Related papers: On Taking Square Roots without Quadratic Nonresidu…

We propose a new method for evaluating NISQ devices. This paper has three distinct parts. First, we present a new quantum algorithm that solves a two hundred year old problem of finding quadratic nonresidues (QNR) in polynomial time. We…

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…

We develop an algebraic framework over arbitrary quadratic fields $L = \mathbb{Q}(\sqrt{D})$ to generalize the Miller-Rabin primality test. Consequently, we present a deterministic primality test for integers of the form $N = K p^{\ell} -…

A key property of an algebraic variety is whether it is absolutely irreducible, meaning that it remains irreducible over the algebraic closure of its defining field, and determining absolute irreducibility is important in algebraic geometry…

Solving quadratic equations over finite fields is a fundamental task in algebraic coding theory and serves as a key subroutine for computing the roots of cubic and quartic polynomials. Notably, any quadratic polynomial over binary extension…

We give a new proof of Fitzgerald's criterion for primitive polynomials over a finite field. Existing proofs essentially use the theory of linear recurrences over finite fields. Here, we give a much shorter and self-contained proof which…

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,…

This paper presents a high-order accurate numerical quadrature algorithm for evaluating integrals over curved surfaces and regions defined implicitly via a level set of a given function restricted to a hyperrectangle. The domain is divided…

Given a nonsingular $n \times n$ matrix of univariate polynomials over a field $\mathbb{K}$, we give fast and deterministic algorithms to compute its determinant and its Hermite normal form. Our algorithms use…

In this paper, I treat quadratic equation over associative $D$-algebra. In quaternion algebra $H$, the equation $x^2=a$ has either $2$ roots, or infinitely many roots. Since $a\in R$, $a<0$, then the equation has infinitely many roots.…

We study real quadratic fields $\mathbb{Q}(\sqrt{D})$ such that, for a given rational integer $m$, all $m$-multiples of totally positive integers are sums of squares. We prove quite sharp necessary and sufficient conditions for this to…

A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly…

We study quantum algorithms for spatial search on finite dimensional grids. Patel et al. and Falk have proposed algorithms based on a quantum walk without a coin, with different operators applied at even and odd steps. Until now, such…

This paper is continuation of the paper "Primitive roots in quadratic field". We consider an analogue of Artin's primitive root conjecture for algebraic numbers which is not a unit in real quadratic fields. Given such an algebraic number,…

Quantum computers are known to be qualitatively more powerful than classical computers, but so far only a small number of different algorithms have been discovered that actually use this potential. It would therefore be highly desirable to…

The unit cost model is both convenient and largely realistic for describing integer decision algorithms over (+,*). Additional operations like division with remainder or bitwise conjunction, although equally supported by computing hardware,…

We explore an algorithm for approximating roots of integers, discuss its motivation and derivation, and analyze its convergence rates with varying parameters and inputs. We also perform comparisons with established methods for approximating…

We present an algorithm to decide the primality of Proth numbers, N=2^e t+1, without assuming any unproven hypothesis. The expected running time and the worst case running time of the algorithm are O ((t log t + log N)log N) and O ((t log t…

In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call m-schemes. We extend the known conditional deterministic subexponential time polynomial factoring…

Efficient methods for computing with matrices over finite fields often involve randomised algorithms, where matrices with a certain property are sought via repeated random selection. Complexity analyses for these algorithms require…