Abstract.We complete the study of [16] and [20] about the Miller's algorithm. The Miller's algorithm is a central step to compute the Weil, Tate and Ate pairings. The aim of this article is to analyse the weakness of the Miller's algorithm when it undergoes a fault attack. We prove that the Miller's algorithm is vulnerable to a fault attack which is valid in all coordinate systems, through the resolution of a nonlinear system. We show that the final exponentiation is no longer a counter measure to this attack for the Tate and Ate pairings.

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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.