Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's pp-rationality conjecture

In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the $p$-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements

A Study of the Relationship between Foreign Aid and Human Development in Africa

Why are some countries more prosperous than others? Why are some countries still poor? What can be done by the West to help the rest to overcome the poverty trap? Finding better answers to these questions still represents the research agenda for development economists and political agenda for government and international institutions. Of course, the first two questions are age‐old ones and have been asked since the beginning of our history. The economic literature has identified important factors that influence the wealth of nations and they include: openness to trade, natural resources, capital accumulation, and innovation. Recent studies have found that cultural aspects and institutional framework tend to play a major role in a nation\u27s development process. The researchers’ work also helps policy makers to find a better answer to the last question. The purpose of this chapter is to evaluate the effectiveness of aid in eradicating poverty and improving life conditions in African countries since 1980. Since we are at the beginning of a new UN development agenda, it is important for all stakeholders (recipient, donors, international agencies, etc.) to identify the conditions that enable aid to work

Familles de courbes adaptées à la factorisation des entiers

Rapport de stage M1Dans la méthode des courbes elliptiques pour factoriser des entiers, on utilise en général des familles de courbes particulières qui permettent d'accélérer les calculs. La famille de Suyama est une de ces familles. Son efficacité est due à la présence d'un grand groupe de torsion. Nous proposons une démarche pour construire de nouvelles familles. En particulier, nous avons trouvé deux familles de courbes, chacune paramétrée par une courbe elliptique de rang 1. Il s'agit de sous-familles de la famille de Suyama qui offrent de meilleures performances

Improvements on the Discrete Logarithm Problem in GF(p)

International audienceThis paper speeds up descrete logarithm algorithms in two ways. First we show how the Factorization Factory can be adapted to the discrete logarithm to drop the complexity from Lp(1/3,1.902) to Lp(1/3,1.639). Next we prove that an early abort strategy can decrease the complexity of the individual logarithm from Lp(1/3,1.447) to Lp(1/3,1.232)

An appendix for a recent paper of Kim

This appendix proposes an improvement of Kim\u27s exTNFS. It has been merged the original paper of Taechan Kim 2015/1027 (version 1)

The Multiple Number Field Sieve for Medium and High Characteristic Finite Fields

International audienceIn this paper, we study the discrete logarithm problem in medium and high characteristic finite fields. We propose a variant of the Number Field Sieve~(NFS) based on numerous number fields. Our improved algorithm computes discrete logarithms in $\mathbb{F}_{p^n}$ for the whole range of applicability of NFS and lowers the asymptotic complexity from $L_{p^n}(1/3,(128/9)^{1/3})$ to $L_{p^n}(1/3,(2^{13}/3^6)^{1/3})$ in the medium characteristic case, and from $L_{p^n}(1/3,(64/9)^{1/3})$ to $L_{p^n}(1/3,((92 + 26 \sqrt{13})/27))^{1/3})$ in the high characteristic case.Version 2 contains an erratum.Dans cet article, nous étudions le problème du logarithme discret dans les corps finis de moyenne et grande caractéristique. Nous proposons une variante du crible algébrique basée sur plusieurs corps de nombres. Nous obtenons une accélération de $L_{p^n}(1/3,(128/9)^{1/3})$ à $L_{p^n}(1/3,(2^{13}/3^6)^{1/3})$ pour la moyenne caractéristique et de $L_{p^n}(1/3,(64/9)^{1/3})$ à $L_{p^n}(1/3,((92 + 26 \sqrt{13})/27))^{1/3})$ pour la grande. La version 2 contient un erratum

Finding ECM-friendly curves through a study of Galois properties

In this paper we prove some divisibility properties of the cardinality of elliptic curves modulo primes. These proofs explain the good behavior of certain parameters when using Montgomery or Edwards curves in the setting of the elliptic curve method (ECM) for integer factorization. The ideas of the proofs help us to find new families of elliptic curves with good division properties which increase the success probability of ECM

The special case of cyclotomic fields in quantum algorithms for unit groups

Unit group computations are a cryptographic primitive for which one has a fast quantum algorithm, but the required number of qubits is $\tilde{O}(m^5)$. In this work we propose a modification of the algorithm for which the number of qubits is $\tilde{O}(m^2)$ in the case of cyclotomic fields. Moreover, under a recent conjecture on the size of the class group of $\mathbb{Q}(\zeta_m+\zeta_m^{-1})$, the quantum algorithms is much simpler because it is a hidden subgroup problem (HSP) algorithm rather than its error estimation counterpart: continuous hidden subgroup problem (CHSP). We also discuss the (minor) speed-up obtained when exploiting Galois automorphisms thnaks to the Buchmann-Pohst algorithm over $\mathcal{O}_K$-lattices

A classification of ECM-friendly families using modular curves: intégré à la thèse de doctorat de Sudarshan Shinde, Sorbonne Université, 10 juillet 2020.

Validé par le jury de thèse de Sudarshan Shinde, Sorbonne Université, 10 juillet 2020.jury :Loïc Mérel (président)Jean-Marc Couveignes (rapporteur)David Zureick Brown (rapporteur)Annick ValibouzeBen SmithPierre-Voncent Koseleff (co-directeur)Razvan Barbulescu (co-drecteur)In this work, we establish a link between the classification of ECM-friendly curves and Mazur's program B, which consists in parameterizing all the families of elliptic curves with exceptional Galois image. Building upon two recent works which treated the case of congruence subgroups of prime-power level which occur for infinitely many $j$-invariants, we prove that there are exactly 1525 families of rational elliptic curves with distinct Galois images which are cartesian products of subgroups of prime-power level. This makes a complete list of rational families of ECM-friendly elliptic curves, out of which less than 25 were known in the literature. We furthermore refine a heuristic of Montgomery to compare these families and conclude that the best 4 families which can be put in $a=-1$ twisted Edwards' form are new