The decimation process in random k-SAT

Let F be a uniformly distributed random k-SAT formula with n variables and m clauses. Non-rigorous statistical mechanics ideas have inspired a message passing algorithm called Belief Propagation Guided Decimation for finding satisfying assignments of F. This algorithm can be viewed as an attempt at implementing a certain thought experiment that we call the Decimation Process. In this paper we identify a variety of phase transitions in the decimation process and link these phase transitions to the performance of the algorithm

Recovering the mass and the charge of a Reissner-Nordstr\"om black hole by an inverse scattering experiment

In this paper, we study inverse scattering of massless Dirac fields that propagate in the exterior region of a Reissner-Nordstr\"om black hole. Using a stationary approach we determine precisely the leading terms of the high-energy asymptotic expansion of the scattering matrix that, in turn, permit us to recover uniquely the mass of the black hole and its charge up to a sign

Genetic diversity and phylogeny of Aedes aegypti, the main arbovirus vector in the Pacific

Background The Pacific region is an area unique in the world, composed of thousands of islands with differing climates and environments. The spreading and establishment of the mosquito Aedes aegypti in these islands might be linked to human migration. Ae. aegypti is the major vector of arboviruses (dengue, chikungunya and Zika viruses) in the region. The intense circulation of these viruses in the Pacific during the last decade led to an increase of vector control measures by local health authorities. The aim of this study is to analyze the genetic relationships among Ae. aegypti populations in this region. Methodology/Principal Finding We studied the genetic variability and population genetics of 270 Ae. aegypti, sampled from 9 locations in New Caledonia, Fiji, Tonga and French Polynesia by analyzing nine microsatellites and two mitochondrial DNA regions (CO1 and ND4). Microsatellite markers revealed heterogeneity in the genetic structure between the western, central and eastern Pacific island countries. The microsatellite markers indicate a statistically moderate differentiation (FST = 0.136; P < = 0.001) in relation to island isolation. A high degree of mixed ancestry can be observed in the most important towns (e.g. Noumea, Suva and Papeete) compared with the most isolated islands (e.g. Ouvea and Vaitahu). Phylogenetic analysis indicated that most of samples are related to Asian and American specimens. Conclusions/Significance Our results suggest a link between human migrations in the Pacific region and the origin of Ae. aegypti populations. The genetic pattern observed might be linked to the island isolation and to the different environmental conditions or ecosystems

Rounding and Chaining LLL: Finding Faster Small Roots of Univariate Polynomial Congruences

International audienceIn a seminal work at EUROCRYPT '96, Coppersmith showed how to find all small roots of a univariate polynomial congruence in polynomial time: this has found many applications in public-key cryptanalysis and in a few security proofs. However, the running time of the algorithm is a high-degree polynomial, which limits experiments: the bottleneck is an LLL reduction of a high-dimensional matrix with extra-large coefficients. We present in this paper the first significant speedups over Coppersmith's algorithm. The first speedup is based on a special property of the matrices used by Coppersmith's algorithm, which allows us to provably speed up the LLL reduction by rounding, and which can also be used to improve the complexity analysis of Coppersmith's original algorithm. The exact speedup depends on the LLL algorithm used: for instance, the speedup is asymptotically quadratic in the bit-size of the small-root bound if one uses the Nguyen-Stehlé L2 algorithm. The second speedup is heuristic and applies whenever one wants to enlarge the root size of Coppersmith's algorithm by exhaustive search. Instead of performing several LLL reductions independently, we exploit hidden relationships between these matrices so that the LLL reductions can be somewhat chained to decrease the global running time. When both speedups are combined, the new algorithm is in practice hundreds of times faster for typical parameters

Bounding basis reduction properties

The paper describes improved analysis techniques for basis reduction that allow one to prove strong complexity bounds and reduced basis guarantees for traditional reduction algorithms and some of their variants. This is achieved by a careful exploitation of the linear equations and inequalities relating various bit sizes before and after one or more reduction steps

Another view of the Gaussian algorithm

We introduce here a rewrite system in the group of unimodular matrices, i.e., matrices with integer entries and with determinant equal to ±1. We use this rewrite system to precisely characterize the mechanism of the Gaussian algorithm, that finds shortest vectors in a two–dimensional lattice given by any basis. Putting together the algorithmic of lattice reduction and the rewrite system theory, we propose a new worst–case analysis of the Gaussian algorithm. There is already an optimal worst–case bound for some variant of the Gaussian algorithm due to Vallée [16]. She used essentially geometric considerations. Our analysis generalizes her result to the case of the usual Gaussian algorithm. An interesting point in our work is its possible (but not easy) generalization to the same problem in higher dimensions, in order to exhibit a tight upper-bound for the number of iterations of LLL–like reduction algorithms in the worst case. Moreover, our method seems to work for analyzing other families of algorithms. As an illustration, the analysis of sorting algorithms are briefly developed in the last section of the paper

Equivalence Classes of Random Boolean Trees and Application to the Catalan Satisfiability Problem

International audienceAn and/or tree is a binary plane tree, with internal nodes labelled by connectives, and with leaves labelled by literals chosen in a fixed set of $k$ variables and their negations. We introduce the first model of such Catalan trees, whose number of variables $k_n$ is a function of $n$, its number of leaves. We describe the whole range of the probability distributions depending on the functions $k_n$, as soon as it tends jointly with $n$ to infinity. As a by-product we obtain a study of the satisfiability problem in the context of Catalan trees.Our study is mainly based on analytic combinatorics and extends the Kozik’s pattern theory, first developed for the fixed-$k$ Catalan tree model