329,809 research outputs found
Random equations in nilpotent groups
In this paper we study satisfiability of random equations in an infinite
finitely generated nilpotent group G. We show that the set SAT(G,k) of all
equations in k > 1 variables over G which are satisfiable in G has an
intermediate asymptotic density in the space of all equations in k variables
over G. When G is a free abelian group of finite rank, we compute this density
precisely; otherwise we give some non-trivial upper and lower bounds. For k = 1
the set SAT(G,k) is negligible. Usually the asymptotic densities of interesting
sets in groups are either zero or one. The results of this paper provide new
examples of algebraically significant sets of intermediate asymptotic density.Comment: 25 page
Counting configuration-free sets in groups
© 2017 Elsevier Ltd. We provide asymptotic counting for the number of subsets of given size which are free of certain configurations in finite groups. Applications include sets without solutions to equations in non-abelian groups, and linear configurations in abelian groups defined from group homomorphisms. The results are obtained by combining the methodology of hypergraph containers joint with arithmetic removal lemmas. Random sparse versions and threshold probabilities for existence of configurations in sets of given density are presented as well.Postprint (updated version
Pseudo-Free Families of Finite Computational Elementary Abelian -Groups
Loosely speaking, a family of computational groups is a family of groups (where is a set of bit strings) whose elements are represented by bit strings in such a way that equality testing, multiplication, inversion, computing the identity element, and sampling random elements in can be performed efficiently when is given. A family of computational groups is called pseudo-free if, given a random index (for an arbitrary value of the security parameter) and random elements , it is computationally hard to find a system of group equations , , and elements such that this system of equations is unsatisfiable in the free group freely generated by (over variables ), but in for all . If a family of computational groups satisfies this definition with the additional requirement that , then this family is said to be weakly pseudo-free. The definition of a (weakly) pseudo-free family of computational groups can be easily generalized to the case when all groups in the family belong to a fixed variety of groups.
In this paper, we initiate the study of (weakly) pseudo-free families of computational elementary abelian -groups, where is an arbitrary fixed prime. We restrict ourselves to families of computational elementary abelian -groups such that for every index , each element of is represented by a single bit string of length polynomial in the length of .
First, we prove that pseudo-freeness and weak pseudo-freeness for families of computational elementary abelian -groups are equivalent. Second, we give some necessary and sufficient conditions for a family of computational elementary abelian -groups to be pseudo-free (provided that at least one of two additional conditions holds). These necessary and sufficient conditions are formulated in terms of collision-intractability or one-wayness of certain homomorphic families of knapsack functions. Third, we establish some necessary and sufficient conditions for the existence of pseudo-free families of computational elementary abelian -groups. With one exception, these conditions are the existence of certain homomorphic collision-intractable families of -ary hash functions or certain homomorphic one-way families of functions.
As an example, we construct a Diffie-Hellman-like key agreement protocol from an arbitrary family of computational elementary abelian -groups. Unfortunately, we do not know whether this protocol is secure under reasonable assumptions
On trivial words in finitely presented groups
We propose a numerical method for studying the cogrowth of finitely presented
groups. To validate our numerical results we compare them against the
corresponding data from groups whose cogrowth series are known exactly.
Further, we add to the set of such groups by finding the cogrowth series for
Baumslag-Solitar groups and prove
that their cogrowth rates are algebraic numbers.Comment: This article has been rewritten as two separate papers, with improved
exposition. The new papers are arXiv:1309.4184 and arXiv:1312.572
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