We introduce a family of combinatorial objects called -schemes, where is a collection of subgroups of a finite group G. A -scheme is a collection of partitions of right coset spaces , indexed by , 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 -schemes to deterministic polynomial factoring over finite fields: suppose and a prime number p are given, such that factorizes into distinct linear factors over the finite field . We show that, assuming the generalized Riemann hypothesis (GRH), can be completely factorized in deterministic polynomial time if the Galois group G of is an almost simple primitive permutation group on the set of roots of , and the socle of G is a subgroup of for k up to . 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 -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.