Abstract: In [FGRS1,FGRS2] the relationship between the universal and elementary theoryof a group ring $R[G]$ and the corresponding universal and elementary theory ofthe associated group $G$ and ring $R$ was examined. Here we assume that $R$ isa commutative ring with identity $1 \ne 0$. Of course, these are relative to anappropriate logical language $L_0,L_1,L_2$ for groups, rings and group ringsrespectively. Axiom systems for these were provided in [FGRS1]. In [FGRS1] itwas proved that if $R[G]$ is elementarily equivalent to $S[H]$ with respect to$L_{2}$, then simultaneously the group $G$ is elementarily equivalent to thegroup $H$ with respect to $L_{0}$, and the ring $R$ is elementarily equivalentto the ring $S$ with respect to $L_{1}$. We then let $F$ be a rank $2$ freegroup and $\mathbb{Z}$ be the ring of integers. Examining the universal theoryof the free group ring ${\mathbb Z}[F]$ the hazy conjecture was made that theuniversal sentences true in ${\mathbb Z}[F]$ are precisely the universalsentences true in $F$ modified appropriately for group ring theory and theconverse that the universal sentences true in $F$ are the universal sentencestrue in ${\mathbb Z}[F]$ modified appropriately for group theory. In this paperwe show this conjecture to be true in terms of axiom systems for ${\mathbbZ}[F]$. PubDate: Mon, 06 Dec 2021 09:51:51 +010

Abstract: For finitely generated subgroups $H$ of a free group $F_m$ of finite rank$m$, we study the language $L_H$ of reduced words that represent $H$ which is aregular language. Using the (extended) core of Schreier graph of $H$, weconstruct the minimal deterministic finite automaton that recognizes $L_H$.Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible andfor such groups explicitly construct ergodic automaton that recognizes $L_H$.This construction gives us an efficient way to compute the cogrowth series$L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the methodand a comparison is made with the method of calculation of $L_H(z)$ based onthe use of Nielsen system of generators of $H$. PubDate: Tue, 16 Nov 2021 05:03:24 +010

Abstract: We generalize a result of Moon on the fibering of certain 3-manifolds overthe circle. Our main theorem is the following: Let $M$ be a closed 3-manifold.Suppose that $G=\pi_1(M)$ contains a finitely generated group $U$ of infiniteindex in $G$ which contains a non-trivial subnormal subgroup $N\neq \mathbb{Z}$of $G$, and suppose that $N$ has a composition series of length $n$ in which atleast $n-1$ terms are finitely generated. Suppose that $N$ intersectsnontrivially the fundamental groups of the splitting tori given by theGeometrization Theorem and that the intersections of $N$ with the fundamentalgroups of the geometric pieces are non-trivial and not isomorphic to$\mathbb{Z}$. Then, $M$ has a finite cover which is a bundle over $\mathbb{S}$with fiber a compact surface $F$ such that $\pi_1(F)$ and $U$ arecommensurable. PubDate: Thu, 11 Nov 2021 06:24:13 +010

Abstract: We find algebraic conditions on a group equivalent to the position of itsDiophantine problem in the Chomsky Hierarchy. In particular, we prove that afinitely generated group has a context-free Diophantine problem if and only ifit is finite. PubDate: Thu, 26 Aug 2021 17:57:47 +020

Abstract: An extension of subgroups $H\leqslant K\leqslant F_A$ of the free group ofrank $ A =r\geqslant 2$ is called onto when, for every ambient free basis $A'$,the Stallings graph $\Gamma_{A'}(K)$ is a quotient of $\Gamma_{A'}(H)$.Algebraic extensions are onto and the converse implication was conjectured byMiasnikov-Ventura-Weil, and resolved in the negative, first byParzanchevski-Puder for rank $r=2$, and recently by Kolodner for general rank.In this note we study properties of this new type of extension among freegroups (as well as the fully onto variant), and investigate their correspondingclosure operators. Interestingly, the natural attempt for a dual notion -- intoextensions -- becomes trivial, making a Takahasi type theorem not possible inthis setting. PubDate: Thu, 15 Apr 2021 17:13:52 +020

Abstract: The elliptic curve discrete logarithm problem is considered a securecryptographic primitive. The purpose of this paper is to propose a paradigmshift in attacking the elliptic curve discrete logarithm problem. In thispaper, we will argue that initial minors are a viable way to solve thisproblem. This paper will present necessary algorithms for this attack. We havewritten a code to verify the conjecture of initial minors using Schurcomplements. We were able to solve the problem for groups of order up to$2^{50}$. PubDate: Tue, 16 Feb 2021 11:31:18 +010

Abstract: We generalize the notions of elliptic pseudoprimes and elliptic Carmichaelnumbers introduced by Silverman to analogues of Euler-Jacobi and strongpseudoprimes. We investigate the relationships among Euler Elliptic Carmichaelnumbers , strong elliptic Carmichael numbers, products of anomalous primes andelliptic Korselt numbers of Type I: The former two of these are introduced inthis paper, and the latter two of these were introduced by Mazur (1973) andSilverman (2012) respectively. In particular, we expand upon a previous work ofBabinkostova et al. by proving a conjecture about the density of certainelliptic Korselt numbers of Type I that are products of anomalous primes. PubDate: Wed, 10 Feb 2021 10:37:09 +010

Abstract: Small cancellation groups form an interesting class with many desirableproperties. It is a well-known fact that small cancellation groups are generic;however, all previously known results of their genericity are asymptotic andprovide no information about "small" group presentations. In this note, we giveclosed-form formulas for both lower and upper bounds on the density of smallcancellation presentations, and compare our results with experimental data. PubDate: Wed, 09 Sep 2020 10:18:41 +020

Abstract: We consider a group-theoretic analogue of the classic subset sum problem. Inthis brief note, we show that the subset sum problem is NP-complete in thefirst Grigorchuk group. More generally, we show NP-hardness of that problem inweakly regular branch groups, which implies NP-completeness if the group is, inaddition, contracting. PubDate: Wed, 24 Jun 2020 11:00:50 +020

Abstract: Much attention has been given to the efficient computation of pairings onelliptic curves with even embedding degree since the advent of pairing-basedcryptography. The few existing works in the case of odd embedding degreesrequire some improvements. This paper considers the computation of optimal atepairings on elliptic curves of embedding degrees $k=9$, $15$, $27$ which havetwists of order three. Our main goal is to provide a detailed arithmetic andcost estimation of operations in the tower extensions field of thecorresponding extension fields. A good selection of parameters enables us toimprove the theoretical cost for the Miller step and the final exponentiationusing the lattice-based method as compared to the previous few works that existin these cases. In particular, for $k=15$, $k=27$, we obtain an improvement, interms of operations in the base field, of up to 25% and 29% respectively in thecomputation of the final exponentiation. We also find that elliptic curves withembedding degree $k=15$ present faster results than BN12 curves at the 128-bitsecurity level. We provide a MAGMA implementation in each case to ensure thecorrectness of the formulas used in this work. PubDate: Fri, 17 Apr 2020 13:40:56 +020

Abstract: A fault injection framework for the decryption algorithm of the Niederreiterpublic-key cryptosystem using binary irreducible Goppa codes and classicaldecoding techniques is described. In particular, we obtain low-degreepolynomial equations in parts of the secret key. For the resulting system ofpolynomial equations, we present an efficient solving strategy and show how toextend certain solutions to alternative secret keys. We also provide estimatesfor the expected number of required fault injections, apply the framework tostate-of-the-art security levels, and propose countermeasures against this typeof fault attack. PubDate: Fri, 20 Mar 2020 12:27:39 +010

Abstract: A $\vee$-complement of a subgroup $H \leqslant \mathbb{F}_n$ is a subgroup $K\leqslant \mathbb{F}_n$ such that $H \vee K = \mathbb{F}_n$. If we also ask $K$to have trivial intersection with $H$, then we say that $K$ is a$\oplus$-complement of $H$. The minimum possible rank of a $\vee$-complement(resp. $\oplus$-complement) of $H$ is called the $\vee$-corank (resp.$\oplus$-corank) of $H$. We use Stallings automata to study these notions andthe relations between them. In particular, we characterize when complementsexist, compute the $\vee$-corank, and provide language-theoretical descriptionsof the sets of cyclic complements. Finally, we prove that the two notions ofcorank coincide on subgroups that admit cyclic complements of both kinds. PubDate: Mon, 02 Mar 2020 12:56:10 +010