Parallel Polynomial Operations on SMPs: an Overview

Abstract

SMP-based parallel algorithms and implementations for polynomial factoring and GCD are overviewed. Topics include polynomial factoring modulo small primes, univariate and multivariate p-adic lifting, and reformulation of lift basis. Sparse polynomial GCD is also covered.

Parallel sparse polynomial multiplication using heaps

“…Early implementations of parallel algorithms for polynomial multiplication include the work of Wang in [18] and Fitch and Norman [13]. For dense univariate polynomials f = a0 + a1x + ... + amx m and g = b0 + b1x + ... + bnx n Wang used the following formula to multiply f × g:…”

Parallel operations of sparse polynomials on multicores

“…As the d k could be computed independently [21], the outer loops of FMAsame (line 4) and FMAcst (line 2) could be easily parallelized. Only a synchronization barrier is required after the loop between the threads which process the body loop.…”

Parallel sparse polynomial division using heaps

“…We divide recursively to obtain a quotient term qi, then we subtract f := f − qig. The recursive coefficient operations could be performed in parallel as suggested by Wang in [15].…”

“…Our algorithm is asynchronous and does not wait between the computation of qi and qi+1. In [15], Wang suggests parallelizing the subtraction of qi · g and synchronizing after each new term of the quotient. No data is provided to assess the effectiveness of this approach but we believe the waiting would be a problem.…”