components p and q can be obtained from the encryption key components a and b like the following:

p = a-1 , where aa-1 mod symbol size = 1
q = - b

The decryption operation, in this case, is:

D(x)
= p(x + q) mod symbol size
= a-1 (x – b) mod symbol size

Please note that:

D(E(x))
= D (ax + b) mod symbol size
= p(ax + b + q) mod symbol size
= a-1 (ax + b – b) mod symbol size
= a-1ax mod symbol size
= x (as aa-1 mod symbol size = 1)

How to implement the affine cipher using Python?

The affine cipher can be implemented using Python in the following way:

import math
from curses.ascii import isalpha


def getKey(symbol_size):

    while True:
        a = input("A: ")
        a = int(a)
        if math.gcd(a, symbol_size) != 1:
            print("gcd(A, 26) should be 1. Please try again.")
            continue
        else:
            break

    while True:
        b = input("B: ")
        b = int(b)
        if b > symbol_size:
            print("B should be less than 26. Please try again.")
            continue
        else:
            break

    k = a*symbol_size + b
    return k


def get_key_components(k, symbol_size):

    a = k // symbol_size
    b = k % symbol_size

    return a, b


def get_decryption_key(k, symbol_size):

    a, b = get_key_components(k, symbol_size)
    p = pow(a, -1, symbol_size)
    q = 0 - b

    return p, q


def encrypt(text, k, symbol_size):

    a, b = get_key_components(k, symbol_size)

    ctext = list()
    for i in range(len(text)):
        if isalpha(text[i]):
            s = (a * (ord(text[i]) - ord('a')) + b) % symbol_size
            ctext.append(chr(s + ord('a')))
        else:
            ctext.append(text[i])

    return "".join(ctext)


def decrypt(ctext, p, q, symbol_size):

    text = list()
    for i in range(len(ctext)):
        if isalpha(ctext[i]):
            s = (p * ((ord(ctext[i]) - ord('a')) + q)) % symbol_size
            text.append(chr(s + ord('a')))
        else:
            text.append(ctext[i])

    return "".join(text)


SYMBOL_SIZE = 26

key = getKey(SYMBOL_SIZE)
print("Your key is ", key)
plaintext = input("Plaintext: ")
ciphertext = encrypt(plaintext, key, SYMBOL_SIZE)
print(ciphertext)
P, Q = get_decryption_key(key, SYMBOL_SIZE)
decrypted_text = decrypt(ciphertext, P, Q, SYMBOL_SIZE)
print(decrypted_text)