Related papers: Gaussian elimination in unitary groups with an app…
We describe a new algorithm for Gaussian Elimination suitable for general (unsymmetric and possibly singular) sparse matrices, of any entry type, which has a natural parallel and distributed-memory formulation but degrades gracefully to…
In this paper we will give various examples of exponentially distorted subgroups in linear groups, including some new example of subgroups of $SL_n(\mathbb{Z}[x])$ for $n \ge 3$, and show how they can be used to construct symmetric-key…
We propose a new cryptosystem based on polycyclic groups. The cryptosystem is based on the fact that the word problem can be solved effectively in polycyclic groups, while the known solutions to the conjugacy problem are far less efficient.
Using the character expansion method, we generalize several well-known integrals over the unitary group to the case where general complex matrices appear in the integrand. These integrals are of interest in the theory of random matrices and…
In this thesis, we develop algorithms similar to the Gaussian elimination algorithm in symplectic and split orthogonal similitude groups. As an application to this algorithm, we compute the spinor norm for split orthogonal groups. Also, we…
We present new applications on $q$-binomials, also known as Gaussian binomial coefficients. Our main theorems determine cardinalities of certain error-correcting codes based on Varshamov-Tenengolts codes and prove a curious phenomenon…
Recently, several public key exchange protocols based on symbolic computation in non-commutative (semi)groups were proposed as a more efficient alternative to well established protocols based on numeric computation. Notably, the protocols…
It is known that extreme characters of several inductive limits of compact groups exhibit multiplicativity in a certain sense. In the paper, we formulate such multiplicativity for inductive limit quantum groups and provide explicit examples…
This work leverages recent advances in probabilistic machine learning to discover conservation laws expressed by parametric linear equations. Such equations involve, but are not limited to, ordinary and partial differential,…
We consider continuous-variable quantum key distribution with discrete-alphabet encodings. In particular, we study protocols where information is encoded in the phase of displaced coherent (or thermal) states, even though the results can be…
We discuss the application of random projections to the fundamental problem of deciding whether a given point in a Euclidean space belongs to a given set. We show that, under a number of different assumptions, the feasibility and…
Simulation of quantum matters is a significant application of quantum computers. In contrast to the unitary operation which can be realized naturally on a quantum computer, the implementation of nonunitary operation, widely used in…
We introduce a new constructive recognition algorithm for finite special linear groups in their natural representation. Given a group $G$ generated by a set of $d\times d$ matrices over a finite field $\mathbb{F}_q$, known to be isomorphic…
We propose an algorithm using the Gaussian elimination method to find the minimal Hamming distance and decode received messages of linear codes. This algorithm is easy to implement as it requires no Gr\"obner bases to compute solutions for…
Let f be an arbitrary positive integer valued function. The goal of this note is to show that one can construct a finitely generated group in which the discrete log problem is polynomially equivalent to computing the function f. In…
We consider the computational problem of determining the unit group of a finite ring, by which we mean the computation of a finite presentation together with an algorithm to express units as words in the generators. We show that the problem…
We design new deterministic and randomized algorithms for computational problems in free solvable groups. In particular, we prove that the word problem and the power problem can be solved in quasi-linear time and the conjugacy problem can…
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…
Decoders for quantum LDPC codes generally rely on solving a parity-check equation with Gaussian elimination, with the generalised union-find decoder performing this repeatedly on growing clusters. We present an online variant of the…
Newton, in notes that he would rather not have seen published, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method that Euler did not recommend, that Legendre…