🤖 AI Summary
This work addresses the challenge of constructing large, pairwise coprime ternary moduli sets for Residue Number Systems (RNS) under fixed bit-width constraints, where efficient modular arithmetic is critical.
Method: We propose a novel coprimality criterion based on polynomial resultants and cyclotomic polynomials, model modulus pairwise coprimality as a compatibility graph, and—uniquely—apply maximum clique search algorithms to identify optimal moduli subsets.
Contribution/Results: We theoretically establish an upper bound on the size of moduli sets of the form $2^n - 2^k + 1$. Experimental validation via the Chinese Remainder Theorem confirms their efficacy in fast modular reduction and scalable parallel integer arithmetic. The approach bridges theoretical rigor and hardware-oriented practicality, yielding an extensible, principled framework for RNS modulus design suitable for VLSI implementation.
📝 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 $n$.