Efficient modular operations using the adapted modular number system

Abstract

The Adapted Modular Number System (AMNS) is a sytem of representation of integers to speed up arithmetic operations modulo a prime p. Such a system can be defined by a tuple (p, n, γ, ρ, E) where E ∈ Z[X]. In [13] conditions are given to build AMNS with E(X) = X n + 1. In this paper, we generalize their results and show how to generate multiple AMNS for a given prime p with E(X) = X n − λ and λ ∈ Z. Moreover, we propose a complete set of algorithms without conditional branching to perform arithmetic and conversion operations in the AMNS, using a Montgomerylike method described in [15].

Randomization of Arithmetic Over Polynomial Modular Number System

“…It is computed through the process. In [7], [14] the authors show that it is always possible to build many PMNS for a given modulus p.…”

“…The choice of the polynomial M to ensure the existence of M 0 is discussed in [14]. The [25], gives a bound on the coe cients of the polynomial computed with the Algorithm 1.…”

“…We assume that the inputs of these algorithms belong to B = (p, n, , ⇢, E), with ⇢ = 2 w , E(X) = X n and = ±2 u with w, u 2 N. It has been shown in [14] that it is always possible to build such PMNS. As an example, with such notations the addition of two c-bit integers requires dc/we w-bit additions.…”

An Alternative Approach to Polynomial Modular Number System Internal Reduction

“…Two approaches have been proposed to perform it based either on Barrett modular reduction algorithm [3] or Montgomery's one [14]. During the past few years improvements have been made on the implementation [10], generation [4], [8], randomization [9] and generalization [6], [11] of PMNS in various contexts. One interesting common feature between all those works is that they all perform the internal reduction step using the Montgomery-like approach.…”

“…In Section III we propose and study two new internal reduction algorithms and two optimized versions of those. In Section IV we give implementation details about our code generation tool targeting 64-bit architectures and we illustrate the efficiency of our approach with experimental results and comparisons with a similar tool based on Montgomery reduction [10].…”