Efficient modular operations using the adapted modular number system

Abstract

The adapted modular number system (AMNS) is an integer number system which aims to speed up arithmetic operations modulo a prime p. Such a system is defined by a tuple $$(p, n, \gamma , \rho , E)$$, where p, n, $$\gamma $$ and $$\rho $$ are integers and $$E\in \mathbb {Z}[X]$$. In El Mrabet and Gama (in: WAIFI, lecture notes in computer science, Springer, 2012) conditions required to build AMNS with $$E(X)=X^n + 1$$ are provided. In this paper, we generalise their approach and provide a method to generate multiple AMNS for a given prime p with $$E(X)=X^n-\lambda $$ and $$\lambda \in \mathbb {Z}{\setminus }\{0\}$$. Moreover, we propose a complete set of algorithms without conditional branching to perform arithmetic and conversion operations in the AMNS, using a Montgomery-like method described in Negre and Plantard (in: Information security and privacy, 13th Australasian conference, ACISP 2008, Wollongong, Australia, 2008). We show that our implementation outperforms GNU MP and OpenSSL libraries. Finally, we highlight some properties of the AMNS which state that it could lead to a helpful countermeasure against some side-channel attacks.