Why does this tiny RSA implementation give wrong results? - c++

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.

Answer 2

It is caused by having m greater than or equal to n like #interjay already answered.
But I found another problem with your code, my gcc4.1.2 compiler output 24 for the encrypted value not 25. It is because you use pow() function and then convert the result (which is type double) to int that causes precision loss.
Don't use pow() function, instead use square and multiply modulo n algorithm to compute c = m^e MOD n and m = c^d MOD n. It is faster than pow() and you won't need to unsafely downcast the result to integer.