Question

I’m trying to implement a simple RSA encryption/decryption process, and I’m pretty sure I’ve got the equations around the right way. Although it doesn’t seem to be printing out the correct decrypted value after the encryption. Any ideas?.

//test program
#include <iostream>
#include <string.h>
#include <math.h>
using namespace std;
int gcd(int a, int b);

int main(){
    char character = 'A'; //character that is to be encrypted


    int p = 7;
    int q = 5;
    int e = 0; // just initializing to 0, assigning actual e value in the 1st for loop 


    int n = p*q;
    int phi = (p-1)*(q-1);
    int d = 0; // " " 2nd for loop

    //---------------------------finding 'e' with phi. where "1 < e < phi(n)"
    for (int i=2; i < phi; i++){
        if (gcd(i,phi) == 1){ //if gcd is 1
            e = i;
            break;
        }
    }
    //----------------------------

    //---------------------------finding 'd' 

    for (int i = 2; i < phi; i++){
        int temp = (e*i)%phi;
        if (temp == 1){
            d = i;
            break;
        }
    }

    printf("n:%d , e:%d , phi:%d , d:%d \n",n,e,phi,d);
    printf("\npublic key is:[%d,%d]\n",e,n);
    printf("private key is:[%d,%d]\n",d,n);

    int m = static_cast<int>(character); //converting to a number
    printf("\nconverted character num:%d\n",m);


    //Encryption part  ie. c = m^e MOD n
    int power = pow(m,e); // m^e
    int c = power%n;      // c = m^e MOD n. ie. encrypted character
    printf("\n\nEncrypted character number:%d\n",c);

    //decryption part,  ie. m = c^d MOD n
    power = pow(c,d);
    int m2 = power%n; 
    printf("\n\ndecrypted character number:%d\n",m2);


    return 0;
}

int gcd(int a, int b){
    int r;
    if (a < 0) a = -a;
    if (b < 0) b = -b;
    if (b > a) { 
        r = b; b = a; a = r;
    }
    while (b > 0) {
        r = a % b;
        a = b;
        b = r;
    }
    return a;
}

(The prime numbers being used are 5 and 7, for the test)

Here I’m converting the character ‘A’ to its numeric value which is of course 65. When I encrypt this value using c = m^e MOD n (where m is the converted value, i.e. 65) it gives me c as 25.

Now, to reverse the process, I do m = c^d MOD n, which gives me m as 30 … which really isn’t correct because it should be 65, no?

Where exactly have I gone wrong?

[edit]

Is my calculation of d correct?

Answer 1

The encrypted message m must be less than n. You can’t use values larger than n, because the calculations are done modulo n. In your case m=65 and n=35. So you are actually getting the correct answer modulo n, because 65 % 35 == 30.