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

<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 secure and efficient way. However, while implementation is simple, parameterization not trivial relies suitable choice of the polynomial which PMNS operates. The initial proposals were based particular binomials trinomials. But these polynomials do always provide systems with characteristics such as small digits, fast reduction, etc.</p><p style='text-indent:20px;'>In this work, we study larger family that can be exploited to design safe PMNS. To so, first state complete existence theorem provides bounds size digits generic polynomial, significantly improving previous bounds. Then, present classes numerous arithmetic.</p>