The Affine cipher (gets it name from the definition of an affine function which is a combination of a translation and scaling) is another example of a substitution cipher where each letter is replaced by another based on some rule.
The affine cipher key consists of a pair of integers (a, b) which is used to are used to form the equation ax+b and subsequently used to generate the mapping of each plaintext character to ciphertext character.
Take (3, 5) as an example. Note a must be a co-prime (have no common factors other than 1) to 26 otherwise different characters may map to the same letter and thus wouldn’t be reversible. In group theory this is because a needs to be cyclic generator of the group Z/26Z
(a) 3*0+5=5 (F) (d) 3*3+5=5 (O) (x) 3*23+5=74=5 (mod 22) (W) abcdefghijklmnopqrstuvwxyz FILORUXADGJMPSVYBEHKNQTWZC
For each letter in your plaintext you replace it it with its corresponding ciphertext letter.
when the clock strikes twelve attack TARS KAR LMVLJ HKEDJRH KTRMQR FKKFLJ
For each letter in your ciphertext you replace it it with its corresponding plaintext letter (the one above it in the mapping).
abcdefghijklmnopqrstuvwxyz - Key (9, 15) PYHQZIRAJSBKTCLUDMVENWFOXG FAZC EAZ HKLHB VEMJBZV EFZKWZ PEEPHB when the clock strikes twelve attack