An affine cipher is encryption using a simple mathematical function. Consider the affine cipher $c_i=a{x}_i+b(mod26)$ where $c_i$ is the $i^{th}$ letter of the ciphertext, obtained by multiplying the $i^{th}$ letter $x_i$ of the plaintext by a fixed integer a modulo $26$ and adding another fixed integer b modulo $26$ such that $0a25,0b25.$ That is$,$ the function is applied to each letter of the plaintext separately, where each alphabet letter of the plaintext and the ciphertext is represented by a number modulo $26$ as below.

A B C DEFGHIJKLMNOPQR S T UVW $,$ I $,$ I $YZ$

a$.$ In modulo $26$ how many possible choices are there for the parameter a$,$ in order to uniquely decrypt an encrypted message? Why?

b$.$ Assume that a malicious adversary tries to break the cipher in part a$.$ That is$,$ he decrypts the message each time with a different key pair $(a,b)$ and checks if he obtains a plaintext that is meaningful. How many keys does he need to try? $($assume that the message is encrypted using a key pair $($a$,$b$)$ that yields to unique decryption$).$

c$.$ Among the pairs $(a,b)$ of the part b$,$ list the one$($s$)$ that encrypt the message to itself.

d$.$ Assuming the adversary has sufficient computational power, all ciphers can be broken. But not one$-$time pad, why?