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

Ivanyos, Gábor; Karpiński, Marek; Rónyai, Lajos; Saxena, Nitin

doi:10.1090/s0025-5718-2011-02505-6

Cited by 12 publications

(9 citation statements)

References 44 publications

(70 reference statements)

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“…We remark that this problem is open even assuming the generalized Riemann hypothesis (GRH) [37]. So far, the techniques used to answer this question are mostly algebraic.…”

Section: Open Problem 10 Find a Polynomial Time Deterministic Algoritmentioning

confidence: 99%

Open Problems for Polynomials over Finite Fields and Applications

Panario

2014

Open Problems in Mathematics and Computational Science

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We survey open problems for univariate polynomials over finite fields. We first comment in some detail on the existence and number of several classes of polynomials. The open problems in that part of the survey are of a more theoretical nature. Then, we center on classes of low-weight (irreducible) polynomials. The conjectures here are more practically oriented. Finally, we give brief descriptions of a selection of open problems from several areas including factorization of polynomials, special polynomials (APN functions, permutations), and relations between rational integers and polynomials.

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“…We remark that this problem is open even assuming the generalized Riemann hypothesis (GRH) [37]. So far, the techniques used to answer this question are mostly algebraic.…”

Section: Open Problem 10 Find a Polynomial Time Deterministic Algoritmentioning

confidence: 99%

Open Problems for Polynomials over Finite Fields and Applications

Panario

2014

Open Problems in Mathematics and Computational Science

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“…For the case K = F q the polynomial time f-algorithm given in [21] by the third author gives a solution. See also [13] for related deterministic methods. Recently the first and the third authors with Schicho [16] found an algorithm for solving the explicit isomorphism problem in the case of number fields.…”

Section: Introductionmentioning

confidence: 99%

Computing Explicit Isomorphisms with Full Matrix Algebras over $$\mathbb {F}_q(x)$$ F q ( x )

Ivanyos

Kutas²,

Rónyai

2017

Found Comput Math

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We propose a polynomial time f -algorithm (a deterministic algorithm which uses an oracle for factoring univariate polynomials over F q ) for computing an isomorphism (if there is any) of a finite dimensional F q (x)-algebra A given by structure constants with the algebra of n by n matrices with entries from F q (x). The method is based on computing a finite F q -subalgebra of A which is the intersection of a maximal F q [x]-order and a maximal R-order, where R is the subring of F q (x) consisting of fractions of polynomials with denominator having degree not less than that of the numerator.

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“…Building on Rónyai's ideas, Evdokimov [18] showed that an arbitrary degree n polynomial f (x) ∈ F q [x] can be factored deterministically in time poly(log q, n log n ) under GRH. This line of approach has since been investigated, in an attempt to either remove GRH [30] or improve the time complexity, leading to several analytic number theory, algebraic-combinatorial conjectures and special case solutions [13,23,31,46].…”

Section: Introductionmentioning

confidence: 99%

Deterministic polynomial factoring and association schemes

Arora¹,

Ivanyos²,

Karpiński³

et al. 2014

LMS J. Comput. Math.

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The problem of finding a nontrivial factor of a polynomial f (x) over a finite field Fq has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the generalized Riemann hypothesis (GRH). In this work we improve the state of the art by focusing on prime degree polynomials; let n be the degree. If (n − 1) has a 'large' r-smooth divisor s, then we find a nontrivial factor of f (x) in deterministic poly(n r , log q) time, assuming GRH and that s = Ω( n/2 r ). Thus, for r = O(1) our algorithm is polynomial time. Further, for r = Ω(log log n) there are infinitely many prime degrees n for which our algorithm is applicable and better than the best known, assuming GRH. Our methods build on the algebraic-combinatorial framework of m-schemes initiated by Ivanyos, Karpinski and Saxena (ISSAC 2009). We show that the m-scheme on n points, implicitly appearing in our factoring algorithm, has an exceptional structure, leading us to the improved time complexity. Our structure theorem proves the existence of small intersection numbers in any association scheme that has many relations, and roughly equal valencies and indistinguishing numbers.

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

Cited by 12 publications

References 44 publications

Open Problems for Polynomials over Finite Fields and Applications

Open Problems for Polynomials over Finite Fields and Applications

Computing Explicit Isomorphisms with Full Matrix Algebras over $$\mathbb {F}_q(x)$$ F q ( x )

Deterministic polynomial factoring and association schemes

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