Constructing a pseudo-free family of finite computational groups under the general integer factoring intractability assumption

We provide a correct version of Remark 3.5 of the paper mentioned in the title. Also, we fix a typo in Remark 4.4 of that paper.

Ramp secret sharing with cheater identification in presence of rushing cheaters

Secret sharing allows one to share a piece of information among n participants in a way that only qualified subsets of participants can recover the secret whereas others cannot. Some of these participants involved may, however, want to forge their shares of the secret(s) in order to cheat other participants. Various cheater identifiable techniques have been devised in order to identify such cheaters in secret sharing schemes. On the other hand, Ramp secret sharing schemes are a practically efficient variant of usual secret sharing schemes with reduced share size and some loss in security. Ramp secret sharing schemes have many applications in secure information storage, information-theoretic private information retrieval and secret image sharing due to producing relatively smaller shares. However, to the best of our knowledge, there does not exist any cheater identifiable ramp secret sharing scheme. In this paper we define the security model for cheater identifiable ramp secret sharing schemes and provide two constructions for cheater identifiable ramp secret sharing schemes. In addition, the second construction is secure against rushing cheaters who are allowed to submit their shares during secret reconstruction after observing other participants’ responses in one round. Also, we do not make any computational assumptions for the cheaters, i.e., cheaters may be equipped with unlimited time and resources, yet, the cheating probability would be bounded above by a very small positive number.

Key agreement based on automaton groups

We suggest several automaton groups as platforms for Anshel–Anshel–Goldfeld key agreement metascheme. They include Grigorchuk and universal Grigorchuk groups, Hanoi 3-towers group, the Basilica group and a subgroup of the affine group {\mathrm{Aff}_{4}(\mathbb{Z})} .

Randomized nonlinear software-oriented MDS diffusion layers

MDS diffusion layers are critical components in the design of symmetric ciphers. In this paper, after introducing some new algebraic structures, we provide new MDS matrices over special types of R-modules. With the help of the proposed methodology, we have more flexibility in designing software-oriented diffusion layers. Most notably, we construct randomized and/or nonlinear MDS diffusion layers, based upon the presented theoretical results, and discuss the resistance of the presented diffusion layers against various kinds of cryptanalysis, compared with classical linear diffusion layers.

On group automorphisms in universal algebraic geometry

In this paper, we study group equations with occurrences of automorphisms. We describe equational domains in this class of equations. Moreover, we solve a number of open problem posed in universal algebraic geometry.

Garside theory and subsurfaces: Some examples in braid groups

Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomial in the length of the braids. In this paper we suggest a more precise bound: for rigid braids with N strands and of Garside length L, the sliding circuit set should have at most {C\cdot L^{N-2}} elements, for some constant C. We construct a family of braids which realise this potential worst case. Our example braids suggest that having a large sliding circuit set is a geometric property of braids, as our examples have multiple subsurfaces with large subsurface projection; thus they are “almost reducible” in multiple ways, and act on the curve graph with small translation distance.

Effective construction of covers of canonical Hom-diagrams for equations over torsion-free hyperbolic groups

We show that, given a finitely generated group G as the coordinate group of a finite system of equations over a torsion-free hyperbolic group Γ, there is an algorithm which constructs a cover of a canonical solution diagram. The diagram encodes all homomorphisms from G to Γ as compositions of factorizations through Γ-NTQ groups and canonical automorphisms of the corresponding NTQ-subgroups. We also give another characterization of Γ-limit groups as iterated generalized doubles over Γ.

An improved version of the AAG cryptographic protocol

An improved version of the Anshel–Anshel–Goldfeld (AAG) algebraic cryptographic key-exchange scheme, that is in particular resistant against the Tsaban linear span cryptanalysis, is established. Unlike the original version, that is based on the intractability of the simultaneous conjugacy search problem for the platform group, the proposed version is based on harder simultaneous membership-conjugacy search problems, and the membership problem needs to be solved for a subset of the platform group that can be easily and efficiently built to be very complicated and without any good structure. A number of other hard problems need to be solved first before start solving the simultaneous membership-conjugacy search problem to obtain the exchanged key.

Conjugacy search problem and the Andrews–Curtis conjecture

We develop new computational methods for studying potential counterexamples to the Andrews–Curtis conjecture, in particular, Akbulut–Kurby examples {\operatorname{AK}(n)}. We devise a number of algorithms in an attempt to disprove the most interesting counterexample {\operatorname{AK}(3)}. That includes an efficient implementation of the folding procedure for pseudo-conjugacy graphs, based on the original modification of a classic disjoint-set data structure. To improve metric properties of the search space (the set of balanced presentations of the trivial group), we introduce a new transformation, called an ACM-move, that generalizes the original Andrews–Curtis transformations and discuss details of a practical implementation. To reduce growth of the search space, we introduce a strong equivalence relation on balanced presentations and study the space modulo automorphisms of the underlying free group. We prove that automorphism moves can be applied to Akbulut–Kurby presentations. The improved technique allows us to enumerate balanced presentations AC-equivalent to {\operatorname{AK}(3)} with relations of lengths up to 20 (previous record was 17).