<div><p>Regardless of the choice of parameters, knowledge of a single signed message, i.e., a pair message/signature, produced by Kahrobaei-Koupparis digital signature scheme, proposed in [D. Kahrobaei and C. Koupparis, 2012], is sufficient to forge a valid signature for any other message. </p></div>

Let $p$ and $n$ be positive integers. Assume additionally that $p\neq 3$ is a prime and that $n>2$. Let $R$ be a field of characteristic $p$. A very special consequence of a result of Bunina and Kunyavskii (2023, arXiv:2308.10076) is that $SL_{n}(R)$ is co-Hopfian as a group if and only if $R$ is co-Hopfian as a ring. In this paper, we prove that if $k$ is the algebraic closure of the $2$ element field, then $SL_{2}(k)$ is a co-Hopfian group. Since this $k$ is trivially seen to be co-Hopfian as a ring our result somewhat extends that of Bunina and Kunyavskii. We apply our result to prove that the class of groups satisfying Turner's Retract Theorem (called Turner groups here) is not closed under elementary equivalence thereby answering a question posed by the authors in (2017, Comm. Algebra). 9 pages. Published in the journal of Groups, Complexity, Cryptology

For any group $G$ and integer $k\ge 2$ the Andrews-Curtis transformations act as a permutation group, termed the Andrews-Curtis group $AC_k(G)$, on the subset $N_k(G) \subset G^k$ of all $k$-tuples that generate $G$ as a normal subgroup (provided $N_k(G)$ is non-empty). The famous Andrews-Curtis Conjecture is that if $G$ is free of rank $k$, then $AC_k(G)$ acts transitively on $N_k(G)$. The set $N_k(G)$ may have a rather complex structure, so it is easier to study the full Andrews-Curtis group $FAC(G)$ generated by AC-transformations on a much simpler set $G^k$. Our goal here is to investigate the natural epimorphism $λ\colon FAC_k(G) \to AC_k(G)$. We show that if $G$ is non-elementary torsion-free hyperbolic, then $FAC_k(G)$ acts faithfully on every nontrivial orbit of $G^k$, hence $λ\colon FAC_k(G) \to AC_k(G)$ is an isomorphism.7 pages. In memory of Ben Fine. Published in journal of Groups, Complexity, Cryptology

In this paper we provide an alternative solution to a result by Juh\'{a}sz that the twisted conjugacy problem for odd dihedral Artin groups is solvable, that is, groups with presentation $G(m) = \langle a,b \; | \; _{m}(a,b) = {}_{m}(b,a) \rangle$, where $m\geq 3$ is odd, and $_{m}(a,b)$ is the word $abab \dots$ of length $m$, is solvable. Our solution provides an implementable linear time algorithm, by considering an alternative group presentation to that of a torus knot group, and working with geodesic normal forms. An application of this result is that the conjugacy problem is solvable in extensions of odd dihedral Artin groups.Comment: Published in the journal of Groups, Complexity, Cryptology

We study both the Submonoid Membership problem and the Rational Subset Membership problem in finitely generated nilpotent groups. We give two reductions with important applications. First, Submonoid Membership in any nilpotent group can be reduced to Rational Subset Membership in smaller groups. As a corollary, we prove the existence of a group with decidable Submonoid Membership and undecidable Rational Subset Membership, confirming a conjecture of Lohrey and Steinberg. Second, the Rational Subset Membership problem in $H_3(\mathbb Z)$ can be reduced to the Knapsack problem in the same group, and is therefore decidable. Combining both results, we deduce that the filiform $3$-step nilpotent group has decidable Submonoid Membership.Comment: v6. 25 pages, 5 figures. Published in the journal of Groups, Complexity, Cryptology

We study the conjugacy class growth function in finitely generated virtually abelian groups. That is, the number of elements in the ball of radius $n$ in the Cayley graph which intersect a fixed conjugacy class. In the class of virtually abelian groups, we prove that this function is always asymptotically equivalent to a polynomial. Furthermore, we show that in any affine Coxeter group, the degree of polynomial growth of a conjugacy class is equivalent to the reflection length of any element of that class.Comment: 14 pages, 2 figures. Published in the journal of Groups, Complexity, Cryptology

For a commutative finite $\mathbb{Z}$-algebra, i.e., for a commutative ring $R$ whose additive group is finitely generated, it is known that the group of units of $R$ is finitely generated, as well. Our main results are algorithms to compute generators and the structure of this group. This is achieved by reducing the task first to the case of reduced rings, then to torsion-free reduced rings, and finally to an order in a reduced ring. The simplified cases are treated via a calculation of exponent lattices and various algorithms to compute the minimal primes, primitive idempotents, and other basic objects. All algorithms have been implemented and are available as a SageMath package. Whenever possible, the time complexity of the described methods is tracked carefully.

We consider a class of groups, called groups of F-type, which includes some known and important classes like Fuchsian groups of geometric rank $\ge 3$, surface groups of genus $\ge 2$, cyclically pinched one-relator groups and torus-knot groups, and discuss algebraic and geometric properties of groups of F-type.

In this paper, we prove a criterion for a predicate structure to be equationally Noetherian.

In this paper we analyze computational properties of the Diophantine problem (and its search variant) for spherical equations $\prod_{i=1}^m z_i^{-1} c_i z_i = 1$ (and its variants) over the class of finite metabelian groups $G_{p,n}=\mathbb{Z}_p^n \rtimes \mathbb{Z}_p^\ast$, where $n\in\mathbb{N}$ and $p$ is prime. We prove that the problem of finding solutions for certain constrained spherical equations is computationally hard on average (assuming that some lattice approximation problem is hard in the worst case).