Elsevier

Journal of Symbolic Computation

Volume 96, January–February 2020, Pages 22-61
Journal of Symbolic Computation

Deterministic polynomial factoring over finite fields: A uniform approach via P-schemes

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Abstract

We introduce a family of combinatorial objects called P-schemes, where P is a collection of subgroups of a finite group G. A P-scheme is a collection of partitions of right coset spaces H\G, indexed by HP, that satisfies a list of axioms. These objects generalize the classical notion of association schemes as well as m-schemes (Ivanyos et al., 2009).

We apply the theory of P-schemes to deterministic polynomial factoring over finite fields: suppose f˜(X)Z[X] and a prime number p are given, such that f(X):=f˜(X)modp factorizes into n=deg(f˜) distinct linear factors over the finite field Fp. We show that, assuming the generalized Riemann hypothesis (GRH), f(X) can be completely factorized in deterministic polynomial time if the Galois group G of f˜(X) is an almost simple primitive permutation group on the set of roots of f˜(X), and the socle of G is a subgroup of Sym(k) for k up to 2O(logn). This is the first deterministic polynomial-time factoring algorithm for primitive Galois groups of superpolynomial order.

We prove our result by developing a generic factoring algorithm and analyzing it using P-schemes. We also show that the main results achieved by known GRH-based deterministic polynomial factoring algorithms can be derived from our generic algorithm in a uniform way.

Finally, we investigate the schemes conjecture in Ivanyos et al. (2009), and formulate analogous conjectures associated with various families of permutation groups. We show that these conjectures form a hierarchy of relaxations of the original schemes conjecture, and their positive resolutions would imply deterministic polynomial-time factoring algorithms for various families of Galois groups under GRH.

MSC

primary
68W30
12Y05
13P05

Keywords

Polynomial factoring
Permutation group
Finite field
Algebraic combinatorics

Cited by (0)

1

Part of this work was done while the author was at Department of Computing and Mathematical Sciences, Caltech.