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

Jean-Claude Bajard; Jérémy Marrez; Thomas Plantard; Pascal Véron

doi:10.3934/amc.2022018

Article Contents

2024, Volume 18, Issue 3: 674-695. Doi: 10.3934/amc.2022018

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

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

Received: September 2021

Revised: February 2022

Early access: March 2022

Published: June 2024

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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.

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Mathematics Subject Classification: Primary: 11T06, 11T71; Secondary: 11H06, 94A60, 68W99.

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Table 1. Elements of $\mathbb{Z}/31\mathbb{Z}$ are represented as polynomials in $\gamma$ , noted as vectors with lowest degree first. The reduction polynomial is $E(X) = X^4-2$ and ${\gamma = 15}$ is a root of $E(X)$ . The digit set is $\{ -1, 0, 1\}$ (i.e., $\rho = 2$ )

0	1	2	3	4	5
(0, 0, 0, 0)	(1, 0, 0, 0)	(-1, 1, -1, 1)	\begin{array}{*{20}{c}} (-1, -1, -1, 1)\\ (-1, 0, 0, -1)\\ (-1, 0, 1, 1)\\ (0, 1, -1, 1) \end{array}	\begin{array}{*{20}{c}} (0, -1, -1, 1)\\ (0, 0, 0, -1)\\ (0, 0, 1, 1)\\ (1, 1, -1, 1) \end{array}	\begin{array}{*{20}{c}} (1, -1, -1, 1)\\ (1, 0, 0, -1)\\ (1, 0, 1, 1) \end{array}
6	7	8	9	10	11
(-1, 1, -1, 0)	\begin{array}{*{20}{c}} (-1, -1, -1, 0)\\ (-1, 0, 1, 0)\\ (0, 1, -1, 0) \end{array}	\begin{array}{*{20}{c}} (0, -1, -1, 0)\\ (0, 0, 1, 0)\\ (1, 1, -1, 0) \end{array}	\begin{array}{*{20}{c}} (1, -1, -1, 0)\\ (1, 0, 1, 0) \end{array}	\begin{array}{*{20}{c}} (-1, 1, -1, -1)\\ (-1, 1, 0, 1) \end{array}	\begin{array}{*{20}{c}} (-1, -1, -1, -1)\\ (-1, -1, 0, 1)\\ (-1, 0, 1, -1)\\ (0, 1, -1, -1)\\ (0, 1, 0, 1) \end{array}
12	13	14	15	16	17
\begin{array}{*{20}{c}} (0, -1, -1, -1)\\ (0, -1, 0, 1)\\ (0, 0, 1, -1)\\ (1, 1, -1, -1)\\ (1, 1, 0, 1) \end{array}	\begin{array}{*{20}{c}} (1, -1, -1, -1)\\ (1, -1, 0, 1)\\ (1, 0, 1, -1) \end{array}	(-1, 1, 0, 0)	\begin{array}{*{20}{c}} (-1, -1, 0, 0)\\ (0, 1, 0, 0) \end{array}	\begin{array}{*{20}{c}} (0, -1, 0, 0)\\ (1, 1, 0, 0) \end{array}	(1, -1, 0, 0)
18	19	20	21	22	23
\begin{array}{*{20}{c}} (-1, 0, -1, 1)\\ (-1, 1, 0, -1)\\ (-1, 1, 1, 1) \end{array}	\begin{array}{*{20}{c}} (-1, -1, 0, -1)\\ (-1, -1, 1, 1)\\ (0, 0, -1, 1)\\ (0, 1, 0, -1)\\ (0, 1, 1, 1) \end{array}	\begin{array}{*{20}{c}} (0, -1, 0, -1)\\ (0, -1, 1, 1)\\ (1, 0, -1, 1)\\ (1, 1, 0, -1)\\ (1, 1, 1, 1) \end{array}	\begin{array}{*{20}{c}} (1, -1, 0, -1)\\ (1, -1, 1, 1) \end{array}	\begin{array}{*{20}{c}} (-1, 0, -1, 0)\\ (-1, 1, 1, 0) \end{array}	\begin{array}{*{20}{c}} (-1, -1, 1, 0)\\ (0, 0, -1, 0)\\ (0, 1, 1, 0) \end{array}
24	25	26	27	28	29
\begin{array}{*{20}{c}} (0, -1, 1, 0)\\ (1, 0, -1, 0)\\ (1, 1, 1, 0) \end{array}	(1, -1, 1, 0)	\begin{array}{*{20}{c}} (-1, 0, -1, -1)\\ (-1, 0, 0, 1)\\ (-1, 1, 1, -1) \end{array}	\begin{array}{*{20}{c}} (-1, -1, 1, -1)\\ (0, 0, -1, -1)\\ (0, 0, 0, 1)\\ (0, 1, 1, -1) \end{array}	\begin{array}{*{20}{c}} (0, -1, 1, -1)\\ (1, 0, -1, -1)\\ (1, 0, 0, 1)\\ (1, 1, 1, -1) \end{array}	(1, -1, 1, -1)
30
(-1, 0, 0, 0)

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