Affine ciphers:-

Let a be an integer between 1 and 26, and let x be the integer corresponding to one of the 26 letters of the alphabet. The corresponding linear cipher transforms the message x into the cipher text y by

For example, if a = 3 and the message is "d," the cipher text is y = 3 × 4 = 12. If the message is "k," the cipher text is y = 3 × 11 mod 26 = 33 mod 26 = 7. To be acceptable, the value of a must be invertible mod 26, such that the correspondence from x to y can be uniquely inverted. A case that does not have this property is a = 4, for then both x = 3 (for "c") and x = 16 (for "p") lead to y = 12. An a is invertible if it has no common factor, other than 1, with 26. Hence, 4 is not invertible since both 4 and 26 share the factor 2. (See section 13.5.)

(a) List the acceptable values of a.

(b) Decipher the following linear cipher text:

(c) An affine cipher is of the form y = ax+b mod 26, where both a and b are integers between 1 and 26, with a being one of the values in part (a). How many keys are there in affine ciphers?

(d) An affine cipher can be combined with a Vigenère cipher by fixing a but using k different values of b and cycling through these b values, letter by letter. How many keys are there in this compound cipher?