Binary Euclidean Algorithm

Objective

The Binary Euclidean Algorithm, also known as Stein’s algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers. Unlike the standard Euclidean algorithm, which relies on division, the binary version replaces division with more efficient bitwise operations such as shifts and subtractions. This makes it highly suitable for implementation in both hardware and low-level software.

Core Idea

$$\gcd(A, B) = \begin{cases} A & \text{ if } B = 0 \\ B & \text{ if } A = 0 \\ 2 \times \gcd \left(\frac{A}{2}, \frac{B}{2} \right) & \text{ if } A \equiv 0 \pmod 2 \land \ B \equiv 0 \pmod 2\\ \gcd \left(\frac{A}{2}, B \right) & \text{ if } A \equiv 0 \pmod 2 \land \ B \equiv 1 \pmod 2 \\ \gcd \left(A, \frac{B}{2} \right) & \text{ if } A \equiv 1 \pmod 2 \land \ B \equiv 0 \pmod 2 \\ \gcd \left(\frac{|A - B|}{2}, \min(A, B) \right) & \text{ if } A \equiv 1 \pmod 2 \land \ B \equiv 1 \pmod 2 \\ \end{cases}$$

$\gcd(A,B)$ is transformed according to the following rules:

1. If $B = 0$, then $\gcd(A, B) = A$

2. If $A = 0$, then $\gcd(A, B) = B$

3. If both $A$ and $B$ are even, then $\gcd(A, B) = 2 \times \gcd\left(\frac{A}{2}, \frac{B}{2}\right)$ because 2 is a common factor

4. If only $A$ is even, then $\gcd(A, B) = \gcd\left(\frac{A}{2}, B\right)$, because $B$ does not have a factor of 2

5. If only $B$ is even, then $\gcd(A, B) = \gcd\left(A, \frac{B}{2}\right)$, because $A$ does not have a factor of 2

6. If both $A$ and $B$ are odd, then $\gcd(A, B) = \gcd\left(\frac{|A - B|}{2}, \min(A, B)\right)$ because $\gcd(A, B) = \gcd(B, A)$, and if $A > B$, then $\gcd(A, B) = \gcd(A - B, B)$. Since $A−B$ is even, we divide it by 2.

Code

def binary_gcd(a, b):
    # Base case
    if a == 0: return b
    if b == 0: return a

    # Case: both even
    if a % 2 == 0 and b % 2 == 0:
        return 2 * binary_gcd(a // 2, b // 2)
    # Case: a even, b odd
    elif a % 2 == 0:
        return binary_gcd(a // 2, b)
    # Case: a odd, b even
    elif b % 2 == 0:
        return binary_gcd(a, b // 2)
    # Case: both odd
    elif a >= b:
        return binary_gcd((a - b) // 2, b)
    else:
        return binary_gcd((b - a) // 2, a)

Written by Ryan Kim of Fractalyze