A geeralization of the Caesar cipher, known as the affine Caesar cipher, has the following form: for each plaintext letter p $($where p can be an integer between $0$ and $25$ inclusive$),$ substitute the ciphertext letter C: C $=$ E$([$a$,$ b$],$ p$)=($ap $+$ b$)$ mod $26$ A basic requirement of any encryption algorithm is that it needs to be one$-$to$-$one, i$.$e$.,$ if p$!=$q$,$ then E$($k$,$p$)!=$ E$($k$,$q$)$ where the key k $=[$a$,$b$].$ Otherwise, decryption is impossible, because more than one plaintext character maps into the same ciphertext character. The affine Caesar cipher is not one$-$to$-$one for all values of a$.$ For example, for a$=2$ and b$=3,$ then E$([$a$,$b$],0)=$ E$([$a$,$b$],13)=3.$

a$.$ Are there any limitations on the value of b for the affine Caesar cipher to be one$-$to$-$one? Explain

why or why not.

b$.$ What are the values of b providing distinct mappings for the affine Caesar cipher? Hint: Because

of the mod$-26$ operation, some b provide equal mapping outputs for the affine Caesar cipher.

c$.$ Determine which values of a are not allowed. Provide a general statement of which values of a

are and are not allowed. Justify your statement.

d$.$ How many keys are there for distinct affine Caesar cipher? Hint: Ordinary Caesar cipher has $26$

keys.

e$.$ A long English book has been encrypted using an affine Caesar cipher. The most frequent letter

of the ciphertext is $"$C$",$ and the second most frequent letter of the ciphertext is $"$D$".$ Break this

code to find the key.