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Factoring high-degree polynomials over ${\mathbf F}_2$
with Niederreiter's algorithm on the IBM SP2

Author: Peter Roelse
Journal: Math. Comp. 68 (1999), 869-880
MSC (1991): Primary 11--04, 11T06, 11Y16.
MathSciNet review: 1604383
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Abstract: A $C$ implementation of Niederreiter's algorithm for factoring polynomials over ${\mathbf F}_2$ is described. The most time-consuming part of this algorithm, which consists of setting up and solving a certain system of linear equations, is performed in parallel. Once a basis for the solution space is found, all irreducible factors of the polynomial can be extracted by suitable $\gcd$-computations. For this purpose, asymptotically fast polynomial arithmetic algorithms are implemented. These include Karatsuba & Ofman multiplication, Cantor multiplication and Newton inversion. In addition, a new efficient version of the half-gcd algorithm is presented. Sequential run times for the polynomial arithmetic and parallel run times for the factorization are given. A new ``world record'' for polynomial factorization over the binary field is set by showing that a pseudo-randomly selected polynomial of degree 300000 can be factored in about 10 hours on 256 nodes of the IBM SP2 at the Cornell Theory Center.

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Additional Information

Peter Roelse
Affiliation: Institute for Experimental Mathematics, University of Essen, Ellernstrasse 29, 45326 Essen, Germany
Address at time of publication: Philips Crypto B.V., De Witbogt 2, 5652 AG Eindhoven, The Netherlands

Keywords: Finite fields, parallel computing, polynomial factorization
Received by editor(s): May 19, 1997
Received by editor(s) in revised form: August 11, 1997
Article copyright: © Copyright 1999 American Mathematical Society