On the computational complexity of the abelian permutation group structure, membership and intersection problems

AbstractAlgorithms on computations on abelian permutation groups are presented here. An algorithm for computing the complete structure, algorithms for membership-inclusion testing and an algorithm for computing the intersection of abelian permutation groups are given. Their worst-case time complexity is a polynomial of degree 4 in terms of n, the number of points moved by the group. The upper bounds on the running time of the algorithms shown here improve the bounds on the above problems cited in the literature

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

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 H ∈ P, 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 pare given, such that f(X) :=f(X) modpfactorizes into n =deg(f)distinct linear factors over the finite field F_p. 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 Gis a subgroup of Sym(k)for kup to 2^O(√log n). 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 conjecturein 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

Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem

In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group.Comment: 10 page