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Generic polynomial
About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.
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3 citations
01 Sep 2002
TL;DR: In this paper, the authors consider the problem of expressing the roots of an irreducible polynomial f in terms of elements of the ground field by rational operations and radicals, and give a practical method for constructing a radical expression of f when f is solvable.
Abstract: We consider a fundamental question of Galois theory: how to express the roots of an irreducible polynomial f , deg(f ) > 4 in terms of elements of the ground field by rational operations and radicals. In general, expressing the roots of f in terms of radicals is impossible when deg(f ) > 4. By Galois theory, however, we can test whether f is solvable by checking solvability of its Galois group. We will give a practical method for constructing a radical expression of the roots of f , when f is solvable, and report its experiment on a real computer.
3 citations
TL;DR: In this article, the universal factorization algebra of the generic polynomial of degree n into the product of two monic polynomials, one of degree r, and the other of degree c, is described.
Abstract: The goal of this paper is to supply an explicit description of the universal factorization algebra of the generic polynomial of degree n into the product of two monic polynomials, one of degree r, ...
3 citations
Posted Content•
TL;DR: This work states a complete existence theorem for PMNS which provides bounds on the size of the digits for a generic polynomial, significantly improving previous bounds and presents classes of suitable polynomials which provide numerous PMNS for safe and efficient arithmetic.
Abstract: Since their introduction in 2004, Polynomial Modular Number Systems (PMNS) have become a very interesting tool for implementing cryptosystems relying on modular arithmetic in a secure and efficient way. However, while their implementation is simple, their parameterization is not trivial and relies on a suitable choice of the polynomial on which the PMNS operates. The initial proposals were based on particular binomials and trinomials. But these polynomials do not always provide systems with interesting characteristics such as small digits, fast reduction, etc.
In this work, we study a larger family of polynomials that can be exploited to design a safe and efficient PMNS. To do so, we first state a complete existence theorem for PMNS which provides bounds on the size of the digits for a generic polynomial, significantly improving previous bounds. Then, we present classes of suitable polynomials which provide numerous PMNS for safe and efficient arithmetic.
3 citations