Related papers: On the matrix code of quadratic relationships for …

We investigate the complexity of isomorphism relations for classes of finitely generated and n-generated computably enumerable (c.e.) algebras, presented via c.e. presentations -- that is, as quotients of term algebras over decidable sets…

Large matrix multiplications are central to large-scale machine learning applications. These operations are often carried out on a distributed computing platform with a master server and multiple workers in the cloud operating in parallel.…

It is a long standing open problem to find search to decision reductions for structured versions of the decoding problem of linear codes. Such results in the lattice-based setting have been carried out using number fields: Polynomial-LWE,…

Tensor operations, such as matrix multiplication, are central to large-scale machine learning applications. For user-driven tasks these operations can be carried out on a distributed computing platform with a master server at the user side…

In studying the depth filtration on multiple zeta values, difficulties quickly arise due to a disparity between it and the coradical filtration. In particular, there are additional relations in the depth graded algebra coming from period…

This Diplom thesis provides an explicit construction of a quantum Goppa code for any hyperelliptic curve over a non-binary field. Hyperelliptic curves have conjugate pairs of rational places. We use these pairs to construct self-orthogonal…

Robust Principal Component Analysis (PCA) (Candes et al., 2011) and low-rank matrix completion (Recht et al., 2010) are extensions of PCA to allow for outliers and missing entries respectively. It is well-known that solving these problems…

In this paper we study recent reaction attacks against QC-LDPC and QC-MDPC code-based cryptosystems, which allow an opponent to recover the private parity-check matrix through its distance spectrum by observing a sufficiently high number of…

Let $\cal M$ denote the set ${\cal S}_{n, q}$ of $n \times n$ symmetric matrices with entries in ${\rm GF}(q)$ or the set ${\cal H}_{n, q^2}$ of $n \times n$ Hermitian matrices whose elements are in ${\rm GF}(q^2)$. Then $\cal M$ equipped…

This paper consists of three parts. The first part presents a large class of new binary quasi-cyclic (QC)-LDPC codes with girth of at least 6 whose parity-check matrices are constructed based on cyclic subgroups of finite fields.…

We present a quantum attack on ML-KEM and related 2-power cyclotomic lattice schemes. Combining with Parts I-III, we provide an algorithm and verify the resulting approximation factor satisfies $\gamma\le 21 < q/2=1664.5$ for ML-KEM-1024,…

We consider the problem of communication efficient secure distributed matrix multiplication. The previous literature has focused on reducing the number of servers as a proxy for minimizing communication costs. The intuition being, that the…

Deep polar codes are pre-transformed polar codes that employ a multi-layered polar kernel transformation strategy to enhance code performance in short blocklength regimes. However, like conventional polar codes, their block length is…

Lattice rules and polynomial lattice rules are quadrature rules for approximating integrals over the $s$-dimensional unit cube. Since no explicit constructions of such quadrature methods are known for dimensions $s > 2$, one usually has to…

This paper compares the efficiency of various algorithms for implementing quantum resistant public key encryption scheme RLCE on 64-bit CPUs. By optimizing various algorithms for polynomial and matrix operations over finite fields, we…

A standard approach to compute the roots of a univariate polynomial is to compute the eigenvalues of an associated \emph{confederate} matrix instead, such as, for instance the companion or comrade matrix. The eigenvalues of the confederate…

We establish new explicit connections between classical (scalar) and matrix Gegenbauer polynomials, which result in new symmetries of the latter and further give access to several properties that have been out of reach before: generating…

Due to the weakness of public key cryptosystems encounter of quantum computers, the need to provide a solution was emerged. The McEliece cryptosystem and its security equivalent, the Niederreiter cryptosystem, which are based on Goppa…

We extend coded distributed computing over finite fields to allow the number of workers to be larger than the field size. We give codes that work for fully general matrix multiplication and show that in this case we serendipitously have…

Maximum Distance Profile (MDP) convolutional codes are an important class of channel codes due to their maximal delay-constrained error correction capabilities. The design of MDP codes has attracted significant attention from the research…