Stein’s Algorithm for finding GCD

Last Updated : 12 Feb, 2025

Stein’s algorithm or binary GCD algorithm is an algorithm that computes the greatest common divisor of two non-negative integers. Stein’s algorithm replaces division with arithmetic shifts, comparisons, and subtraction.

Examples:

Input: a = 17, b = 34
Output : 17

Input: a = 50, b = 49
Output: 1

Algorithm to find GCD using Stein’s algorithm gcd(a, b)

The algorithm is mainly an optimization over standard Euclidean Algorithm for GCD

If both a and b are 0, gcd is zero gcd(0, 0) = 0.
gcd(a, 0) = a and gcd(0, b) = b because everything divides 0.
If a and b are both even, gcd(a, b) = 2*gcd(a/2, b/2) because 2 is a common divisor. Multiplication with 2 can be done with bitwise shift operator.
If a is even and b is odd, gcd(a, b) = gcd(a/2, b). Similarly, if a is odd and b is even, then
gcd(a, b) = gcd(a, b/2). It is because 2 is not a common divisor.
If both a and b are odd, then gcd(a, b) = gcd(|a-b|/2, b). Note that difference of two odd numbers is even
Repeat steps 3–5 until a = b, or until a = 0. In either case, the GCD is power(2, k) * b, where power(2, k) is 2 raise to the power of k and k is the number of common factors of 2 found in step 3.

C++

// Iterative C++ program to
// implement Stein's Algorithm
#include <bits/stdc++.h>
using namespace std;

// Function to implement
// Stein's Algorithm
int gcd(int a, int b)
{
    /* GCD(0, b) == b; GCD(a, 0) == a,
       GCD(0, 0) == 0 */
    if (a == 0)
        return b;
    if (b == 0)
        return a;

    /*Finding K, where K is the
      greatest power of 2
      that divides both a and b. */
    int k;
    for (k = 0; ((a | b) & 1) == 0; ++k) 
    {
        a >>= 1;
        b >>= 1;
    }

    /* Dividing a by 2 until a becomes odd */
    while ((a & 1) == 0)
        a >>= 1;

    /* From here on, 'a' is always odd. */
    do
    {
        /* If b is even, remove all factor of 2 in b */
        while ((b & 1) == 0)
            b >>= 1;

        /* Now a and b are both odd.
           Swap if necessary so a <= b,
           then set b = b - a (which is even).*/
        if (a > b)
            swap(a, b); // Swap u and v.

        b = (b - a);
    }while (b != 0);

    /* restore common factors of 2 */
    return a << k;
}

// Driver code
int main()
{
    int a = 34, b = 17;
    printf("Gcd of given numbers is %d\n", gcd(a, b));
    return 0;
}

Java

// Iterative Java program to
// implement Stein's Algorithm
import java.io.*;

class GFG {

    // Function to implement Stein's
    // Algorithm
    static int gcd(int a, int b)
    {
        // GCD(0, b) == b; GCD(a, 0) == a,
        // GCD(0, 0) == 0
        if (a == 0)
            return b;
        if (b == 0)
            return a;

        // Finding K, where K is the greatest
        // power of 2 that divides both a and b
        int k;
        for (k = 0; ((a | b) & 1) == 0; ++k) 
        {
            a >>= 1;
            b >>= 1;
        }

        // Dividing a by 2 until a becomes odd
        while ((a & 1) == 0)
            a >>= 1;

        // From here on, 'a' is always odd.
        do 
        {
            // If b is even, remove
            // all factor of 2 in b
            while ((b & 1) == 0)
                b >>= 1;

            // Now a and b are both odd. Swap
            // if necessary so a <= b, then set
            // b = b - a (which is even)
            if (a > b) 
            {
                // Swap u and v.
                int temp = a;
                a = b;
                b = temp;
            }

            b = (b - a);
        } while (b != 0);

        // restore common factors of 2
        return a << k;
    }

    // Driver code
    public static void main(String args[])
    {
        int a = 34, b = 17;

        System.out.println("Gcd of given "
                           + "numbers is " + gcd(a, b));
    }
}

// This code is contributed by Nikita Tiwari

Python

# Iterative Python 3 program to
# implement Stein's Algorithm

# Function to implement
# Stein's Algorithm


def gcd(a, b):

    # GCD(0, b) == b; GCD(a, 0) == a,
    # GCD(0, 0) == 0
    if (a == 0):
        return b

    if (b == 0):
        return a

    # Finding K, where K is the
    # greatest power of 2 that
    # divides both a and b.
    k = 0

    while(((a | b) & 1) == 0):
        a = a >> 1
        b = b >> 1
        k = k + 1

    # Dividing a by 2 until a becomes odd
    while ((a & 1) == 0):
        a = a >> 1

    # From here on, 'a' is always odd.
    while(b != 0):

        # If b is even, remove all
        # factor of 2 in b
        while ((b & 1) == 0):
            b = b >> 1

        # Now a and b are both odd. Swap if
        # necessary so a <= b, then set
        # b = b - a (which is even).
        if (a > b):

            # Swap u and v.
            temp = a
            a = b
            b = temp

        b = (b - a)

    # restore common factors of 2
    return (a << k)


# Driver code
a = 34
b = 17

print("Gcd of given numbers is ", gcd(a, b))

# This code is contributed by Nikita Tiwari.

// Iterative C# program to implement
// Stein's Algorithm
using System;

class GFG {

    // Function to implement Stein's
    // Algorithm
    static int gcd(int a, int b)
    {

        // GCD(0, b) == b; GCD(a, 0) == a,
        // GCD(0, 0) == 0
        if (a == 0)
            return b;
        if (b == 0)
            return a;

        // Finding K, where K is the greatest
        // power of 2 that divides both a and b
        int k;
        for (k = 0; ((a | b) & 1) == 0; ++k) 
        {
            a >>= 1;
            b >>= 1;
        }

        // Dividing a by 2 until a becomes odd
        while ((a & 1) == 0)
            a >>= 1;

        // From here on, 'a' is always odd
        do 
        {
            // If b is even, remove
            // all factor of 2 in b
            while ((b & 1) == 0)
                b >>= 1;

            /* Now a and b are both odd. Swap
            if necessary so a <= b, then set
            b = b - a (which is even).*/
            if (a > b) {

                // Swap u and v.
                int temp = a;
                a = b;
                b = temp;
            }

            b = (b - a);
        } while (b != 0);

        /* restore common factors of 2 */
        return a << k;
    }

    // Driver code
    public static void Main()
    {
        int a = 34, b = 17;

        Console.Write("Gcd of given "
                      + "numbers is " + gcd(a, b));
    }
}

// This code is contributed by nitin mittal

JavaScript

<script>

// Iterative JavaScript program to
// implement Stein's Algorithm

// Function to implement
// Stein's Algorithm
function gcd( a,  b)
{
    /* GCD(0, b) == b; GCD(a, 0) == a,
       GCD(0, 0) == 0 */
    if (a == 0)
        return b;
    if (b == 0)
        return a;

    /*Finding K, where K is the
      greatest power of 2
      that divides both a and b. */
    let k;
    for (k = 0; ((a | b) & 1) == 0; ++k) 
    {
        a >>= 1;
        b >>= 1;
    }

    /* Dividing a by 2 until a becomes odd */
    while ((a & 1) == 0)
        a >>= 1;

    /* From here on, 'a' is always odd. */
    do
    {
        /* If b is even, remove all factor of 2 in b */
        while ((b & 1) == 0)
            b >>= 1;

        /* Now a and b are both odd.
           Swap if necessary so a <= b,
           then set b = b - a (which is even).*/
        if (a > b){
        let t = a;
        a = b;
        b = t;
        }

        b = (b - a);
    }while (b != 0);

    /* restore common factors of 2 */
    return a << k;
}

// Driver code

    let a = 34, b = 17;
    document.write("Gcd of given numbers is "+ gcd(a, b));

// This code contributed by gauravrajput1 

</script>

PHP

<?php
// Iterative php program to 
// implement Stein's Algorithm

// Function to implement 
// Stein's Algorithm
function gcd($a, $b)
{
    // GCD(0, b) == b; GCD(a, 0) == a,
    // GCD(0, 0) == 0
    if ($a == 0)
        return $b;
    if ($b == 0)
        return $a;

    // Finding K, where K is the greatest
    // power of 2 that divides both a and b.
    $k;
    for ($k = 0; (($a | $b) & 1) == 0; ++$k)
    {
        $a >>= 1;
        $b >>= 1;
    }

    // Dividing a by 2 until a becomes odd 
    while (($a & 1) == 0)
        $a >>= 1;

    // From here on, 'a' is always odd.
    do
    {
        
        // If b is even, remove 
        // all factor of 2 in b 
        while (($b & 1) == 0)
            $b >>= 1;

        // Now a and b are both odd. Swap
        // if necessary so a <= b, then set 
        // b = b - a (which is even)
        if ($a > $b)
            swap($a, $b); // Swap u and v.

        $b = ($b - $a);
    } while ($b != 0);

    // restore common factors of 2
    return $a << $k;
}

// Driver code
$a = 34; $b = 17;
echo "Gcd of given numbers is " . 
                     gcd($a, $b);

// This code is contributed by ajit
?>

Output

Gcd of given numbers is 17

Time Complexity: O(N*N)
Auxiliary Space: O(1)

[Expected Approach 2] Recursive Implementation – O(NN) Time and O(NN) Space

C++

// Recursive C++ program to
// implement Stein's Algorithm
#include <bits/stdc++.h>
using namespace std;

// Function to implement
// Stein's Algorithm
int gcd(int a, int b)
{
    if (a == b)
        return a;

    // GCD(0, b) == b; GCD(a, 0) == a,
    // GCD(0, 0) == 0
    if (a == 0)
        return b;
    if (b == 0)
        return a;

    // look for factors of 2
    if (~a & 1) // a is even
    {
        if (b & 1) // b is odd
            return gcd(a >> 1, b);
        else // both a and b are even
            return gcd(a >> 1, b >> 1) << 1;
    }

    if (~b & 1) // a is odd, b is even
        return gcd(a, b >> 1);

    // reduce larger number
    if (a > b)
        return gcd((a - b) >> 1, b);

    return gcd((b - a) >> 1, a);
}

// Driver code
int main()
{
    int a = 34, b = 17;
    printf("Gcd of given numbers is %d\n", gcd(a, b));
    return 0;
}

Java

// Recursive Java program to
// implement Stein's Algorithm
import java.io.*;

class GFG {

    // Function to implement
    // Stein's Algorithm
    static int gcd(int a, int b)
    {
        if (a == b)
            return a;

        // GCD(0, b) == b; GCD(a, 0) == a,
        // GCD(0, 0) == 0
        if (a == 0)
            return b;
        if (b == 0)
            return a;

        // look for factors of 2
        if ((~a & 1) == 1) // a is even
        {
            if ((b & 1) == 1) // b is odd
                return gcd(a >> 1, b);

            else // both a and b are even
                return gcd(a >> 1, b >> 1) << 1;
        }

        // a is odd, b is even
        if ((~b & 1) == 1)
            return gcd(a, b >> 1);

        // reduce larger number
        if (a > b)
            return gcd((a - b) >> 1, b);

        return gcd((b - a) >> 1, a);
    }

    // Driver code
    public static void main(String args[])
    {
        int a = 34, b = 17;
        System.out.println("Gcd of given"
                           + "numbers is " + gcd(a, b));
    }
}

// This code is contributed by Nikita Tiwari

Python

# Recursive Python 3 program to
# implement Stein's Algorithm

# Function to implement
# Stein's Algorithm


def gcd(a, b):

    if (a == b):
        return a

    # GCD(0, b) == b; GCD(a, 0) == a,
    # GCD(0, 0) == 0
    if (a == 0):
        return b

    if (b == 0):
        return a

    # look for factors of 2
    # a is even
    if ((~a & 1) == 1):

        # b is odd
        if ((b & 1) == 1):
            return gcd(a >> 1, b)
        else:
            # both a and b are even
            return (gcd(a >> 1, b >> 1) << 1)

    # a is odd, b is even
    if ((~b & 1) == 1):
        return gcd(a, b >> 1)

    # reduce larger number
    if (a > b):
        return gcd((a - b) >> 1, b)

    return gcd((b - a) >> 1, a)


# Driver code
a, b = 34, 17
print("Gcd of given numbers is ",
      gcd(a, b))

# This code is contributed
# by Nikita Tiwari.

// Recursive C# program to
// implement Stein's Algorithm
using System;

class GFG {

    // Function to implement
    // Stein's Algorithm
    static int gcd(int a, int b)
    {
        if (a == b)
            return a;

        // GCD(0, b) == b;
        // GCD(a, 0) == a,
        // GCD(0, 0) == 0
        if (a == 0)
            return b;
        if (b == 0)
            return a;

        // look for factors of 2
        // a is even
        if ((~a & 1) == 1) {

            // b is odd
            if ((b & 1) == 1)
                return gcd(a >> 1, b);

            else

                // both a and b are even
                return gcd(a >> 1, b >> 1) << 1;
        }

        // a is odd, b is even
        if ((~b & 1) == 1)
            return gcd(a, b >> 1);

        // reduce larger number
        if (a > b)
            return gcd((a - b) >> 1, b);

        return gcd((b - a) >> 1, a);
    }

    // Driver code
    public static void Main()
    {
        int a = 34, b = 17;
        Console.Write("Gcd of given"
                      + "numbers is " + gcd(a, b));
    }
}

// This code is contributed by nitin mittal.

JavaScript

<script>

// JavaScript program to
// implement Stein's Algorithm

     // Function to implement
    // Stein's Algorithm
    function gcd(a, b)
    {
        if (a == b)
            return a;
 
        // GCD(0, b) == b; GCD(a, 0) == a,
        // GCD(0, 0) == 0
        if (a == 0)
            return b;
        if (b == 0)
            return a;
 
        // look for factors of 2
        if ((~a & 1) == 1) // a is even
        {
            if ((b & 1) == 1) // b is odd
                return gcd(a >> 1, b);
 
            else // both a and b are even
                return gcd(a >> 1, b >> 1) << 1;
        }
 
        // a is odd, b is even
        if ((~b & 1) == 1)
            return gcd(a, b >> 1);
 
        // reduce larger number
        if (a > b)
            return gcd((a - b) >> 1, b);
 
        return gcd((b - a) >> 1, a);
    }

// Driver Code

        let a = 34, b = 17;
        document.write("Gcd of given "
                           + "numbers is " + gcd(a, b));
                        
</script>

PHP

<?php
// Recursive PHP program to
// implement Stein's Algorithm

// Function to implement
// Stein's Algorithm
function gcd($a, $b)
{
    if ($a == $b)
        return $a;

    /* GCD(0, b) == b; GCD(a, 0) == a,
       GCD(0, 0) == 0 */
    if ($a == 0)
        return $b;
    if ($b == 0)
        return $a;

    // look for factors of 2
    if (~$a & 1) // a is even
    {
        if ($b & 1) // b is odd
            return gcd($a >> 1, $b);
        else // both a and b are even
            return gcd($a >> 1, $b >> 1) << 1;
    }

    if (~$b & 1) // a is odd, b is even
        return gcd($a, $b >> 1);

    // reduce larger number
    if ($a > $b)
        return gcd(($a - $b) >> 1, $b);

    return gcd(($b - $a) >> 1, $a);
}

// Driver code
$a = 34; $b = 17;
echo "Gcd of given numbers is: ", 
                     gcd($a, $b);

// This code is contributed by aj_36
?>

Output

Gcd of given numbers is 17

Time Complexity: O(N*N) where N is the number of bits in the larger number.
Auxiliary Space: O(N*N) where N is the number of bits in the larger number.

You may also like – Basic and Extended Euclidean Algorithm

Advantages over Euclid’s GCD Algorithm

Stein’s algorithm is optimized version of Euclid’s GCD Algorithm.
it is more efficient by using the bitwise shift operator.

Program to find LCM of two numbers

Aarti_Rathi and Rahul Agrawal

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Stein’s Algorithm for finding GCD

[Expected Approach 2] Recursive Implementation – O(N*N) Time and O(N*N) Space

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