Trading GRH for algebra: Algorithms for factoring polynomials and related structures

Gábor Ivanyos, Marek Karpinski, L. Rónyai, Nitin Saxena

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

In this paper we develop a general technique to eliminate the assumption of the Generalized Riemann Hypothesis (GRH) from various deterministic polynomial factoring algorithms over finite fields. It is the first bona fide progress on that issue for more than 25 years of study of the problem. Our main results are basically of the following form: either we construct a nontrivial factor of a given polynomial or compute a nontrivial automorphism of the factor algebra of the given polynomial. Probably the most notable application of such automorphisms is efficiently finding zero divisors in noncommutative algebras. The proof methods used in this paper exploit virtual roots of unity and lead to efficient actual polynomial factoring algorithms in special cases.

Original languageEnglish
Pages (from-to)493-531
Number of pages39
JournalMathematics of Computation
Volume81
Issue number277
DOIs
Publication statusPublished - 2011

Fingerprint

Riemann hypothesis
Factoring
Algebra
Polynomials
Polynomial
Noncommutative Algebra
Zero-divisor
Roots of Unity
Automorphism
Galois field
Automorphisms
Eliminate

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Applied Mathematics
  • Computational Mathematics

Cite this

Trading GRH for algebra : Algorithms for factoring polynomials and related structures. / Ivanyos, Gábor; Karpinski, Marek; Rónyai, L.; Saxena, Nitin.

In: Mathematics of Computation, Vol. 81, No. 277, 2011, p. 493-531.

Research output: Contribution to journalArticle

Ivanyos, Gábor ; Karpinski, Marek ; Rónyai, L. ; Saxena, Nitin. / Trading GRH for algebra : Algorithms for factoring polynomials and related structures. In: Mathematics of Computation. 2011 ; Vol. 81, No. 277. pp. 493-531.
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