Find Square Root under Modulo p | Set 2 (Shanks Tonelli algorithm)

Given a number ‘n’ and a prime ‘p’, find square root of n under modulo p if it exists.
Examples:

Input: n = 2, p = 113
Output: 62
62^2 = 3844  and  3844 % 113 = 2

Input:  n = 2, p = 7
Output: 3 or 4
3 and 4 both are square roots of 2 under modulo
7 because (3*3) % 7 = 2 and (4*4) % 7 = 2

Input:  n = 2, p = 5
Output: Square root doesn't exist

We have discussed Euler’s criterion to check if square root exists or not. We have also discussed a solution that works only when p is in form of 4*i + 3

In this post, Shank Tonelli’s algorithm is discussed that works for all types of inputs.

Algorithm steps to find modular square root using shank Tonelli’s algorithm :

1) Calculate n ^ ((p – 1) / 2) (mod p), it must be 1 or p-1, if it is p-1, then modular square root is not possible.

2) Then after write p-1 as (s * 2^e) for some integer s and e, where s must be an odd number and both s and e should be positive.

3) Then find a number q such that q ^ ((p – 1) / 2) (mod p) = -1

4) Initialize variable x, b, g and r by following values

   x = n ^ ((s + 1) / 2 (first guess of square root)
   b = n ^ s                
   g = q ^ s
   r = e   (exponent e will decrease after each updation)

5) Now loop until m > 0 and update value of x, which will be our final answer.

   Find least integer m such that b^(2^m) = 1(mod p)  and  0 <= m <= r – 1 
   If m = 0, then we found correct answer and return x as result
   Else update x, b, g, r as below
       x = x * g ^ (2 ^ (r – m - 1))
       b = b * g ^(2 ^ (r - m))
       g = g ^ (2 ^ (r - m))
       r = m

so if m becomes 0 or b becomes 1, we terminate and print the result. This loop guarantees to terminate because value of m is decreased each time after updation.

Following is the implementation of above algorithm.

C++

// C++ program to implement Shanks Tonelli algorithm for 
// finding Modular  Square Roots 
#include <bits/stdc++.h> 
using namespace std; 
  
//  utility function to find pow(base, exponent) % modulus 
int pow(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; 
} 
  
//  utility function to find gcd 
int gcd(int a, int b) 
{ 
    if (b == 0) 
        return a; 
    else
        return gcd(b, a % b); 
} 
  
//  Returns k such that b^k = 1 (mod p) 
int order(int p, int b) 
{ 
    if (gcd(p, b) != 1) 
    { 
        printf("p and b are not co-prime.\n"); 
        return -1; 
    } 
  
    //  Initializing k with first odd prime number 
    int k = 3; 
    while (1) 
    { 
        if (pow(b, k, p) == 1) 
            return k; 
        k++; 
    } 
} 
  
//  function return  p - 1 (= x argument) as  x * 2^e, 
//  where x will be odd  sending e as reference because 
//  updation is needed in actual e 
int convertx2e(int x, int& e) 
{ 
    e = 0; 
    while (x % 2 == 0) 
    { 
        x /= 2; 
        e++; 
    } 
    return x; 
} 
  
//  Main function for finding the modular square root 
int STonelli(int n, int p) 
{ 
    //  a and p should be coprime for finding the modular 
    // square root 
    if (gcd(n, p) != 1) 
    { 
        printf("a and p are not coprime\n"); 
        return -1; 
    } 
  
    //  If below expression return (p - 1)  then modular 
    // square root is not possible 
    if (pow(n, (p - 1) / 2, p) == (p - 1)) 
    { 
        printf("no sqrt possible\n"); 
        return -1; 
    } 
  
    //  expressing p - 1, in terms of s * 2^e,  where s 
    // is odd number 
    int s, e; 
    s = convertx2e(p - 1, e); 
  
    //  finding smallest q such that q ^ ((p - 1) / 2) 
    //  (mod p) = p - 1 
    int q; 
    for (q = 2; ; q++) 
    { 
        // q - 1 is in place of  (-1 % p) 
        if (pow(q, (p - 1) / 2, p) == (p - 1)) 
            break; 
    } 
  
    //  Initializing variable x, b and g 
    int x = pow(n, (s + 1) / 2, p); 
    int b = pow(n, s, p); 
    int g = pow(q, s, p); 
  
    int r = e; 
  
    // keep looping until b become 1 or m becomes 0 
    while (1) 
    { 
        int m; 
        for (m = 0; m < r; m++) 
        { 
            if (order(p, b) == -1) 
                return -1; 
  
            //  finding m such that b^ (2^m) = 1 
            if (order(p, b) == pow(2, m)) 
                break; 
        } 
        if (m == 0) 
            return x; 
  
        // updating value of x, g and b according to 
        // algorithm 
        x = (x * pow(g, pow(2, r - m - 1), p)) % p; 
        g = pow(g, pow(2, r - m), p); 
        b = (b * g) % p; 
  
        if (b == 1) 
            return x; 
        r = m; 
    } 
} 
  
//  driver program to test above function 
int main() 
{ 
    int n = 2; 
  
    // p should be prime 
    int p = 113; 
  
    int x = STonelli(n, p); 
  
    if (x == -1) 
        printf("Modular square root is not exist\n"); 
    else
        printf("Modular square root of %d and %d is %d\n", 
                n, p, x); 
} 

Java

// Java program to implement Shanks 
// Tonelli algorithm for finding  
// Modular Square Roots  
class GFG 
{ 
    static int z = 0; 
      
// utility function to find 
// pow(base, exponent) % modulus  
static int pow1(int base1,  
    int exponent, int modulus)  
{  
    int result = 1;  
    base1 = base1 % modulus;  
    while (exponent > 0)  
    {  
        if (exponent % 2 == 1)  
            result = (result * base1) % modulus;  
        exponent = exponent >> 1;  
        base1 = (base1 * base1) % modulus;  
    }  
    return result;  
}  
  
// utility function to find gcd  
static int gcd(int a, int b)  
{  
    if (b == 0)  
        return a;  
    else
        return gcd(b, a % b);  
}  
  
// Returns k such that b^k = 1 (mod p)  
static int order(int p, int b)  
{  
    if (gcd(p, b) != 1)  
    {  
        System.out.println("p and b are" +  
                            "not co-prime.");  
        return -1;  
    }  
  
    // Initializing k with first 
    // odd prime number  
    int k = 3;  
    while (true)  
    {  
        if (pow1(b, k, p) == 1)  
            return k;  
        k++;  
    }  
}  
  
// function return p - 1 (= x argument) 
// as x * 2^e, where x will be odd  
// sending e as reference because  
// updation is needed in actual e  
static int convertx2e(int x)  
{  
    z = 0; 
    while (x % 2 == 0)  
    {  
        x /= 2;  
        z++;  
    }  
    return x;  
}  
  
// Main function for finding  
// the modular square root  
static int STonelli(int n, int p)  
{  
    // a and p should be coprime for   
    // finding the modular square root  
    if (gcd(n, p) != 1)  
    {  
        System.out.println("a and p are not coprime");  
        return -1;  
    }  
  
    // If below expression return (p - 1) then modular  
    // square root is not possible  
    if (pow1(n, (p - 1) / 2, p) == (p - 1))  
    {  
        System.out.println("no sqrt possible");  
        return -1;  
    }  
  
    // expressing p - 1, in terms of   
    // s * 2^e, where s is odd number  
    int s, e;  
    s = convertx2e(p - 1); 
    e = z; 
  
    // finding smallest q such that q ^ ((p - 1) / 2)  
    // (mod p) = p - 1  
    int q;  
    for (q = 2; ; q++)  
    {  
        // q - 1 is in place of (-1 % p)  
        if (pow1(q, (p - 1) / 2, p) == (p - 1))  
            break;  
    }  
  
    // Initializing variable x, b and g  
    int x = pow1(n, (s + 1) / 2, p);  
    int b = pow1(n, s, p);  
    int g = pow1(q, s, p);  
  
    int r = e;  
  
    // keep looping until b  
    // become 1 or m becomes 0  
    while (true)  
    {  
        int m;  
        for (m = 0; m < r; m++)  
        {  
            if (order(p, b) == -1)  
                return -1;  
  
            // finding m such that b^ (2^m) = 1  
            if (order(p, b) == Math.pow(2, m))  
                break;  
        }  
        if (m == 0)  
            return x;  
  
        // updating value of x, g and b 
        // according to algorithm  
        x = (x * pow1(g, (int)Math.pow(2,  
                            r - m - 1), p)) % p;  
        g = pow1(g, (int)Math.pow(2, r - m), p);  
        b = (b * g) % p;  
  
        if (b == 1)  
            return x;  
        r = m;  
    }  
}  
  
// Driver code  
public static void main (String[] args)  
{ 
  
    int n = 2;  
  
    // p should be prime  
    int p = 113;  
  
    int x = STonelli(n, p);  
  
    if (x == -1) 
        System.out.println("Modular square" +  
                        "root is not exist\n");  
    else
        System.out.println("Modular square root of " + 
                            n + " and " + p + " is " + 
                            x + "\n");  
}  
} 
  
// This code is contributed by mits 

Python3

# Python3 program to implement Shanks Tonelli 
# algorithm for finding Modular Square Roots  
  
# utility function to find pow(base,  
# exponent) % modulus  
def pow1(base, exponent, modulus):  
  
    result = 1;  
    base = base % modulus;  
    while (exponent > 0):  
        if (exponent % 2 == 1): 
            result = (result * base) % modulus;  
        exponent = int(exponent) >> 1;  
        base = (base * base) % modulus;  
  
    return result;  
  
# utility function to find gcd  
def gcd(a, b):  
    if (b == 0):  
        return a;  
    else: 
        return gcd(b, a % b);  
  
# Returns k such that b^k = 1 (mod p)  
def order(p, b):  
  
    if (gcd(p, b) != 1): 
        print("p and b are not co-prime.\n");  
        return -1;  
  
    # Initializing k with first  
    # odd prime number  
    k = 3;  
    while (True):  
        if (pow1(b, k, p) == 1):  
            return k;  
        k += 1;  
  
# function return p - 1 (= x argument) as  
# x * 2^e, where x will be odd sending e  
# as reference because updation is needed 
# in actual e  
def convertx2e(x): 
    z = 0;  
    while (x % 2 == 0): 
        x = x / 2;  
        z += 1;  
          
    return [x, z];  
  
# Main function for finding the 
# modular square root  
def STonelli(n, p):  
  
    # a and p should be coprime for 
    # finding the modular square root  
    if (gcd(n, p) != 1): 
        print("a and p are not coprime\n");  
        return -1;  
  
    # If below expression return (p - 1) then 
    # modular square root is not possible  
    if (pow1(n, (p - 1) / 2, p) == (p - 1)): 
        print("no sqrt possible\n");  
        return -1;  
  
    # expressing p - 1, in terms of s * 2^e,  
    # where s is odd number  
    ar = convertx2e(p - 1); 
    s = ar[0]; 
    e = ar[1]; 
  
    # finding smallest q such that  
    # q ^ ((p - 1) / 2) (mod p) = p - 1  
    q = 2;  
    while (True): 
          
        # q - 1 is in place of (-1 % p)  
        if (pow1(q, (p - 1) / 2, p) == (p - 1)):  
            break; 
        q += 1; 
  
    # Initializing variable x, b and g  
    x = pow1(n, (s + 1) / 2, p);  
    b = pow1(n, s, p);  
    g = pow1(q, s, p);  
  
    r = e;  
  
    # keep looping until b become  
    # 1 or m becomes 0  
    while (True): 
        m = 0;  
        while (m < r): 
            if (order(p, b) == -1):  
                return -1;  
  
            # finding m such that b^ (2^m) = 1  
            if (order(p, b) == pow(2, m)):  
                break; 
            m += 1; 
  
        if (m == 0):  
            return x;  
  
        # updating value of x, g and b  
        # according to algorithm  
        x = (x * pow1(g, pow(2, r - m - 1), p)) % p;  
        g = pow1(g, pow(2, r - m), p);  
        b = (b * g) % p;  
  
        if (b == 1):  
            return x;  
        r = m;  
  
# Driver Code 
n = 2;  
  
# p should be prime  
p = 113;  
  
x = STonelli(n, p);  
  
if (x == -1):  
    print("Modular square root is not exist\n");  
else: 
    print("Modular square root of", n,  
          "and", p, "is", x);  
          
# This code is contributed by mits 

C#

// C# program to implement Shanks 
// Tonelli algorithm for finding  
// Modular Square Roots  
using System; 
  
class GFG 
{ 
      
static int z=0; 
      
// utility function to find  
// pow(base, exponent) % modulus  
static int pow1(int base1,  
        int exponent, int modulus)  
{  
    int result = 1;  
    base1 = base1 % modulus;  
    while (exponent > 0)  
    {  
        if (exponent % 2 == 1)  
            result = (result * base1) % modulus;  
        exponent = exponent >> 1;  
        base1 = (base1 * base1) % modulus;  
    }  
    return result;  
}  
  
// utility function to find gcd  
static int gcd(int a, int b)  
{  
    if (b == 0)  
        return a;  
    else
        return gcd(b, a % b);  
}  
  
// Returns k such that b^k = 1 (mod p)  
static int order(int p, int b)  
{  
    if (gcd(p, b) != 1)  
    {  
        Console.WriteLine("p and b are" + 
                            "not co-prime.");  
        return -1;  
    }  
  
    // Initializing k with  
    // first odd prime number  
    int k = 3;  
    while (true)  
    {  
        if (pow1(b, k, p) == 1)  
            return k;  
        k++;  
    }  
}  
  
// function return p - 1 (= x argument)  
// as x * 2^e, where x will be odd sending 
// e as reference because updation is  
// needed in actual e  
static int convertx2e(int x)  
{  
    z = 0; 
    while (x % 2 == 0)  
    {  
        x /= 2;  
        z++;  
    }  
    return x;  
}  
  
// Main function for finding 
// the modular square root  
static int STonelli(int n, int p)  
{  
    // a and p should be coprime for  
    // finding the modular square root  
    if (gcd(n, p) != 1)  
    {  
        Console.WriteLine("a and p are not coprime");  
        return -1;  
    }  
  
    // If below expression return (p - 1) then  
    // modular square root is not possible  
    if (pow1(n, (p - 1) / 2, p) == (p - 1))  
    {  
        Console.WriteLine("no sqrt possible");  
        return -1;  
    }  
  
    // expressing p - 1, in terms of s * 2^e, 
    //  where s is odd number  
    int s, e;  
    s = convertx2e(p - 1); 
    e=z; 
  
    // finding smallest q such that q ^ ((p - 1) / 2)  
    // (mod p) = p - 1  
    int q;  
    for (q = 2; ; q++)  
    {  
        // q - 1 is in place of (-1 % p)  
        if (pow1(q, (p - 1) / 2, p) == (p - 1))  
            break;  
    }  
  
    // Initializing variable x, b and g  
    int x = pow1(n, (s + 1) / 2, p);  
    int b = pow1(n, s, p);  
    int g = pow1(q, s, p);  
  
    int r = e;  
  
    // keep looping until b become  
    // 1 or m becomes 0  
    while (true)  
    {  
        int m;  
        for (m = 0; m < r; m++)  
        {  
            if (order(p, b) == -1)  
                return -1;  
  
            // finding m such that b^ (2^m) = 1  
            if (order(p, b) == Math.Pow(2, m))  
                break;  
        }  
        if (m == 0)  
            return x;  
  
        // updating value of x, g and b   
        // according to algorithm  
        x = (x * pow1(g, (int)Math.Pow(2, r - m - 1), p)) % p;  
        g = pow1(g, (int)Math.Pow(2, r - m), p);  
        b = (b * g) % p;  
  
        if (b == 1)  
            return x;  
        r = m;  
    }  
}  
  
// Driver code  
static void Main()  
{  
    int n = 2;  
  
    // p should be prime  
    int p = 113;  
  
    int x = STonelli(n, p);  
  
    if (x == -1) 
        Console.WriteLine("Modular square root" + 
                            "is not exist\n");  
    else
        Console.WriteLine("Modular square root of" +  
                        "{0} and {1} is {2}\n", n, p, x);  
}  
} 
  
// This code is contributed by mits 

PHP

<?php 
// PHP program to implement Shanks Tonelli  
// algorithm for finding Modular Square Roots  
  
// utility function to find pow(base, 
// exponent) % modulus  
function pow1($base, $exponent, $modulus)  
{  
    $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;  
}  
  
// utility function to find gcd  
function gcd($a, $b)  
{  
    if ($b == 0)  
        return $a;  
    else
        return gcd($b, $a % $b);  
}  
  
// Returns k such that b^k = 1 (mod p)  
function order($p, $b)  
{  
    if (gcd($p, $b) != 1)  
    {  
        print("p and b are not co-prime.\n");  
        return -1;  
    }  
  
    // Initializing k with first 
    // odd prime number  
    $k = 3;  
    while (1)  
    {  
        if (pow1($b, $k, $p) == 1)  
            return $k;  
        $k++;  
    }  
}  
  
// function return p - 1 (= x argument) as 
// x * 2^e, where x will be odd sending e  
// as reference because updation is needed 
// in actual e  
function convertx2e($x, &$e)  
{  
    $e = 0;  
    while ($x % 2 == 0)  
    {  
        $x = (int)($x / 2);  
        $e++;  
    }  
    return $x;  
}  
  
// Main function for finding the  
// modular square root  
function STonelli($n, $p)  
{  
    // a and p should be coprime for  
    // finding the modular square root  
    if (gcd($n, $p) != 1)  
    {  
        print("a and p are not coprime\n");  
        return -1;  
    }  
  
    // If below expression return (p - 1) then  
    // modular square root is not possible  
    if (pow1($n, ($p - 1) / 2, $p) == ($p - 1))  
    {  
        printf("no sqrt possible\n");  
        return -1;  
    }  
  
    // expressing p - 1, in terms of s * 2^e,  
    // where s is odd number  
    $e = 0;  
    $s = convertx2e($p - 1, $e);  
  
    // finding smallest q such that  
    // q ^ ((p - 1) / 2) (mod p) = p - 1  
    $q = 2;  
    for (; ; $q++)  
    {  
        // q - 1 is in place of (-1 % p)  
        if (pow1($q, ($p - 1) / 2, $p) == ($p - 1))  
            break;  
    }  
  
    // Initializing variable x, b and g  
    $x = pow1($n, ($s + 1) / 2, $p);  
    $b = pow1($n, $s, $p);  
    $g = pow1($q, $s, $p);  
  
    $r = $e;  
  
    // keep looping until b become 
    // 1 or m becomes 0  
    while (1)  
    {  
        $m = 0;  
        for (; $m < $r; $m++)  
        {  
            if (order($p, $b) == -1)  
                return -1;  
  
            // finding m such that b^ (2^m) = 1  
            if (order($p, $b) == pow(2, $m))  
                break;  
        }  
        if ($m == 0)  
            return $x;  
  
        // updating value of x, g and b  
        // according to algorithm  
        $x = ($x * pow1($g, pow(2, $r - $m - 1), $p)) % $p;  
        $g = pow1($g, pow(2, $r - $m), $p);  
        $b = ($b * $g) % $p;  
  
        if ($b == 1)  
            return $x;  
        $r = $m;  
    }  
}  
  
// Driver Code 
$n = 2;  
  
// p should be prime  
$p = 113;  
  
$x = STonelli($n, $p);  
  
if ($x == -1)  
    print("Modular square root is not exist\n");  
else
    print("Modular square root of " .  
          "$n and $p is $x\n");  
      
// This code is contributed by mits 
?> 

Output :

Modular square root of 2 and 113 is 62

For more detail about above algorithm please visit :
http://cs.indstate.edu/~sgali1/Shanks_Tonelli.pdf

For detail of example (2, 113) see :
http://www.math.vt.edu/people/brown/class_homepages/shanks_tonelli.pdf

This article is contributed by Utkarsh Trivedi. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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