关于 $\mathbb{Z}/p\mathbb{Z}$ 上的多项式模数系
On Polynomial Modular Number Systems over $\mathbb{Z}/p\mathbb{Z}$
AI总结：
本文研究更广泛的多项式族以设计安全高效的PMNS，提出完整存在性定理并显著改进数字大小界限，同时给出合适的多项式类别。
摘要：
自2004年提出以来，多项式模数系（PMNS）已成为一种非常有用的工具，用于以安全且高效的方式实现依赖模算术的密码系统。然而，尽管其实现简单，其参数化却并非易事，并且依赖于对PMNS所基于的多项式的合适选择。最初的方案基于特定的二项式和三项式。但这些多项式并不总能提供具有理想特性的系统，例如较小的数字、快速约简等。在本工作中，我们研究了一类更广泛的多项式，可用于设计安全且高效的PMNS。为此，我们首先给出了PMNS的完整存在性定理，该定理为一般多项式提供了数字大小的界限，显著改进了先前的界限。然后，我们提出了合适的多项式类别，为安全且高效的算术提供了大量PMNS。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
