= Solution
<Perfect secrecy> means that observing the <ciphertext> changes no <probability> concerning the <plaintext>. For every distribution of the message $X$, and every $x,c$ with $\Pr(C=c)>0$,
$$
\boxed{\Pr(X=x\mid C=c)=\Pr(X=x).}
$$
Equivalently, for every pair $x,x'$ the conditional <probability distributions> of the <ciphertext> are identical:
$$
\Pr(C=c\mid X=x)=\Pr(C=c\mid X=x').
$$
The randomness here comes from independent key generation and any <encryption> randomness. The claim is information theoretic: unlimited computation cannot extract information about $X$ from $C$ beyond what was known beforehand.
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