This week my professor in class taught us the definition of perfect secrecy. He said that for any ciphertext the probability that it might have come from any message in the message space should be equal.
$\Pr_{k\leftarrow Gen}[Enc_k(m)=c]=\Pr_{k\leftarrow Gen}[Enc_k(m')=c]$
He then said to think of the possibility where we vary the ciphertext and not the message. I think what he said might be interpreted as that the probability that a message encrypts into a ciphertext should be equal for all ciphertexts.
$\Pr_{k\leftarrow Gen}[Enc_k(m)=c]=\Pr_{k\leftarrow Gen}[Enc_k(m)=c']$
I thought about this but haven't been able to relate it to the original definition. I wanted to prove the equivalence or get a better inference of what he said. I don't know if this is even true. So a counterexample might be more useful.
If I haven't written any detail please let me know.