Stein’s Algorithm for finding GCD

04 Jul, 2022
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) 
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 evenRepeat 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.
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.
Iterative Implementation
C++
Java
Python3
C#
PHP
Javascript
// Iterative C++ program to// implement Stein's Algorithm#include <bits/stdc++.h>using namespace std; // Function to implement// Stein's Algorithmint 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 codeint main(){    int a = 34, b = 17;    printf("Gcd of given numbers is %d\n", gcd(a, b));    return 0;}
// Iterative Java program to// implement Stein's Algorithmimport 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
# 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 codea = 34b = 17 print("Gcd of given numbers is ", gcd(a, b)) # This code is contributed by Nikita Tiwari.
// Iterative C# program to implement// Stein's Algorithmusing 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// Iterative php program to// implement Stein's Algorithm // Function to implement// Stein's Algorithmfunction 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?>
<script> // Iterative JavaScript program to// implement Stein's Algorithm // Function to implement// Stein's Algorithmfunction 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>
Gcd of given numbers is 17
Time Complexity: O(N*N)Auxiliary Space: O(1)
Recursive Implementation
C++
Java
Python3
C#
PHP
Javascript
// Recursive C++ program to// implement Stein's Algorithm#include <bits/stdc++.h>using namespace std; // Function to implement// Stein's Algorithmint 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 codeint main(){    int a = 34, b = 17;    printf("Gcd of given numbers is %d\n", gcd(a, b));    return 0;}
// Recursive Java program to// implement Stein's Algorithmimport 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
# 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 codea, b = 34, 17print("Gcd of given numbers is ",      gcd(a, b)) # This code is contributed# by Nikita Tiwari.
// Recursive C# program to// implement Stein's Algorithmusing 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// Recursive PHP program to// implement Stein's Algorithm // Function to implement// Stein's Algorithmfunction 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?>
<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>
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.
nitin mittal
jit_t
sodiumcloridep
GauravRajput1
code_hunt
amartyaghoshgfg
germanshephered48
amansinghkushwaha2003
codewithrathi
bencivengadante
GCD-LCM
Mathematical
Mathematical
Writing code in comment?
Please use ide.geeksforgeeks.org,
generate link and share the link here.
Merge two sorted arrays
Operators in C / C++
Prime Numbers
Minimum number of jumps to reach end
Find minimum number of coins that make a given value
The Knight's tour problem | Backtracking-1
Algorithm to solve Rubik's Cube
Program for Decimal to Binary Conversion
Modulo 10^9+7 (1000000007)
Modulo Operator (%) in C/C++ with Examples