Volume 6, Issue 1

Abstract. We describe an algorithm that, on input of a recursive presentation P of a group, outputs a recursive presentation of a torsion-free quotient of P , isomorphic to P whenever P is itself torsion-free. Using this, we show the existence of a universal finitely presented torsion-free group; one into which all finitely presented torsion-free groups embed (first proved by Belegradek). We apply our techniques to show that recognising embeddability of finitely presented groups is Π20$\Pi ^{0}_{2}$-hard, Σ20$\Sigma ^{0}_{2}$-hard, and lies in Σ30$\Sigma ^{0}_{3}$. We also show that the sets of orders of torsion elements of finitely presented groups are precisely the Σ20$\Sigma ^{0}_{2}$ sets which are closed under taking factors.

Abstract. We consider the class of groups whose word problem is poly-context-free ; that is, an intersection of finitely many context-free languages. We show that any group which is virtually a finitely generated subgroup of a direct product of free groups has poly-context-free word problem, and conjecture that the converse also holds. We prove our conjecture for several classes of soluble groups, including metabelian groups and torsion-free soluble groups, and present progress towards resolving the conjecture for soluble groups in general. Some of the techniques introduced for proving languages not to be poly-context-free may be of independent interest.

Abstract. We propose a polynomial time quantum algorithm for solving the discrete logarithm problem (DLP) in matrices over finite group rings. The hardness of this problem was recently employed in the design of a key-exchange protocol proposed by D. Kahrobaei, C. Koupparis and V. Shpilrain [Groups Complex. Cryptol. 5 (2013), 97–115]. Our result implies that the Kahrobaei–Koupparis–Shpilrain protocol does not belong to the realm of post-quantum cryptography.

Abstract. We provide conditions under which the set of Rijndael-like functions considered as permutations of the state space and based on operations of the finite field GF (pk)${\mathrm {GF}(p^k)}$ (p≥2${p\ge 2}$) is not closed under functional composition. These conditions justify using a sequential multiple encryption to strengthen the Advanced Encryption Standard (AES), a Rijndael cipher with specific block sizes. In [Discrete Appl. Math. 156 (2008), 3139–3149], R. Sparr and R. Wernsdorf provided conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field GF (2k)${\mathrm {GF}(2^k)}$ is equal to the alternating group on the state space. In this paper we provide conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field GF (pk)${\mathrm {GF}(p^k)}$ (p≥2${p\ge 2}$) is equal to the symmetric group or the alternating group on the state space.

Abstract. This is the second paper in a series of four, where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Here, for an arbitrary group G of infinite words over an ordered abelian group Λ we construct a Λ-tree ΓG$\Gamma _G$ equipped with a free action of G . Moreover, we show that ΓG$\Gamma _G$ is a universal tree for G in the sense that it isometrically and equivariantly embeds into every Λ-tree equipped with a free G -action compatible with the original length function on G . Also, for a group G acting freely on a Λ-tree Γ we show how one can easily obtain an embedding of G into the set of reduced infinite words R(Λ,X)$R(\Lambda , X)$, where the alphabet X is obtained from the action G on Γ.