Related papers: Multiplication in Cyclotomic Rings and its Applica…
We present an algorithm for computing the integral closure of a reduced ring that is finitely generated over a finite field.
Thanks to a new construction of the so-called Chudnovsky-Chudnovsky multiplication algorithm, we design efficient algorithms for both the exponentiation and the multiplication in finite fields. They are tailored to hardware implementation…
In this paper, we present a review of three widely-used practical square root algorithms. We then describe a unifying framework where each of these well-known algorithms can be seen as a special case of it. The framework with singular…
We present a uniform methodology for computing with finitely generated matrix groups over any infinite field. As one application, we completely solve the problem of deciding finiteness in this class of groups. We also present an algorithm…
A finite semifield is a division algebra over a finite field where multiplication is not necessarily associative. We consider here the complexity of the multiplication in small semifields and finite field extensions. For this operation, the…
Several problems in computer algebra can be efficiently solved by reducing them to calculations over finite fields. In this paper, we describe an algorithm for the reconstruction of multivariate polynomials and rational functions from their…
Cyclotomic fast Fourier transforms (CFFTs) are efficient implementations of discrete Fourier transforms over finite fields, which have widespread applications in cryptography and error control codes. They are of great interest because of…
In this paper, we study the arithmetics of skew polynomial rings over finite fields, mostly from an algorithmic point of view. We give various algorithms for fast multiplication, division and extended Euclidean division. We give a precise…
We study the complexity of multiplication of two elements in a finite field extension given by their coordinates in a normal basis. We show how to control this complexity using the arithmetic and geometry of algebraic curves.
In this paper we discuss the problem of performing elementary finite field arithmetic on a quantum computer. Of particular interest, is the controlled-multiplication operation, which is the only group-specific operation in Shor's algorithms…
We use cyclotomy to design new classes of permutation polynomials over finite fields. This allows us to generate many classes of permutation polynomials in an algorithmic way. Many of them are permutation polynomials of large indices.
In this paper we look at the automorphisms of the multiplicative group of finite nearfields. We find partial results for the actual automorphism groups. We find counting techniques for the size of all finite nearfields. We then show that…
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…
Finite field multiplier is mainly used in error-correcting codes and signal processing. Finite field multiplier is regarded as the bottleneck arithmetic unit for such applications and it is the most complicated operation over finite field…
With the help of hypergeometric functions over finite fields, we study some arithmetic properties of cyclotomic matrices involving characters and binary quadratic forms over finite fields. Also, we confirm some related conjectures posed by…
In this paper we define a pair of faithful functors that map isomorphic and isotopic finite-dimensional algebras over finite fields to isomorphic graphs. These functors reduce the cost of computation that is usually required to determine…
We set new speed records for multiplying long polynomials over finite fields of characteristic two. Our multiplication algorithm is based on an additive FFT (Fast Fourier Transform) by Lin, Chung, and Huang in 2014 comparing to previously…
Cyclic codes over finite fields are widely implemented in data storage systems, communication systems, and consumer electronics, as they have very efficient encoding and decoding algorithms. They are also important in theory, as they are…
Let $\mathbb{F}_q[T]$ be the polynomial ring over a finite field $\mathbb{F}_q$. We study the endomorphism rings of Drinfeld $\mathbb{F}_q[T]$-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings…
In this paper, we introduce distributed matrix multiplication (DMM)-friendly algebraic function fields for polynomial codes and Matdot codes, and present several constructions for such function fields through extensions of the rational…