We consider exponent equations in finitely generated groups. These are equations, where the variables appear as exponents of group elements and take values from the natural numbers. Solvability of such (systems of) equations has been intensively studied for various classes of groups in recent years. In many cases, it turns out that the set of all solutions on an exponent equation is a semilinear set that can be constructed effectively. Such groups are called knapsack semilinear. Examples of knapsack semilinear groups are hyperbolic groups, virtually special groups, co-context-free groups and free solvable groups. Moreover, knapsack semilinearity is preserved by many group theoretic constructions, e.g., finite extensions, graph products, wreath products, amalgamated free products with finite amalgamated subgroups, and HNN-extensions with finite associated subgroups. On the other hand, arbitrary HNN-extensions do not preserve knapsack semilinearity. In this paper, we consider the knapsack semilinearity of HNN-extensions, where the stable letter $t$ acts trivially by conjugation on the associated subgroup $A$ of the base group $G$. We show that under some additional technical conditions, knapsack semilinearity transfers from base group $G$ to the HNN-extension $H$. These additional technical conditions are satisfied in many cases, e.g., when $A$ is a centralizer in $G$ or $A$ is a quasiconvex subgroup of the hyperbolic group $G$.
In this paper, we study geodesic growth of numbered graph products; these are a generalization of right-angled Coxeter groups, defined as graph products of finite cyclic groups. We first define a graph-theoretic condition called link-regularity, as well as a natural equivalence amongst link-regular numbered graphs, and show that numbered graph products associated to link-regular numbered graphs must have the same geodesic growth series. Next, we derive a formula for the geodesic growth of right-angled Coxeter groups associated to link-regular graphs. Finally, we find a system of equations that can be used to solve for the geodesic growth of numbered graph products corresponding to link-regular numbered graphs that contain no triangles and have constant vertex numbering.
Let $S$ be a pool of $s$ parties and Alice be the dealer. In this paper, we propose a scheme that allows the dealer to encrypt messages in such a way that only one authorized coalition of parties (which the dealer chooses depending on the message) can decrypt. At the setup stage, each of the parties involved in the process receives an individual key from the dealer. To decrypt information, an authorized coalition of parties must work together to use their keys. Based on this scheme, we propose a threshold encryption scheme. For a given message $f$ the dealer can choose any threshold $m = m(f).$ More precisely, any set of parties of size at least $m$ can evaluate $f$; any set of size less than $m$ cannot do this. Similarly, the distribution of keys among the included parties can be done in such a way that authorized coalitions of parties will be given the opportunity to put a collective digital signature on any documents. This primitive can be generalized to the dynamic setting, where any user can dynamically join the pool $S$. In this case the new user receives a key from the dealer. Also any user can leave the pool $S$. In both cases, already distributed keys of other users do not change. The main feature of the proposed schemes is that for a given $s$ the keys are distributed once and can be used multiple times. The proposed scheme is based on the idea of hidden multipliers in encryption. As a platform, one can use both multiplicative groups of finite fields and groups of […]