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,\gamma ,\rho, E) where E \in Z[X] and E(\gamma) = 0 (mod p). In a PMNS, an element a of Z/pZ is represented by a polynomial A such that: A(\gamma) = a (mod p), deg A < n and max a_i < \rho. 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.
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Laurent-Stéphane Didier, Fangan-Yssouf Dosso, Nadia El Mrabet, Jérémy Marrez, Pascal Véron. Randomization of Arithmetic over Polynomial Modular Number System. 26th IEEE International Symposium on Computer Arithmetic, Jun 2019, Kyoto, Japan. pp.199-206, ⟨10.1109/ARITH.2019.00048⟩. ⟨hal-02099713⟩

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