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Stein’s Algorithm for finding GCD

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  • Difficulty Level : Hard
  • Last Updated : 04 Jul, 2022
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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) 

  1. If both a and b are 0, gcd is zero gcd(0, 0) = 0.
  2. gcd(a, 0) = a and gcd(0, b) = b because everything divides 0.
  3. 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.
  4. 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.
  5. 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
  6. 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.

Iterative Implementation

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

Python3




# 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.

C#




// 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

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
?>

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>

Output

Gcd of given numbers is 17

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

Recursive Implementation

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

Python3




# 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.

C#




// 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.

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
?>

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>

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.

This article is contributed by Aarti_Rathi and Rahul Agrawal. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.


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