Constructing r -th nonresidue over a €nite €eld is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree r e (where r is a prime) over a given €nite €eld Fq of characteristic p (equivalently, constructing the bigger €eld Fqr e ). Both these problems have famous randomized algorithms but the derandomization is an open question. We give some ne...

For a one-dimensional complex-valued function/this paper deals with iterative root finding methods using divided differences of /. Assuming that / is given in a Newtonian representation we show how Homer-like transformations ("shift transformations") yield the divided differences needed in each iteration step. In particular, we consider an iteration method given by Larkin [5] and derive an equi...

Floating point arithmetic plays a central role in science, engineering, and €nance by enabling developers to approximately compute with real numbers. To address numerical issues in large ƒoating-point applications, developers must identify root causes, which is dicult because ƒoating point errors are generally silent, non-local, and non-compositional. Œis paper presents Herbgrind, a tool to he...

We show that all algebraic-geometric codes possess a succinct representation that allows for the list decoding algorithms of [9, 6] to run in polynomial time. We do this by presenting a root-finding algorithm for univariate polynomials over function fields when their coefficients lie in finite-dimensional linear spaces, and proving that there is a polynomial size representation given which the ...

Polynomial root-finders usually consist of two stages. At first a crude approximation to a root is slowly computed; then it is much faster refined by means of the same or distinct iteration. The efficiency of computing an initial approximation resists formal study, and the users rely on empirical data. In contrast, the efficiency of refinement is formally measured by the classical concept q whe...

Let Fq be a finite field of q elements of characteristic p. The classical algorithm of Berlekamp [1] reduces the problem of factoring polynomials of degree n over Fq to the problem of factoring squarefree polynomials of degree n over Fp that fully split in Fp, see also [8, Chapter 14]. Shoup [15, Theorem 3.1] has given a deterministic algorithm that fully factors any polynomial of degree n over...

Let A0, A1, . . . , An be given square matrices of size m with rational coefficients. The paper focuses on the exact computation of one point in each connected component of the real determinantal variety {x ∈ Rn : det(A0 +x1A1 + · · ·+xnAn) = 0}. Such a problem finds applications in many areas such as control theory, computational geometry, optimization, etc. Using standard complexity results t...

In this work we analyze four previously published respectively trivial approaches to the task of finding the roots of the error locator polynomial during the decryption operation of code-based encryption schemes. We compare the performance of these algorithms and show that optimizations concerning finite field element representations play a key role for the speed of software implementations. Fu...

This article presents an analytic mode-matching approach suitable for modelling the propagation of sound in a two-dimensional, three-part, ducting system. The approach avoids the need to find roots of the characteristic equation for the middle section of the duct (the component) and is readily applicable to a broad class of problems. It is demonstrated that the system of equations, derived via ...