On the design and implementation of Brown's algorithm over the integers and number fields

Abstract

We study the design and implementation of the dense modular GCD algorithm of Brown applied to bivariate polynomial GCDs over the integers and number fields. We present an improved design of Brown's algorithm and compare it asymptotically with Brown's original algorithm, with GCD-HEU, the heuristic GCD algorithm, and with the EEZGCD algorithm. We also make an empirical comparison based on Maple implementations of the algorithms. Our findings show that a careful implementation of our improved version of Brown's algorithm is much better than the other algorithms in theory and in practice.