Volume 1, Issue 2 - Special Issue: A Short Biography of Ben Fine Dedicated to Ben Fine on the occasion of his 60th birthday

After reviewing the importance of the Word Problem and the Conjugacy Problem for group-based cryptosystems, this paper offers an efficient method for solving both problems in the Shuffle Group.

The object of this paper is to give a proof of the following theorem: S / P ≅ Λ mn ⊆ ℚ + , where S / P is a certain torsion-free factor group of the Baumslag-Solitar group 〈 a, b ; a –1 b m a = b n | m ≠ 0, n ≠ 0, m, n ∈ ℤ〉, with m and n are relatively prime, and Λ mn is a subgroup of the additive group of the rational numbers ℚ + .

If two groups are residually-𝑃, their free product is not necessarily so; however, it is known that the free product of residually torsion-free nilpotent groups is again residually torsion-free nilpotent. In this paper it is shown that the free metabelian product of a free nilpotent group of class two with a free abelian group is residually torsion-free nilpotent.

Ben Fine observed that a theorem of Magnus on normal closures of elements in free groups is first order expressible and thus holds in every elementarily free group. This classical theorem, vintage 1931, asserts that if two elements in a free group have the same normal closure, then either they are conjugate or one is conjugate to the inverse of the other in the free group. An examination of a set of sentences capturing this theorem reveals that the sentences are universal-existential. Consequently the theorem holds in the almost locally free groups of Gaglione and Spellman. We give a sufficient condition for the theorem to hold in every fully residually free group as well as a sufficient condition for the theorem to hold, even more generally, in every residually free group.

We propose an authentication scheme where forgery (a.k.a. impersonation) seems infeasible without finding the prover's long-term private key. The latter would follow from solving the conjugacy search problem in the platform (noncommutative) semigroup, i.e., to recovering X from X –1 AX and A . The platform semigroup that we suggest here is the semigroup of n × n matrices over truncated multivariable polynomials over a ring.

A generalized tetrahedron group is defined to be a group admitting the following presentation: , 2 ≤ l, m, n, p, q, r , where each W i ( a, b ) is a cyclically reduced word involving both a and b . These groups appear in many contexts, not least as fundamental groups of certain hyperbolic orbifolds or as subgroups of generalized triangle groups. In this paper, we build on previous work to show that the Tits alternative holds for Tsaranov's generalized tetrahedron groups, that is, if G is a Tsaranov generalized tetrahedron group then G contains a non-abelian free subgroup or is solvable-by-finite. The term Tits alternative comes from the respective property for finitely generated linear groups over a field (see [Tits, J. Algebra 20: 250–270, 1972]).

A method for non-abelian Cramer-Shoup cryptosystem is presented. The role of decision and search is explored, and the platform of solvable / polycyclic group is suggested. In the process we review recent progress in non-abelian cryptography and post some open problems that naturally arise from this path of research.

It is an open problem whether the shifted conjugacy (decision) problem in B ∞ is solvable. We settle this problem by reduction to an instance of the simultaneous conjugacy problem in B n for some n ∈ ℕ.

This is the first in a two-part survey of current techniques in algebraic cryptanalysis. After introducing the basic setup of algebraic attacks and discussing several attack scenarios for symmetric cryptosystems, public key cryptosystems, and stream ciphers, we discuss a number of individual methods. The XL, XSL, and MutantXL attacks are based on linearization techniques for multivariate polynomial systems. Then we look at Gröbner basis and border bases methods. In the last section we introduce attacks based on integer programming techniques and try them in some concrete cases.

This paper is concerned with the question of determining which groups have their word problems lying in a given complexity class. Our main results give sufficient conditions for the existence of groups whose word problem is contained in some specified space complexity class but is not contained in some other given space complexity class.

In this paper we modify the technique of cyclic permutations to work with the shifted conjugacy problem. We apply this technique to design a heuristic attack on the cryptographic authentication scheme based on shifted conjugacy of braids proposed by Dehornoy in [Using shifted conjugacy in braid-based cryptography, American Mathematical Society, 2006] and report experimental results.

If R is a binomial ring, then a nilpotent R -powered group G is termed power-commutative if for any α ∈ R , [ g α , h ] = 1 implies [ g, h ] = 1 whenever g α ≠ 1. In this paper, we further contribute to the theory of nilpotent R -powered groups. In particular, we prove that if G is a nilpotent R -powered group of finite type which is not of finite π -type for any prime π ∈ R , then G is PC if and only if it is an abelian R -group.

We show that the universal theory of torsion groups is strongly contained in the universal theory of finite groups. This answers a question of Dyson. We also prove that the universal theory of some natural classes of torsion groups is undecidable. Finally we observe that the universal theory of the class of hyperbolic groups is undecidable and use this observation to construct a lacunary hyperbolic group with undecidable universal theory. Surprisingly, torsion groups play an important role in the proof of the latter results.