Decrypting an Affine Cipher with Modulus - encryption

Question

I'm trying to decrypt the ciphertext vczkh which I know was encoded using an affine cipher with the equation 7x + 8(mod 26). This makes my decryption function p = (c – b) * a^-1 (mod 26) where b = 8, a = 7, c = number corresponding with cipher character starting from 0, and p is the same for plaintext. Since I can't have a fraction I calculated that 11 is congruent to 7 making my function p = (c - 8) * 11. Running this for all five letters gives me NMFWP but I know the answer is supposed to be NOVEL. I do not know what I'm doing wrong.

Answer 1

In order to decrypt and affine cipher given a and b you need to use Dk = a^-1(y-b) mod m where m depends in the cardinality of the alphabet you are currently using (English 26, Italian 21, ...), a^-1 = m-a and k = (a, b).
For instance, vczkh with a=7 and b=8 gets decrypted into nqlmh given a^-1 = m - a = 26 - 7 = 19
So for v, since v is at position 21 in the english alphabet:
v -> 19(21-8) mod 26 -> 247 mod 26 -> 13 that corresponds to n
Here's a Javascript script I wrote
//Getting Args from console
var args = {
"operation" : process.argv[2],
"a" : parseInt(process.argv[3]),
"b" : parseInt(process.argv[4]),
"word" : process.argv[5]
};
var encryptedWord = [];
var decryptedWord = [];
if(!args.operation || !args.a || !args.b || !args.word){
console.log("Arguments are missing, please, use: node \"encrypt/decrypt\" a b word");
return;
} else {
if(typeof args.a === 'number' || typeof args.b === 'number'){
if(typeof args.word !== 'string'){
console.log("Word must be a string");
return;
} else {
// If a and m are coprimes
if(gcdCalc(args.a, 26) === 1){
if(args.operation === "encrypt"){
encryptWord().then(function(encrWord){
console.log("Word "+args.word+" got encrypted into "+encrWord);
});
} else if(args.operation === "decrypt"){
decryptWord().then(function(decrWord){
console.log("Ciphetext "+args.word+" got decrypted into "+decrWord);
});
} else {
console.log("Invalid operation specified. Use encrypt or decrypt.");
return;
}
} else {
console.log("a "+args.a+ " and m 26 are not coprimes");
return;
}
}
} else {
console.log("You must assign an Integer number to a and b. Remember that a must be coprime with m (26)");
return;
}
}
function gcdCalc(a, b) {
if (b) {
return gcdCalc(b, a % b);
} else {
return Math.abs(a);
}
};
function encryptWord(){
return new Promise( function(resolve){
var chars = args.word.split("");
var currInt = 0;
var currEnc = "";
chars.forEach( function( currChar){
currInt = parseInt(currChar, 36) - 10;
// E(a,b)(n) = an + b mod 26
currEnc = mod((args.a * currInt + args.b), 26);
encryptedWord.push(String.fromCharCode(97 + currEnc));
});
return resolve(encryptedWord.join(""));
});
}
function decryptWord(){
return new Promise( function(resolve){
var chars = args.word.split("");
var currInt = 0;
var currEnc = "";
//a^-1 = m - a
var a_1 = 26 - args.a;
chars.forEach( function( currChar){
currInt = parseInt(currChar, 36) - 10;
// D(y) = a^-1 * (y - b) mod 26
currEnc = mod((a_1 * (currInt - args.b)), 26);
decryptedWord.push(String.fromCharCode(97 + currEnc));
});
return resolve(decryptedWord.join(""));
});
}
function mod(n, m) {
var remain = n % m;
return Math.floor(remain >= 0 ? remain : remain + m);
};
To run it you need node, then, you can use it:
Encrypt : node affine-cipher.js encrypt 5 8 affine that becomes ihhwvc
Decrypt : node affine-cipher.js decrypt 5 8 ihhwvc that becomes affine
Notice that a and m MUST be coprime. For instance gcd(a, m) MUST be 1.
The total number of possible keys is 312 since a may vary in 12 different numbers which are coprime with 26 and b may vary in all 26 different number, so 12*26=312.
Hope I've been helpful.