The Affine cipher is a type of monoalphabetic substitution cipher, wherein each letter in an alphabet is mapped to its numeric equivalent, encrypted using a simple mathematical function, and converted back to a letter. The formula used means that each letter encrypts to one other letter, and back again, meaning the cipher is essentially a standard substitution cipher with a rule governing which letter goes to which.
The whole process relies on working modulo m (the length of the alphabet used). In the affine cipher, the letters of an alphabet of size m are first mapped to the integers in the range 0 … m-1.
The ‘key’ for the Affine cipher consists of 2 numbers, we’ll call them a and b. The following discussion assumes the use of a 26 character alphabet (m = 26). a should be chosen to be relatively prime to m (i.e. a should have no factors in common with m).
Encryption
It uses modular arithmetic to transform the integer that each plaintext letter corresponds to into another integer that correspond to a ciphertext letter. The encryption function for a single letter is
E ( x ) = ( a x + b ) mod m modulus m: size of the alphabet a and b: key of the cipher. a must be chosen such that a and m are coprime.
Decryption
In deciphering the ciphertext, we must perform the opposite (or inverse) functions on the ciphertext to retrieve the plaintext. Once again, the first step is to convert each of the ciphertext letters into their integer values. The decryption function is
D ( x ) = a^-1 ( x - b ) mod m a^-1 : modular multiplicative inverse of a modulo m. i.e., it satisfies the equation 1 = a a^-1 mod m .
To find a multiplicative inverse
We need to find a number x such that:
If we find the number x such that the equation is true, then x is the inverse of a, and we call it a^-1. The easiest way to solve this equation is to search each of the numbers 1 to 25, and see which one satisfies the equation.
[g,x,d] = gcd(a,m); % we can ignore g and d, we dont need them x = mod(x,m);
If you now multiply x and a and reduce the result (mod 26), you will get the answer 1. Remember, this is just the definition of an inverse i.e. if a*x = 1 (mod 26), then x is an inverse of a (and a is an inverse of x)
Example:
Implementation:
C++
//CPP program to illustrate Affine Cipher#include<bits/stdc++.h>using namespace std;//Key values of a and bconst int a = 17;const int b = 20;string encryptMessage(string msg){ ///Cipher Text initially empty string cipher = ""; for (int i = 0; i < msg.length(); i++) { // Avoid space to be encrypted if(msg[i]!=' ') /* applying encryption formula ( a x + b ) mod m {here x is msg[i] and m is 26} and added 'A' to bring it in range of ascii alphabet[ 65-90 | A-Z ] */ cipher = cipher + (char) ((((a * (msg[i]-'A') ) + b) % 26) + 'A'); else //else simply append space character cipher += msg[i]; } return cipher;}string decryptCipher(string cipher){ string msg = ""; int a_inv = 0; int flag = 0; //Find a^-1 (the multiplicative inverse of a //in the group of integers modulo m.) for (int i = 0; i < 26; i++) { flag = (a * i) % 26; //Check if (a*i)%26 == 1, //then i will be the multiplicative inverse of a if (flag == 1) { a_inv = i; } } for (int i = 0; i < cipher.length(); i++) { if(cipher[i]!=' ') /*Applying decryption formula a^-1 ( x - b ) mod m {here x is cipher[i] and m is 26} and added 'A' to bring it in range of ASCII alphabet[ 65-90 | A-Z ] */ msg = msg + (char) (((a_inv * ((cipher[i]+'A' - b)) % 26)) + 'A'); else //else simply append space character msg += cipher[i]; } return msg;}//Driver Programint main(void){ string msg = "AFFINE CIPHER"; //Calling encryption function string cipherText = encryptMessage(msg); cout << "Encrypted Message is : " << cipherText<<endl; //Calling Decryption function cout << "Decrypted Message is: " << decryptCipher(cipherText); return 0;} |
Java
// Java program to illustrate Affine Cipherclass GFG { // Key values of a and b static int a = 17; static int b = 20; static String encryptMessage(char[] msg) { /// Cipher Text initially empty String cipher = ""; for (int i = 0; i < msg.length; i++) { // Avoid space to be encrypted /* applying encryption formula ( a x + b ) mod m {here x is msg[i] and m is 26} and added 'A' to bring it in range of ascii alphabet[ 65-90 | A-Z ] */ if (msg[i] != ' ') { cipher = cipher + (char) ((((a * (msg[i] - 'A')) + b) % 26) + 'A'); } else // else simply append space character { cipher += msg[i]; } } return cipher; } static String decryptCipher(String cipher) { String msg = ""; int a_inv = 0; int flag = 0; //Find a^-1 (the multiplicative inverse of a //in the group of integers modulo m.) for (int i = 0; i < 26; i++) { flag = (a * i) % 26; // Check if (a*i)%26 == 1, // then i will be the multiplicative inverse of a if (flag == 1) { a_inv = i; } } for (int i = 0; i < cipher.length(); i++) { /*Applying decryption formula a^-1 ( x - b ) mod m {here x is cipher[i] and m is 26} and added 'A' to bring it in range of ASCII alphabet[ 65-90 | A-Z ] */ if (cipher.charAt(i) != ' ') { msg = msg + (char) (((a_inv * ((cipher.charAt(i) + 'A' - b)) % 26)) + 'A'); } else //else simply append space character { msg += cipher.charAt(i); } } return msg; } // Driver code public static void main(String[] args) { String msg = "AFFINE CIPHER"; // Calling encryption function String cipherText = encryptMessage(msg.toCharArray()); System.out.println("Encrypted Message is : " + cipherText); // Calling Decryption function System.out.println("Decrypted Message is: " + decryptCipher(cipherText)); }}// This code contributed by Rajput-Ji |
Python
# Implementation of Affine Cipher in Python# Extended Euclidean Algorithm for finding modular inverse# eg: modinv(7, 26) = 15def egcd(a, b): x,y, u,v = 0,1, 1,0 while a != 0: q, r = b//a, b%a m, n = x-u*q, y-v*q b,a, x,y, u,v = a,r, u,v, m,n gcd = b return gcd, x, ydef modinv(a, m): gcd, x, y = egcd(a, m) if gcd != 1: return None # modular inverse does not exist else: return x % m# affine cipher encryption function # returns the cipher textdef affine_encrypt(text, key): ''' C = (a*P + b) % 26 ''' return ''.join([ chr((( key[0]*(ord(t) - ord('A')) + key[1] ) % 26) + ord('A')) for t in text.upper().replace(' ', '') ])# affine cipher decryption function # returns original textdef affine_decrypt(cipher, key): ''' P = (a^-1 * (C - b)) % 26 ''' return ''.join([ chr((( modinv(key[0], 26)*(ord(c) - ord('A') - key[1])) % 26) + ord('A')) for c in cipher ])# Driver Code to test the above functionsdef main(): # declaring text and key text = 'AFFINE CIPHER' key = [17, 20] # calling encryption function affine_encrypted_text = affine_encrypt(text, key) print('Encrypted Text: {}'.format( affine_encrypted_text )) # calling decryption function print('Decrypted Text: {}'.format ( affine_decrypt(affine_encrypted_text, key) ))if __name__ == '__main__': main()# This code is contributed by# Bhushan Borole |
C#
// C# program to illustrate Affine Cipherusing System; class GFG { // Key values of a and b static int a = 17; static int b = 20; static String encryptMessage(char[] msg) { /// Cipher Text initially empty String cipher = ""; for (int i = 0; i < msg.Length; i++) { // Avoid space to be encrypted /* applying encryption formula ( a x + b ) mod m {here x is msg[i] and m is 26} and added 'A' to bring it in range of ascii alphabet[ 65-90 | A-Z ] */ if (msg[i] != ' ') { cipher = cipher + (char) ((((a * (msg[i] - 'A')) + b) % 26) + 'A'); } else // else simply append space character { cipher += msg[i]; } } return cipher; } static String decryptCipher(String cipher) { String msg = ""; int a_inv = 0; int flag = 0; //Find a^-1 (the multiplicative inverse of a //in the group of integers modulo m.) for (int i = 0; i < 26; i++) { flag = (a * i) % 26; // Check if (a*i)%26 == 1, // then i will be the multiplicative inverse of a if (flag == 1) { a_inv = i; } } for (int i = 0; i < cipher.Length; i++) { /*Applying decryption formula a^-1 ( x - b ) mod m {here x is cipher[i] and m is 26} and added 'A' to bring it in range of ASCII alphabet[ 65-90 | A-Z ] */ if (cipher[i] != ' ') { msg = msg + (char) (((a_inv * ((cipher[i] + 'A' - b)) % 26)) + 'A'); } else //else simply append space character { msg += cipher[i]; } } return msg; } // Driver code public static void Main(String[] args) { String msg = "AFFINE CIPHER"; // Calling encryption function String cipherText = encryptMessage(msg.ToCharArray()); Console.WriteLine("Encrypted Message is : " + cipherText); // Calling Decryption function Console.WriteLine("Decrypted Message is: " + decryptCipher(cipherText)); }}/* This code contributed by PrinciRaj1992 */ |
Javascript
//Javascript program to illustrate Affine Cipher//Key values of a and blet a = 17;let b = 20;function encryptMessage(msg){ ///Cipher Text initially empty let cipher = ""; for (let i = 0; i < msg.length; i++) { // Avoid space to be encrypted if(msg[i] !=' ') /* applying encryption formula ( a x + b ) mod m {here x is msg[i] and m is 26} and added 'A' to bring it in range of ascii alphabet[ 65-90 | A-Z ] */ cipher = cipher + String.fromCharCode((((a * (msg[i].charCodeAt(0)-65) ) + b) % 26) + 65); else //else simply append space character cipher += msg[i]; } return cipher;}function decryptCipher(cipher){ let msg = ""; let a_inv = 0; let flag = 0; //Find a^-1 (the multiplicative inverse of a //in the group of integers modulo m.) for (let i = 0; i < 26; i++) { flag = (a * i) % 26; //Check if (a*i)%26 == 1, //then i will be the multiplicative inverse of a if (flag == 1) { a_inv = i; } } for (let i = 0; i < cipher.length; i++) { if(cipher[i]!=' ') /*Applying decryption formula a^-1 ( x - b ) mod m {here x is cipher[i] and m is 26} and added 'A' to bring it in range of ASCII alphabet[ 65-90 | A-Z ] */ msg = msg + String.fromCharCode(((a_inv * ((cipher[i].charCodeAt(0)+65 - b)) % 26)) + 65); else //else simply append space character msg += cipher[i]; } return msg;}//Driver Programlet msg = "AFFINE CIPHER";//Calling encryption functionlet cipherText = encryptMessage(msg);console.log("Encrypted Message is : " + cipherText);//Calling Decryption functionconsole.log("Decrypted Message is: " + decryptCipher(cipherText));// The code is contributed by Arushi Jindal. |
Output
Encrypted Message is : UBBAHK CAPJKX Decrypted Message is: AFFINE CIPHER
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