Résultats de l’Enquête internationale 2013 « Quelles activités, quels processus, quelles méthodes, quels systèmes »

International audienc

Management Control, International Observatory 2012

International audienc

Elliptic Curve Scalar Multiplication Combining Yao’s Algorithm and Double Bases

Abstract. In this paper we propose to take one step back in the use of double base number systems for elliptic curve point scalar multiplication. Using a mod-ified version of Yao’s algorithm, we go back from the popular double base chain representation to a more general double base system. Instead of representing an integer k as Pn i=1 2 bi3ti where (bi) and (ti) are two decreasing sequences, we only set a maximum value for both of them. Then, we analyze the efficiency of our new method using different bases and optimal parameters. In particular, we pro-pose for the first time a binary/Zeckendorf representation for integers, providing interesting results. Finally, we provide a comprehensive comparison to state-of-the-art methods, including a large variety of curve shapes and latest point addition formulae speed-ups

A Formal Library for Elliptic Curves in the Coq Proof Assistant

International audienceA preliminary step towards the verification of elliptic curve cryptographic algorithms is the development of formal libraries with the corresponding mathematical theory. In this paper we present a formaliza-tion of elliptic curves theory, in the SSReflect extension of the Coq proof assistant. Our central contribution is a library containing many of the objects and core properties related to elliptic curve theory. We demonstrate the applicability of our library by formally proving a non-trivial property of elliptic curves: the existence of an isomorphism between a curve and its Picard group of divisors

Management control – International Observatory 2014

International audienc

Testing data types implementations from algebraic specifications

Algebraic specifications of data types provide a natural basis for testing data types implementations. In this framework, the conformance relation is based on the satisfaction of axioms. This makes it possible to formally state the fundamental concepts of testing: exhaustive test set, testability hypotheses, oracle. Various criteria for selecting finite test sets have been proposed. They depend on the form of the axioms, and on the possibilities of observation of the implementation under test. This last point is related to the well-known oracle problem. As the main interest of algebraic specifications is data type abstraction, testing a concrete implementation raises the issue of the gap between the abstract description and the concrete representation. The observational semantics of algebraic specifications bring solutions on the basis of the so-called observable contexts. After a description of testing methods based on algebraic specifications, the chapter gives a brief presentation of some tools and case studies, and presents some applications to other formal methods involving datatypes

An exercise on the Average Number of Real Zeros of Random Real Polynomials.

International audienc

On the Enumeration of Double-Base Chains with Applications to Elliptic Curve Cryptography

The Double-Base Number System (DBNS) uses two bases, 2 and 3, in order to represent any integer n. A Double-Base Chain (DBC) is a special case of a DBNS expansion. DBCs have been introduced to speed up the scalar multiplication [n]P on certain families of elliptic curves used in cryptography. In this context, our contributions are twofold. First, given integers n, a, and b, we outline a recursive algorithm to compute the number of different DBCs with a leading factor dividing 2a3b and representing n. A simple modification of the algorithm allows to determine the number of DBCs with a specified length as well as the actual expansions. In turn, this gives rise to a method to compute an optimal DBC representing n, i.e. an expansion with minimal length. Our implementation is able to return an optimal expansion for most integers up to 2⁶⁰ bits in a few minutes. Second, we introduce an original and potentially more efficient approach to compute a random scalar multiplication [n]P, based on the concept of controlled DBC. Instead of generating a random integer n and then trying to find an optimal, or at least a short DBC to represent it, we propose to directly generate n as a random DBC with a chosen leading factor 2a3b and length ℓ. To inform the selection of those parameters, in particular ℓ, which drives the trade-off between the efficiency and the security of the underlying cryptosystem, we enumerate the total number of DBCs having a given leading factor 2a3b and a certain length ℓ. The comparison between this total number of DBCs and the total number of integers that we wish to represent a priori provides some guidance regarding the selection of suitable parameters. Experiments indicate that our new Near Optimal Controlled DBC approach provides a speedup of at least 10% with respect to the NAF for sizes from 192 to 512 bits. Computations involve elliptic curves defined over Fp, using the Inverted Edwards coordinate system and state of the art scalar multiplication techniques.20 page(s