论文标题
在$ \ mathbb {z}/p \ mathbb {z} $的多项式模块化系统上
On Polynomial Modular Number Systems over $\mathbb{Z}/p\mathbb{Z}$
论文作者
论文摘要
自2004年推出以来，多项式模块化数字系统（PMN）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靠模块化算术的密码系统。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。最初的建议基于特定的二项式和三项官方。但是这些多项式并不总是为系统提供有趣的特征，例如小数，快速减少等。 在这项工作中，我们研究了一个可以利用的多项式家庭来设计安全有效的PMN。为此，我们首先陈述了PMN的完整存在定理，该定理为通用多项式的数字大小提供了界限，从而显着改善了以前的界限。然后，我们提出合适的多项式类别的类别，这些类别为安全有效的算术提供了许多PMN。
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.
