Dyadically resolving trinomials for fast modular arithmetic

TL;DR

This paper explores a family of trinomial moduli for residue number systems, providing methods to construct large coprime sets and analyzing their theoretical bounds for efficient modular arithmetic.

Contribution

It introduces a structured approach to generate large pairwise coprime trinomial moduli using graph theory and polynomial resultants, advancing residue number system design.

Findings

Established a coprimality condition based on polynomial resultants.

Developed a graph-theoretic model to construct large sets of moduli.

Proved upper bounds on the size of coprime sets using cyclotomic polynomial properties.

Abstract

Residue number systems based on pairwise relatively prime moduli are a powerful tool for accelerating integer computations via the Chinese Remainder Theorem. We study a structured family of moduli of the form $2^n - 2^k + 1$, originally proposed for their efficient arithmetic and bit-level properties. These trinomial moduli support fast modular operations and exhibit scalable modular inverses. We investigate the problem of constructing large sets of pairwise relatively prime trinomial moduli of fixed bit length. By analyzing the corresponding trinomials $x^n - x^k + 1$, we establish a sufficient condition for coprimality based on polynomial resultants. This leads to a graph-theoretic model where maximal sets correspond to cliques in a compatibility graph, and we use maximum clique-finding algorithms to construct large examples in practice. Using the theory of graph colorings, resultants, and properties of cyclotomic polynomials, we also prove upper bounds on the size of such sets as a function of $n$.