Failure of the Point Blinding Countermeasure Against Fault Attack in Pairing-Based Cryptography

Article published in the proceedings of the C2SI conference, May 2015.Pairings are mathematical tools that have been proven to be very useful in the construction of many cryptographic protocols. Some of these protocols are suitable for implementation on power constrained devices such as smart cards or smartphone which are subject to side channel attacks. In this paper, we analyse the efficiency of the point blinding countermeasure in pairing based cryptography against side channel attacks. In particular,we show that this countermeasure does not protect Miller's algorithm for pairing computation against fault attack. We then give recommendation for a secure implementation of a pairing based protocol using the Miller algorithm

Harmonic Analysis and a Bentness-Like Notion in Certain Finite Abelian Groups Over Some Finite Fields

Article published in Malaysian Journal of Mathematical SciencesIt is well-known that degree two finite field extensions can be equipped with a Hermitian-like structure similar to the extension of the complex field over the reals. In this contribution, using this structure, we develop a modular character theory and the appropriate Fourier transform for some particular kind of finite Abelian groups. Moreover we introduce the notion of bent functions for finite field valued functions rather than usual complex-valued functions, and we study several of their properties

On Near Prime-Order Elliptic Curves with Small Embedding Degrees

Article published in the proceeding of the conference CAI 2015 http://www.ims.uni-stuttgart.de/events/CAI2015In this paper, we generalize the method of Scott and Barreto in order to construct a family of pairing-friendly elliptic curve. We present an explicit algorithm to obtain generalized MNT families curves with any cofactors. We also analyze the complex multiplication equations of these curves and transform them into generalized Pell equation. As an example, we describe a way to generate Edwards curves with embedding degree 6

Computing Optimal Ate Pairings on Elliptic Curves with Embedding Degree 9,159,15 and 2727

Much attention has been given to efficient computation of pairings on elliptic curves with even embedding degree since the advent of pairing-based cryptography. The existing few works in the case of odd embedding degrees require some improvements. This paper considers the computation of optimal ate pairings on elliptic curves of embedding degrees k=9, 15 \mbox{ and } 27 which have twists of order three. Mainly, we provide a detailed arithmetic and cost estimation of operations in the tower extensions field of the corresponding extension fields. A good selection of parameters enables us to improve the theoretical cost for the Miller step and the final exponentiation using the lattice-based method comparatively to the previous few works that exist in these cases. In particular for $k=15$ and $k=27$ we obtained an improvement, in terms of operations in the base field, of up to $25\%$ and $29\%$ respectively in the computation of the final exponentiation. Also, we obtained that elliptic curves with embedding degree $k=15$ present faster results than BN$12$ curves at the $128$-bit security levels. We provided a MAGMA implementation in each case to ensure the correctness of the formulas used in this work

Differential Power Analysis against the Miller Algorithm

Article en cours de publicationPairings permit several protocol simplications and original scheme creation, for example Identity Based Cryptography protocols. Initially, the use of pairings did not involve any secret entry, consequently, side channel attacks were not a threat for pairing based cryptography. On the contrary, in an Identity Based Cryptographic protocol, one of the two entries to the pairing is secret. Side Channel Attacks can be therefore applied to nd this secret. We realize a Differential Power Analysis(DPA) against the Miller algorithm, the central step to compute the Weil, Tate and Ate pairing. Keywords: Pairing, Miller Algorithm, Pairing Based Cryptography, SCA, DPA

Choosing and generating parameters for low level pairing implementation on BN curves

Many hardware and software pairing implementations can be found in the literature and some pairing friendly parameters are given. However, depending on the situation, it could be useful to generate other nice parameters (e.g. resistance to subgroup attacks, larger security levels, database of pairing friendly curves). The main purpose of this paper is to describe explicitly and exhaustively what should be done to generate the best possible parameters and to make the best choices depending on the implementation context (in terms of pairing algorithm, ways to build the tower field, $\mathbb{F}_{p^{12}}$ arithmetic, groups involved and their generators, system of coordinates). We focus on low level implementations, assuming that $\mathbb{F}_p$ additions have a significant cost compared to other $\mathbb{F}_p$ operations. However, the results obtained are still valid in the case where $\mathbb{F}_p$ additions can be neglected. We also explain why the best choice for the polynomials defining the tower field $\mathbb{F}_{p^{12}}$ is only depending on the value of the BN parameter $u$ modulo small integers like $12$ as a nice application of old elementary arithmetic results. Moreover, we use this opportunity to give some new improvements on $\mathbb{F}_{p^{12}}$ arithmetic (in a pairing context) in terms of $\mathbb{F}_p$-addition allowing to save around $10\%$ of them depending on the context

On near prime-order elliptic curves with small embedding degrees (Full version)

In this paper, we extend the method of Scott and Barreto and present an explicit and simple algorithm to generate families of generalized MNT elliptic curves. Our algorithm allows us to obtain all families of generalized MNT curves with any given cofactor. Then, we analyze the complex multiplication equations of these families of curves and transform them into generalized Pell equation. As an example, we describe a way to generate Edwards curves with embedding degree 6, that is, elliptic curves having cofactor h = 4

PMNS revisited for consistent redundancy and equality test

The Polynomial Modular Number System (PMNS) is a non-positional number system for modular arithmetic. A PMNS is defined by a tuple $(p, n, \gamma, \rho, E)$, where $p$, $n$, $\gamma$ and $\rho$ are positive non-zero integers and $E\in\mathbb{Z}_{n}[X]$ is a monic polynomial such that $E(\gamma) \equiv 0 \pmod p$. The PMNS is a redundant number system. In~\cite{rando-pmns-arith26}, Didier et al. used this redundancy property to randomise the data during the Elliptic Curve Scalar Multiplication (ECSM). In this paper, we refine the results on redundancy and propose several new results on PMNS. More precisely, we study a generalisation of the Montgomery-like internal reduction method proposed by Negre and Plantard, along with some improvements on parameter bounds for smaller memory cost to represent elements in this system. We also show how to perform equality test in the PMNS