Binary Euclidean Algorithm

Synonyms

Binary GCD algorithm; Stein’s algorithm

Related Concepts

Euclidean Algorithm; Modular Inverse Computation

Definition

The binary euclidean algorithm is a technique for computing the greatest common divisor and the euclidean coefficients of two nonnegative integers.

Background

The principles behind this algorithm were first published by R. Silver and J. Tersian, and independently by J. Stein [1]. Knuth claims [2] that the same algorithm may have been known in ancient China based on it’s appearance in verbal form in the first century A.D. text Nine Chapters on Arithmetic by Chiu Chang Suan Shu.

Theory

The binary GCD algorithm is based on the following observations on two arbitrary positive integers u and v:

• – If u and v are both even, then $\gcd (u,v) = 2\gcd (u/2,v/2)$;

• – If u is even and v is odd, then $\gcd (u,v) =\gcd (u/2,v)$;

• – Otherwise both are odd, and $\gcd (u,v) =\gcd (\vert u - v\vert /2,v)$.

The three conditions cover all possible cases for u and v. The Binary...