Two-sources Randomness Extractors for Elliptic Curves

This paper studies the task of two-sources randomness extractors for elliptic curves defined over finite fields $K$, where $K$ can be a prime or a binary field. In fact, we introduce new constructions of functions over elliptic curves which take in input two random points from two differents subgroups. In other words, for a ginven elliptic curve $E$ defined over a finite field $\mathbb{F}_q$ and two random points $P \in \mathcal{P}$ and $Q\in \mathcal{Q}$, where $\mathcal{P}$ and $\mathcal{Q}$ are two subgroups of $E(\mathbb{F}_q)$, our function extracts the least significant bits of the abscissa of the point $P\oplus Q$ when $q$ is a large prime, and the $k$-first $\mathbb{F}_p$ coefficients of the asbcissa of the point $P\oplus Q$ when $q = p^n$, where $p$ is a prime greater than $5$. We show that the extracted bits are close to uniform. Our construction extends some interesting randomness extractors for elliptic curves, namely those defined in \cite{op} and \cite{ciss1,ciss2}, when $\mathcal{P} = \mathcal{Q}$. The proposed constructions can be used in any cryptographic schemes which require extraction of random bits from two sources over elliptic curves, namely in key exchange protole, design of strong pseudo-random number generators, etc

Incomplete exponential sums and Diffie-Hellman triples

http://www.math.missouri.edu/~bbanks/papers/index.htmlLet p be a prime and 79 an integer of order t in the multiplicative group modulo p. In this paper, we continue the study of the distribution of Diffie-Hellman triples (V-x, V-y, V-xy) by considering the closely related problem of estimating exponential sums formed from linear combinations of the entries in such triples. We show that the techniques developed earlier for complete sums can be combined, modified and developed further to treat incomplete sums as well. Our bounds imply uniformity of distribution results for Diffie-Hellman triples as the pair (x, y) varies over small boxes

Extracteur aléatoires multi-sources sur les corps finis et les courbes elliptiques

International audienceWe propose two-sources randomness extractors over finite fields and on elliptic curves that can extract from two sources of information without consideration of other assumptions that the starting algorithmic assumptions with a competitive level of security. These functions have several applications. We propose here a description of a version of a Diffie-Hellman key exchange protocol and key extraction.Nous proposons des extracteurs d'aléas 2-sources sur les corps finis et sur les courbes elliptiques capables d'extraire à partir de plusieurs sources d'informations sans considération d'autres hypothèses que les hypothèses algorithmiques de départ avec un niveau de sécurité compétitif. Ces fonctions possèdent plusieurs applications. Nous proposons ici une version du protocole d'échange de clé Diffie-Hellman incluant la phase d'extraction

Quantum algorithms for algebraic problems

Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation, and in particular, on problems with an algebraic flavor.Comment: 52 pages, 3 figures, to appear in Reviews of Modern Physic