Related papers: Algorithmic problems in Engel groups and cryptogra…
We give an efficient algorithm for Lang's Theorem in split connected reductive groups defined over finite fields of characteristic greater than 3. This algorithm can be used to construct many important structures in finite groups of Lie…
Sampling algorithms, hypergraph degree sequences, and polytopes play a crucial role in statistical analysis of network data. This article offers a brief overview of open problems in this area of discrete mathematics from the point of view…
This paper classifies the derivations of group algebras in terms of the generators and defining relations of the group. If $RG$ is a group ring, where $R$ is commutative and $S$ is a set of generators of $G$ then necessary and sufficient…
Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same…
We provide a survey on the Hidden Subgroup Problem (HSP), which plays an important role in studying the security of public-key cryptosystems. We first review the abelian case, where Kitaev's algorithm yields an efficient quantum solution to…
We study properties of Engel elements in weakly branch groups, lying in the group of automorphisms of a spherically homogeneous rooted tree. More precisely, we prove that the set of bounded left Engel elements is always trivial in weakly…
The discrete logarithm problem is one of the backbones in public key cryptography. In this paper we study the discrete logarithm problem in the group of circulant matrices over a finite field. This gives rise to secure and fast public key…
An important open problem in supersingular isogeny-based cryptography is to produce, without a trusted authority, concrete examples of "hard supersingular curves" that is, equations for supersingular curves for which computing the…
This is a list of some problems and conjectures related to various types of algebras, that is to algebraic operads. Some comments and hints are included.
In this paper, we survey the status of attacks on the ring and polynomial learning with errors problems (RLWE and PLWE). Recent work on the security of these problems [Eisentr\"ager-Hallgren-Lauter, Elias-Lauter-Ozman-Stange] gives rise to…
We survey group-theoretic algorithms for finding (some or all) subgroups of a finite group and discuss the implementation of these algorithms in the computer algebra system GAP
The braid group has recently attracted much attention. This is primarily based upon the discovery of its usage in various cryptosystems [AAG],[KLCHKP]. One major focus of current research has been in solving decision problems in braid…
We introduce the notion of a diagram category and discuss its application to the invariant theory of classical groups and super groups, with some indications concerning extensions to quantum groups and quantum super groups. Tensor functors…
The study of the relation between Lie algebras and groups, and especially the derivation of new algebras from them, is a problem of great interest in mathematics and physics, because finding a new Lie group from an already known one also…
The notion of entropy appears in many fields and this paper is a survey about entropies in several branches of Mathematics. We are mainly concerned with the topological and the algebraic entropy in the context of continuous endomorphisms of…
Although recent advances of blockchain systems, notably in the form of cryptocurrency, have drawn tremendous interests from both researchers and practitioners, limited studies existed toward the theoretical foundation of blockchains. This…
We introduce a generalization of the Euclidean algorithm for rings equipped with an involution, and completely enumerate all isomorphism classes of orders over definite, rational quaternion algebras equipped with an orthogonal involution…
The theory of quadratic forms and class numbers has connections to many classical problems in number theory. Recently, class numbers have appeared in the study of black holes in string theory. We describe this connection and raise questions…
This is a survey of some problems in geometric group theory which I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some…
We study the query complexity of quantum learning problems in which the oracles form a group $G$ of unitary matrices. In the simplest case, one wishes to identify the oracle, and we find a description of the optimal success probability of a…