Given any probability distribution P r on the set $\{k ∈ \mathbb{Z}_{26} : \gcd(k, 26) = 1\}$. Show that the Affine Cipher achieves perfect secrecy if the key (k, a) is used with probability $Pr(k) ·(1/26)$.
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An encryption scheme is said to achieve perfect secrecy if for every probability distribution over M (the messsage space), every message m$\in$*M* and every ciphertext c$\in$C for which Pr[C = c]>0:
Pr[M = m | C = c] = Pr[M = m] or, equivalently, Pr[E(K,m)=c]=Pr[E(K,m')=c] where E is the encryption algorithm that encrypt the message M according to the key K.
So I think you should define/identify also the probability distribution over the message space(M) before starting to talking about perfect secrecy. However, I think the AffineCipher cannot achieve perfect secrecy except in particular cases in which the probaiblity distributions are too much flat and the message to be enciphered very short.
