Perfect secrecy of affine cipher

Question

How can I show that the affine cipher has perfect secrecy if the key $(k,a)$, where $k\in\{1,3,5,7,9,11,15,17,19,21,23,25\}$?

I know to show perfect secrecy I need to show that $Pr(Y)=\sum_{e_k(x)=y}Pr(X)Pr(K)$.

So I am getting that $Pr(Y)=\frac{1}{12}\sum_{e_k(x)=y} Pr(X)$, but not sure how to calculate Pr(X)?

note: X is set of plaintexts and Y is set of ciphertexts, both have cardinality 26

Answer 1

Hint:

For PR x you will have 1 because you know it will be reciprocal of the cardinality of the key size.