$ it holds that

$P{r}_{\text{k}\text{l}\text{a}\text{r}\text{r}\{0,1{\}}^n}[Enc(k,m)=c]=\frac{1}{2^n}.$

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 $C.$