Volume 7, Issue 2

We give an algorithm which computes a presentation for a subgroup, denoted 𝒜ℳg,p,1${\mathcal {AM}_{g,p,1}}$, of the automorphism group of a free group. It is known that 𝒜ℳg,p,1${\mathcal {AM}_{g,p,1}}$ is isomorphic to the mapping class group of an orientable genus- g surface with p punctures and one boundary component. We define a variation of the Auter space.

Lattice rounding in Euclidean space can be viewed as finding the nearest point in the orbit of an action by a discrete group, relative to the norm inherited from the ambient space. Using this point of view, we initiate the study of non-abelian analogs of lattice rounding involving matrix groups. In one direction, we consider an algorithm for solving a normed word problem when the inputs are random products over a basis set, and give theoretical justification for its success. In another direction, we prove a general inapproximability result which essentially rules out strong approximation algorithms (i.e., whose approximation factors depend only on dimension) analogous to LLL in the general case.

The definition of graph automatic groups by Kharlampovich, Khoussainov and Miasnikov and its extension to 𝒞-graph automatic by Elder and the first author raise the question of whether Thompson's group F is graph automatic. We define a language of normal forms based on the combinatorial “caret types”, which arise when elements of F are considered as pairs of finite rooted binary trees. The language is accepted by a finite state machine with two counters, and forms the basis of a 3-counter graph automatic structure for the group.

We propose a novel probabilistic public-key encryption, based on the RSA cryptosystem. We prove that in contrast to the (standard model) RSA cryptosystem each user can choose his own encryption exponent from a more extensive set of positive integers than it can be done by the creator of the concrete RSA cryptosystem who chooses and distributes encryption keys among all users. Moreover, we show that the proposed encryption remains secure even in the case when the adversary knows the factors of the modulus n=pq${n=pq}$, where p and q are distinct primes. So, the security assumptions are stronger for the proposed encryption than for the RSA cryptosystem. More exactly, the adversary can break the proposed scheme if he can solve the general prime factorization problem for positive integers, in particular for the modulus n=pq${n=pq}$ and the Euler function ϕ(n)=(p-1)(q-1)${\varphi (n)=(p-1)(q-1)}$. In fact, the proposed encryption does not use any extra tools or functions compared to the RSA cryptosystem.

We give an algorithm to decide whether a given braid with four strings is a product of three factors which are conjugates of standard generators of the braid group. The algorithm is of polynomial time. It is based on the Garside theory. We give also a polynomial algorithm to decide if a given braid with any number of strings is a product of two factors which are conjugates of given powers of the standard generators (in my previous paper this problem was solved without polynomial estimates).

In several recent works of D. Kahrobaei, C. Koupparis, and V. Shpilrain, public-key protocols have been proposed which depend on the difficulty of computing discrete logarithms in matrix rings over group rings. In particular, the specific ring of 3×3 matrices over 𝔽7S5${\mathbb {F}_7S_5}$ has been proposed for use in some of these protocols. In this paper, we show that the discrete logarithm problem in this matrix ring can be solved on a modern PC in seconds, and we give a solution to the challenge problem over 𝔽2S5${\mathbb {F}_2S_5}$ proposed in one of the aforementioned works.

For the class of differentiable modulo p n functions, the numbers of all bijective and transitive functions are found. A recurrent formula for calculating inverse functions is constructed, and the bijectivity and transitivity conditions are formulated.

A generic-case approach to algorithmic problems was suggested by Myasnikov, Kapovich, Schupp and Shpilrain in 2003. This approach studies the behavior of an algorithm on typical inputs and ignores the rest of the inputs. In this paper we consider generic complexity of the searching graph isomorphism problem. We fit this problem in the frameworks of generic complexity and prove that its natural subproblem is generically hard provided that the searching graph isomorphism problem is hard in the worst case.

A semiring is an algebraic structure satisfying the usual axioms for a not necessarily commutative ring, but without the requirement that addition be invertible. Aside from rings, well-studied instances in cryptographic applications include the Boolean semiring and the tropical semiring . The latter, in particular, behaves to a large extent like a field and exhibits interesting properties in the cryptographic context. This short note explores a GPU-based highly parallel implementation of a protocol recently proposed by Grigoriev and Shpilrain [Comm. Algebra 42 (2014), 2624–2632], in the context of Diffie–Hellman key agreements.