246 research outputs found
Double coset problem for parabolic subgroups of braid groups
We provide the first solution to the double coset problem (DCP) for a large
class of natural subgroups of braid groups, namely for all parabolic subgroups
which have a connected associated Coxeter graph. Update: We succeeded to solve
the DCP for all parabolic subgroups of braid groups.Comment: 8 pages. Update remark adde
A new key exchange protocol based on the decomposition problem
In this paper we present a new key establishment protocol based on the
decomposition problem in non-commutative groups which is: given two elements
of the platform group and two subgroups (not
necessarily distinct), find elements such that . Here we introduce two new ideas that improve the security of key
establishment protocols based on the decomposition problem. In particular, we
conceal (i.e., do not publish explicitly) one of the subgroups , thus
introducing an additional computationally hard problem for the adversary,
namely, finding the centralizer of a given finitely generated subgroup.Comment: 7 page
Generalizing Boolean Satisfiability III: Implementation
This is the third of three papers describing ZAP, a satisfiability engine
that substantially generalizes existing tools while retaining the performance
characteristics of modern high-performance solvers. The fundamental idea
underlying ZAP is that many problems passed to such engines contain rich
internal structure that is obscured by the Boolean representation used; our
goal has been to define a representation in which this structure is apparent
and can be exploited to improve computational performance. The first paper
surveyed existing work that (knowingly or not) exploited problem structure to
improve the performance of satisfiability engines, and the second paper showed
that this structure could be understood in terms of groups of permutations
acting on individual clauses in any particular Boolean theory. We conclude the
series by discussing the techniques needed to implement our ideas, and by
reporting on their performance on a variety of problem instances
On conjugacy separability of fibre products
In this paper we study conjugacy separability of subdirect products of two
free (or hyperbolic) groups. We establish necessary and sufficient criteria and
apply them to fibre products to produce a finitely presented group in
which all finite index subgroups are conjugacy separable, but which has an
index overgroup that is not conjugacy separable. Conversely, we construct a
finitely presented group which has a non-conjugacy separable subgroup of
index such that every finite index normal overgroup of is conjugacy
separable. The normality of the overgroup is essential in the last example, as
such a group will always posses an index overgroup that is not
conjugacy separable.
Finally, we characterize -conjugacy separable subdirect products of two
free groups, where is a prime. We show that fibre products provide a
natural correspondence between residually finite -groups and -conjugacy
separable subdirect products of two non-abelian free groups. As a consequence,
we deduce that the open question about the existence of an infinite finitely
presented residually finite -group is equivalent to the question about the
existence of a finitely generated -conjugacy separable full subdirect
product of infinite index in the direct product of two free groups.Comment: v2: 38 pages; this is the version accepted for publicatio
The counting complexity of group-definable languages
AbstractA group family is a countable family B={Bn}n>0 of finite black-box groups, i.e., the elements of each group Bn are uniquely encoded as strings of uniform length (polynomial in n) and for each Bn the group operations are computable in time polynomial in n. In this paper we study the complexity of NP sets A which has the following property: the set of solutions for every x∈A is a subgroup (or is the right coset of a subgroup) of a group Bi(|x|) from a given group family B, where i is a polynomial. Such an NP set A is said to be defined over the group family B.Decision problems like Graph Automorphism, Graph Isomorphism, Group Intersection, Coset Intersection, and Group Factorization for permutation groups give natural examples of such NP sets defined over the group family of all permutation groups. We show that any such NP set defined over permutation groups is low for PP and C=P.As one of our main results we prove that NP sets defined over abelian black-box groups are low for PP. The proof of this result is based on the decomposition theorem for finite abelian groups. As an interesting consequence of this result we obtain new lowness results: Membership Testing, Group Intersection, Group Factorization, and some other problems for abelian black-box groups are low for PP and C=P.As regards the corresponding counting problem for NP sets over any group family of arbitrary black-box groups, we prove that exact counting of number of solutions is in FPAM. Consequently, none of these counting problems can be #P-complete unless PH collapses
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