Rheinische Friedrich-Wilhelms-Universität Bonn Institut für Informatik
 
Abteilung V

 
Universität Bonn -> Institut für Informatik -> Abteilung V
CS-Reports 2012 Copyright 2012 Universität Bonn, Institut für Informatik, Abt. V
85329

25.05.2012

Deterministic Polynomial Factoring and Association Schemes
Manuel Arora, Gábor Ivanyos, Marek Karpinski and Nitin Saxena
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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(nr,log q) time; assuming GRH and that s=Ω(n/2r). 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.

Last Change: 05/25/12 at 13:36:31
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Universität Bonn -> Institut für Informatik -> Abteilung V

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