**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 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 2.

**Iterative Implementation**

// 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; }

Output:

Gcd of given numbers is 17

**Recursive Implementation**

//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; }

Output:

Gcd of given numbers is 17

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

You may also like – Basic and Extended Euclidian Algorithm

**References**:

- https://en.wikipedia.org/wiki/Binary_GCD_algorithm
- http://andreinc.net/2010/12/12/binary-gcd-steins-algorithm-in-c/
- http://www.cse.unt.edu/~tarau/teaching/PP/NumberTheoretical/GCD/Binary%20GCD%20algorithm.pdf

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