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Published 2014 | Published
Journal Article Open

Deterministic polynomial factoring and association schemes

Abstract

The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q 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(loglog 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.

Additional Information

© 2014 The Author(s). Received 1 February 2013; revised 3 September 2013. This work was done while M.A. and N.S. were employed in the Hausdorff Center for Mathematics, Bonn (HCM). We would like to thank HCM and the Department of Computer Science, University of Bonn for its support. Especially, for hosting G.I. for a crucial part of the research, and for helping organize a related workshop on algebraic-combinatorial techniques. We thank Sergei Evdokimov, Akihide Hanaki, Mikhail Muzychuk, Ilya Ponomarenko, Igor Shparlinski and Paul-Hermann Zieschang for the many fruitful conversations. Especially, M.A. is grateful to Ilya for the numerous, still ongoing, discussions, explanations and pointers.

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