An Alternative Approach to Polynomial Modular Number System Internal Reduction

Abstract

The Polynomial Modular Number System (PMNS) is an alternative to the binary multi-precision representation that allows to transport the arithmetic of a finite field to a polynomial ring. The most important operation in that system is the internal reduction that follows any arithmetic operation. All recent works on the subject use the same algorithm derived from Montgomery's modular multiplications to perform this internal reduction. This paper designs and analyzes two alternative algorithms to perform the internal reduction, both based on Babai's Closest Vector algorithms. It allows to significantly reduce the number of additions needed to perform this operation. A comprehensive experimental analysis shows that one of those algorithms is also faster in practice. For that matter, a C code generation tool has been developed in order to produce implementations for any prime field.