Big numbers p-adic arithmetic: A parallel approach

Abstract

In the past years, great attention has been payed to approximated p-adic arithmetic expressed in the form of Hensel codes and several contributions have made this p-adic arithmetic really effective, just according to this new consideration of it. However it has been shown that the appliability of p-adic arithmetic is strongly constrained by the size of rational numbers which constitutes the output of the given computation. The key idea we like to present here, it is based on the intention of getting any advantage out the variation of the prime p, choosing it as a very large number. This choise suggests a definition of a new algorithm which will benefit from a parallel execution. Thus our aim is to perform “big” rational numbers arithmetic applying the so called g-adic approach, based on the theory of g-adic numbers. This paper describes a general schema of g-adic computation and then presents two algorithms to perform the inverse mapping together with the related complexity analysis.

This work has been partially supported by M.P.I.: Progetti di Manipolazione Algebrica, Calcolo Algebrico; and by C.N.R.: Progetto Strategico Matematica computazionale.