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;
}
// 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
# 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
<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
// 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
?>
OutputGcd of given numbers is 17
Time Complexity: O(N*N)
Auxiliary Space: O(1)
[Expected Approach 2] Recursive Implementation - O(N*N) Time and O(N*N) 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;
}
// 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
# 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.
<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
// 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
?>
OutputGcd 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.
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