Problem $1$

Prove that the shift cipher for a single character is perfectly secure$.$ Particularly, consider the following private$-$key encryption scheme:

Message space: English alphabets lowercase letters a $-$ z$,$ represented by $0$ to $25,$ respectively. Ciphertext ${?}^{\text{I}}$ and secret key spaces: the same as the message space.

KeyGen: pick a random lower case letter, i$.$e$.,$ pick sk larr$[0,25]$ uniformly at random, and interpret the number as the english letter.

Enc$($sk$,m)$: given a message min$[0,25],$ output $c=m+$ sk mod$26.$

Dec$(sk,c)$ : given a ciphertext cin$[0,25],$ output $m=c-$skmod$26.$

Note: all lowercase letters and numbers from $0$ to $25$ are used interchangeably.

For this problem, you need to write $(1)$ the definition of "perfectly secure$"$ which you are going to use next, and $(2)$ a proof that the construction satisfies the definition. You don't need to prove correctness as that is very trivial.

Problem $1$