Algebraic Computations with Continued Fractions

AbstractGeneral algorithms, viewed as transducers, are introduced for computing rational expressions with continued fraction expansions. Moreover, expansions of some algebraic numbers, like2or those related to primitive matrices are considered

Unifom Generators and Combinatorial Design

International audienceThe concept of randomness is fundamental in many domains and in particular in cryptography. Intuitively, a system, which is unpredictable is more difficult to attack and as a consequence, creating sequences that look like random represents a major issue. In this paper, we first study theoretically how a source of symbols with positive entropy can be turned into a true random generator called Bernoulli. We concentrate on a special type of generators, which consists in randomly choosing k elements out of n elements. After studying some existing algorithms, which are of Las Vegas type, we introduce new constructions from a binary generator taken as a primary random source of symbols. Our method is based on combinatorial block designs and we construct algorithms of Monte Carlo type involving random walks. We analyze in detail properties of our general method. Several explicit constructions of k-out-of-n generators are given. We show that the speed of convergence to the uniform distribution is better than any known method using algorithms with bounded running times

Log-linear Convergence and Optimal Bounds for the (1+1)(1+1)-ES

International audienceThe $(1+1)$-ES is modeled by a general stochastic process whose asymptotic behavior is investigated. Under general assumptions, it is shown that the convergence of the related algorithm is sub-log-linear, bounded below by an explicit log-linear rate. For the specific case of spherical functions and scale-invariant algorithm, it is proved using the Law of Large Numbers for orthogonal variables, that the linear convergence holds almost surely and that the best convergence rate is reached. Experimental simulations illustrate the theoretical results

Distribution functions of the sequence phi(n)/n, n in (k,k+N]

International audienceIt is well known that the sequence $\varphi(n)/n$, n=1,2,... has a singular asymptotic distribution function. P. Erdös in 1946 found a sufficient condition on the sequence of intervals (k,k+N], such that phi(n)/n, n in (k,k+N], has the same singular function. In this note we prove a sufficient and necessary condition. For simplifying the necessary condition we express the sum \sum_{k n*k+N(!(n) ¡ log logN)2, where !(n) is the number of di®erent primes divided n

AES Side-Channel Countermeasure using Random Tower Field Constructions

International audienceMasking schemes to secure AES implementations against side-channel attacks is a topic of ongoing research. The most sensitive part of the AES is the non-linear SubBytes operation, in particular, the inversion in GF(2^8), the Galois field of 2^8 elements. In hardware implementations, it is well known that the use of the tower of extensions GF(2) ⊂ GF(2^2) ⊂ GF(2^4) ⊂ GF(2^8) leads to a more efficient inversion. We propose to use a random isomorphism instead of a fixed one. Then, we study the effect of this randomization in terms of security and efficiency. Considering the field extension GF(2^8)/GF(2^4), the inverse operation leads to computation of its norm in GF(2^4). Hence, in order to thwart side-channel attack, we manage to spread the values of norms over GF(2^4). Combined with a technique of boolean masking in tower fields, our countermeasure strengthens resistance against first-order differential side-channel attacks

The dynamical point of view of low-discrepancy sequences

International audienceIn this overview we show by examples, how to associate certain sequences in the higher-dimensional unit cube to suitable dynamical systems. We present methods and notions from ergodic theory that serve as tools for the study of low-discrepancy sequences and discuss an important technique, cutting- and-stacking of intervals