How to prove that permutation and substitution ciphers satisfy H(X) = H(Y) in Shanon Entropy?

Question

I was not able to mathematically prove that all permutation and substitution ciphers satisfy H(X)=H(Y) if we say that Y is the set of ciphertexts while X is the corresponding set of plaintexts in Shanon Entropy?

More generally, how is it possible to mathematically prove that Shannon entropy does not change when applying any bijective function to X?

Answer 1

For any one to one encryption mapping, which these ciphers are assumed to be, say $E:{\cal M} \rightarrow {\cal C}$ under whatever key, we have: $$ H(Y) =-\sum_{y \in {\cal C}}p(y)\log p(y)= -\sum_{y \in {\cal C}} p(E^{-1}(y)) \log p(E^{-1}(y)) $$ which can be rewritten as $$ H(Y)=-\sum_{x \in {\cal M}} p(\sigma(x)) \log p(\sigma(x)) $$ for some permutation $\sigma$ of the messages.

Note that the decryption mapping exists since $E$ is one to one.