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