# Perfect Secrecy and Message distributions

It is know that for a perfect secrecy scheme, for all m1,m2 and c: $Pr[C=c | M=m1] = Pr[C=c | M=m2]$

I also know that for all distributions of M, m1,m2 and c: $Pr[M=m1 | C=c] = Pr[M=m2 | C=c]$ Does not imply perfect secrecy (by counter example).

My question is, how does the "for all distributions of M" affect the answer?
1. If I removed the "for all distributions of M" would it imply perfect secrecy?
2. If I added the first condition "for all distributions of M" would it change anything?

• So you would replace the "for all" in the first question by an "exists"? And would you mind explicitly formulating the second part as a formula / statement? – SEJPM Nov 12 '16 at 14:56
• I do not understand what is the problem with the second statement... The difference is that I switched the inner variables, and added the sentence "for all distributions". I'm trying to understand if, without the sentence, the statement would imply perfect secrecy (just like the first one)? – Jjang Nov 12 '16 at 16:26