Further analysis of the binary Euclidean algorithm

TL;DR

This paper analyzes the binary Euclidean algorithm, correcting previous errors, discussing existing results, and providing numerical evidence supporting a conjecture about its average case behavior.

Contribution

It offers a corrected and detailed analysis of the binary Euclidean algorithm's average case, including numerical support for Vallée's conjecture.

Findings

Corrected errors in existing literature

Discussed Vallée's results on the algorithm

Numerical computations support Vallée's conjecture

Abstract

The binary Euclidean algorithm is a variant of the classical Euclidean algorithm. It avoids multiplications and divisions, except by powers of two, so is potentially faster than the classical algorithm on a binary machine. We describe the binary algorithm and consider its average case behaviour. In particular, we correct some errors in the literature, discuss some results of Vall\'ee, and describe a numerical computation which supports a conjecture of Vall\'ee.