Randomization of Arithmetic Over Polynomial Modular Number System

Abstract

The Polynomial Modular Number System (PMNS) is an integer number system designed to speed up arithmetic operations modulo a prime p. Such a system is defined by a tuple B = (p, n, , ⇢, E) where E 2 Z[X] and E() ⌘ 0 (mod p). In a PMNS, an element a of Z/pZ is represented by a polynomial A such that: A() ⌘ a (mod p), deg A < n and k Ak 1 < ⇢. In [6], the authors mentioned that PMNS can be highly redundant but they didn't really take advantage of this possibility. In this paper we use, for the first time, the redundancy of PMNS to protect algorithms against Side Channel Attacks (SCA). More precisely, we focus on elliptic curve cryptography. We show how to randomize the modular multiplication in order to be safe against existing SCA and we demonstrate the resistance of our construction. We describe the generation of a PMNS while guaranteeing, for all elements of Z/pZ, the minimum number of distinct representations we want. We also show how to reach all these representations.

On Polynomial Modular Number Systems over $\mathbb{Z}/p\mathbb{Z}$

“…2n − 1 (see Prop. 2.3 of [7]). As a consequence, if G(X) and F (X) are two elements of the PMNS, i.e.…”

An Alternative Approach to Polynomial Modular Number System Internal Reduction

“…In both cases E should be chosen so that the associated matrix E n−1 has the smallest norm. The rest of this work focuses on polynomials of the form X n − λ with |λ| as small as possible in order to compare the algorithms presented here to that used in [9]. In that case the following theorem holds.…”

“…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 this paper we propose two new approaches to perform the internal reduction based on two algorithms initially proposed by Babai to solve the Closest Vector Problem for integer lattices [1]. Those two algorithms have been mentioned before in the literature in order to obtain theoretical bounds on the size of the coefficients [4] but discarded as a practical tools due their high computational complexities [9]. We show that it is actually possible to design competitive algorithms based on Babai's ideas achieving faster internal reduction in many cases.…”

On Polynomial Modular Number Systems over $ \mathbb{Z}/{p}\mathbb{Z} $

“…A first hardware implementation of the AMNS is described in [17]. To end, it is shown in [22,49] that some "random steps" can be injected in AMNS multiplication in order to resist to a side channel analysis.…”