<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>

中文翻译：

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关于 $ \mathbb{Z}/{p}\mathbb{Z} 上的多项式模数系统

<p style='text-indent:20px;'>自 2004 年推出以来，多项式模数系统 (PMNS) 已成为一种非常有趣的工具，用于以安全有效的方式实现依赖于模运算的密码系统。但是，尽管它们的实现很简单，但它们的参数化并不是微不足道的，并且依赖于PMN运行的多项式的合适选择。最初的提议是基于特定的二项式和三项式。但这些多项式并不总是为系统提供有趣的特征，例如小数字、快速缩减等。</p><p style='text-indent:20px;'>在这项工作中，我们研究了一个更大的多项式族可用于设计安全高效的 PMNS。为此，我们首先陈述了一个完整的 PMNS 存在定理，它为通用多项式的数字大小提供了界限，显着改善了先前的界限。然后，我们提出了合适的多项式类别，这些多项式为安全高效的算术提供了大量的 PMNS。</p>