Find Square Root under Modulo p (Shanks Tonelli algorithm) in C++

In this problem, we are given two values n and a prime number p. Our task is to find Square Root under Modulo p.

Let's take an example to understand the problem,

Input : n = 4, p = 11
Output : 9

Solution Approach

Here, we will be using Tonelli-Shanks Algorithm.

Tonelli-Shanks Algorithm is used in modular arithmetic to solve for a value x in congruence of the form x2 = n (mod p).

The algorithm to find square root modulo using shank's Tonelli Algorithm −

Step 1 − Find the value of $(n^{((p-1)/2)})(mod\:p)$, if its value is p -1, then modular square root is not possible.

Step 2 − Then, we will use the value p - 1 as (s * 2^e). Where s is odd and positive and e is positive.

Step 3 − Calculate the value q^((p-1)/2)(mod p) = -1

Step 4 − use loop for m greater than 0 and update the value of x,

Find m such that b^(2^m) - 1(mod p) where 0 <= m <= r-1.

If M is 0, return x otherwise update values,

x = x * (g^(2 ^ (r - m - 1))
b = b * (g^(2 ^ (r - m))
g = (g^(2 ^ (r - m - 1))
r = m

Example

Program to illustrate the working of our solution,

#include <iostream>
#include <math.h>
using namespace std;
int powerMod(int base, int exponent, int modulus) {
   int result = 1;
   base = base % modulus;
   while (exponent > 0) {
      if (exponent % 2 == 1)
      result = (result * base)% modulus;
      exponent = exponent >> 1;
      base = (base * base) % modulus;
   }
   return result;
}
int gcd(int a, int b) {
   if (b == 0)
   return a;
   else
   return gcd(b, a % b);
}
int orderValues(int p, int b) {
   if (gcd(p, b) != 1) {
      return -1;
   }
   int k = 3;
   while (1) {
      if (powerMod(b, k, p) == 1)
      return k;
      k++;
   }
}
int findx2e(int x, int& e) {
   e = 0;
   while (x % 2 == 0) {
      x /= 2;
      e++;
   }
   return x;
}
int calcSquareRoot(int n, int p) {
   if (gcd(n, p) != 1) {
      return -1;
   }
   if (powerMod(n, (p - 1) / 2, p) == (p - 1)) {
      return -1;
   }
   int s, e;
   s = findx2e(p - 1, e);
   int q;
   for (q = 2; ; q++) {
      if (powerMod(q, (p - 1) / 2, p) == (p - 1))
      break;
   }
   int x = powerMod(n, (s + 1) / 2, p);
   int b = powerMod(n, s, p);
   int g = powerMod(q, s, p);
   int r = e;
   while (1) {
      int m;
      for (m = 0; m < r; m++) {
         if (orderValues(p, b) == -1)
         return -1;
         if (orderValues(p, b) == pow(2, m))
         break;
      }
      if (m == 0)
      return x;
      x = (x * powerMod(g, pow(2, r - m - 1), p)) % p;
      g = powerMod(g, pow(2, r - m), p);
      b = (b * g) % p;
      if (b == 1)
      return x;
      r = m;
   }
}
int main() {
   int n = 3;
   int p = 13;
   int sqrtVal = calcSquareRoot(n, p);
   if (sqrtVal == -1)
      cout<<"Modular square root is not exist";
   else
      cout<<"Modular square root of the number is "<<sqrtVal;
}

Output

Modular square root of the number is 9