On Polynomial Modular Number Systems over $ \mathbb{Z}/{p}\mathbb{Z} $

Author:

Bajard Jean-Claude1,Marrez Jérémy2,Plantard Thomas3,Véron Pascal4

Affiliation:

1. Sorbonne Université, CNRS, Inria, Institut de Mathématiques de Jussieu, Paris Rive Gauche, France

2. Sorbonne Université, CNRS, Laboratoire d'informatique de Paris 6, Paris, France

3. Paypal, Emerging Technology Research Group, San Jose, USA

4. Laboratoire IMath, Université de Toulon, La Garde, France

Abstract

<p style='text-indent:20px;'>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.</p><p style='text-indent:20px;'>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.</p>

Publisher

American Institute of Mathematical Sciences (AIMS)

Subject

Applied Mathematics,Discrete Mathematics and Combinatorics,Computer Networks and Communications,Algebra and Number Theory,Applied Mathematics,Discrete Mathematics and Combinatorics,Computer Networks and Communications,Algebra and Number Theory

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. An Alternative Approach to Polynomial Modular Number System Internal Reduction;IEEE Transactions on Emerging Topics in Computing;2022-07-01

2. PMNS for Efficient Arithmetic and Small Memory Cost;IEEE Transactions on Emerging Topics in Computing;2022-07-01

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