Related papers: Complexity of Computing Quadratic Nonresidues
We present a deterministic polynomial-time algorithm that determines whether a finite module over a finite commutative ring is cyclic, and if it is, outputs a generator.
We propose polynomial-time algorithms for finding nontrivial zeros of quadratic forms with four variables over rational function fields of characteristic 2. We apply these results to find prescribed quadratic subfields of quaternion…
We propose an algorithm for determining the irreducible polynomials over finite fields, based on the use of the companion matrix of polynomials and the generalized Jordan normal form of square matrices.
We present an explicit algorithmic method for computing square roots in quaternion algebras over global fields of characteristic different from 2.
We extend our techniques developed in our earlier paper appeared in Computational Complexity, 2017 (preprint: arXiv:1508.00690) to obtain a deterministic polynomial time algorithm for computing the non-commutative rank together with…
Given a non-empty genus in $n$ dimensions with determinant $d$, we give a randomized algorithm that outputs a quadratic form from this genus. The time complexity of the algorithm is poly$(n,\log d)$; assuming Generalized Riemann Hypothesis…
A study of real quadratic maps with real critical points, emphasizing the effective construction of critically finite maps with specified combinatorics. We discuss the behavior of the Thurston algorithm in obstructed cases, and in one…
This article deals with the computation of the characteristic polynomial of dense matrices over small finite fields and over the integers. We first present two algorithms for the finite fields: one is based on Krylov iterates and Gaussian…
A randomized algorithm for a search problem is *pseudodeterministic* if it produces a fixed canonical solution to the search problem with high probability. In their seminal work on the topic, Gat and Goldwasser posed as their main open…
We first show a deterministic algorithm for taking $r$-th roots over $\F_q$ without being given any $r$-th nonresidue, where $\F_q$ is a finite field with $q$ elements and $r$ is a small prime such that $r^2$ divides of $q-1$. As…
In this paper, we propose a new construction of quadratic bent functions in polynomial forms. Right Euclid algorithm in skew-polynomial rings over finite fields of characteristic 2 is applied in the proof.
It's well known that the quadratic residue code over finite fields is an interesting class of cyclic codes for its higher minimum distance. Let $g$ be a positive integer and $p,p_{1},\ldots, p_{g}$ be distinct odd primes, the present paper…
We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the…
When modeling scientific and industrial problems, geometries are typically modeled by explicit boundary representations obtained from computer-aided design software. Unfitted (also known as embedded or immersed) finite element methods offer…
A novel method for computation of the discrete Fourier transform over a finite field with reduced multiplicative complexity is described. If the number of multiplications is to be minimized, then the novel method for the finite field of…
An algorithm is presented for the efficient and accurate computation of the coefficients of the characteristic polynomial of a general square matrix. The algorithm is especially suited for the evaluation of canonical traces in determinant…
We give an overview of the existing algorithms to compute nonunique factorization invariants in finitely generated monoids.
Let S be an infinite set of non-empty, finite subsets of the nonnegative integers. If p is an odd prime, let c(p) denote the cardinality of the set {T {\in} S : T {\subseteq} {1,...,p-1} and T is a set of quadratic residues (respectively,…
Let $p$ be a large prime, and let $k\ll \log p$. A new proof of the existence of any pattern of $k$ consecutive quadratic residues and quadratic nonresidues is introduced in this note. Further, an application to the least quadratic…
An \emph{indexing} of a finite set $S$ is a bijection $D : \{1,...,|S|\} \rightarrow S$. We present an indexing for the set of quadratic residues modulo $N$ that is decodable in polynomial time on the size of $N$, given the factorization of…