Polynomial Modular Number System (PMNS) is a convenient number system for modular arithmetic, introduced in 2004. The main motivation was to accelerate arithmetic modulo an integer $p$. An existence theorem of PMNS with specific properties was given. The construction of such systems relies on sparse polynomials whose roots modulo $p$ can be chosen as radices of this kind of positional representation. However, the choice of those polynomials and the research of their roots are not trivial. In this paper, we introduce a general theorem on the existence of PMNS and we provide bounds on the size of the digits used to represent an integer modulo $p$. Then, we present classes of suitable polynomials to obtain systems with an efficient arithmetic. Finally, given a prime $p$, we evaluate the number of roots of polynomials modulo $p$ in order to give a number of PMNS bases we can reach. Hence, for a fixed prime $p$, it is possible to get numerous PMNS, which can be used efficiently for different applications based on large prime finite fields, such as those we find in cryptography, like RSA, Diffie-Hellmann key exchange and ECC (Elliptic Curve Cryptography).

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关于 $\mathbb{Z}/p\mathbb{Z}$ 上的多项式模数系统

多项式模数系统 (PMNS) 是一种方便的模算术数系统，于 2004 年推出。主要动机是加速算术模整数 $p$。给出了具有特定性质的PMNS的存在定理。这种系统的构建依赖于稀疏多项式，其根模 $p$ 可以被选为这种位置表示的基数。然而，这些多项式的选择及其根的研究并非易事。在本文中，我们介绍了关于 PMNS 存在的一般定理，并提供了用于表示整数模 $p$ 的数字大小的界限。然后，我们提出了合适的多项式类以获得具有有效算法的系统。最后，给定素数 $p$，我们评估多项式模 $p$ 的根数，以便给出我们可以达到的 PMNS 基数。因此，对于固定的质数 $p$，可以得到大量 PMNS，这些 PMNS 可以有效地用于基于大型质数有限域的不同应用，例如我们在密码学中发现的那些，如 RSA、Diffie-Hellmann 密钥交换和ECC（椭圆曲线密码术）。