Parallel univariate polynomial factorization on shared-memory multiprocessors

Abstract

Using parallelism afforded by shared-memory multiprocessors to speed up systems for polynomial factorization is discussed. The approach is to take the fastest known factoring algorithm for practical purposes and parallelize key parts of it. The univariate factoring algorithm consists of two major tasks (a) factoring modulo small integer primes and (b) EEZ lifting and recovery of true factors. A C coded system PFACTOR that implements (a) in parallel is described in detail. PFAC-TOR is a stand-alone parallel factorizer that can take input from a file, a pipe or a socket connection over a network. It can also be used interactively as a UNIX command. PFACTOR consists of parallel selection of primes, automatic balancing of work, parallel Berlekamp algorithm, and parallel reconciliation of degrees of factors modulo different primes. Actual timings on the Encore Multimax show the effectiveness of the approach. Work on part (b) is still on going.

Parallel Polynomial Operations on SMPs: an Overview

“…Experiences and timing data obtained in, Wang (1990Wang ( , 1992Wang ( , 1994 as well as other reports indicate that SMP-based implementations produce good speed up when np is small. As the number of processes gets larger the parallel programs often become less effective.…”

“…A C-coded system PFACTOR (Wang, 1990) implements factoring modulo small primes. PFACTOR is a stand-alone parallel factorizer that can take input from a file, a pipe or a socket connection over a network.…”

Factoring Polynomials Over Finite Fields: A Survey

“…We list some works that were not cited in the text: Tonelli (1891), Schwarz (1939Schwarz ( , 1940, Golomb et al (1959), Prange (1959), Schwarz (1960Schwarz ( , 1961, Lloyd (1964), Lloyd and Remmers (1966), Chien et al (1969), Prešić (1970), Agou (1976a,b), Adleman et al (1977), Agou (1977), Chen and Li (1977), Willett (1978), Agou (1980), Mignotte (1980), Camion (1981), Gunji and Arnon (1981), Camion (1982), Adleman and Lenstra (1986), Kaltofen (1987), Evdokimov (1989), Poli and Gennero (1989), Knopfmacher and Knopfmacher (1990), Lenstra (1990), Shoup (1990a), Wang (1990), Trevisan and Wang (1991), Evdokimov (1993), Knopfmacher and Knopfmacher (1993), Niederreiter and Göttfert (1993), Shparlinski (1993a,b), Davis (1994), Niederreiter (1994a,b), Knopfmacher (1995), Knopfmacher and Warlimont (1995), Niederreiter and Göttfert (1995), Fleischmann and Roelse (1996), Rónyai and Szánto (1996), Gao et al (1999), …”

Modular Algorithm for Sparse Multivariate Polynomial Interpolationand its Parallel Implementation

“…That algorithm is requested to work with multiple moduli, in order to expand parallelism in numeric evaluations and in the algorithm itself. In general, multi-modular computations are easy to parallelize (Wang 1990;Villard, 1989;Wang, 1992).…”