Discrete logarithms in curves over finite fields

A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields

The complexity of class polynomial computation via floating point approximations

We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmetic-geometric mean. It runs in time $O (|D| \log^5 |D| \log \log |D|) = O (|D|^{1 + \epsilon}) = O (h^{2 + \epsilon})$ for any $\epsilon > 0$, where $D$ is the CM discriminant and $h$ is the degree of the class polynomial. Another fast algorithm uses multipoint evaluation techniques known from symbolic computation; its asymptotic complexity is worse by a factor of $\log |D|$. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary-quadratic order and on a rigorously proven upper bound for the height of class polynomials

Generalised Weber Functions

A generalised Weber function is given by \w_N(z) = \eta(z/N)/\eta(z), where $\eta(z)$ is the Dedekind function and $N$ is any integer; the original function corresponds to $N=2$. We classify the cases where some power \w_N^e evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formula for the degree of the modular equation relating \w_N(z) and $j(z)$. Our ultimate goal is the use of these invariants in constructing reductions of elliptic curves over finite fields suitable for cryptographic use

FastECPP over MPI

The FastECPP algorithm is currently the fastest approach to prove theprimality of general numbers, and has the additional benefit of creatingcertificates that can be checked independently and with a lower complexity.This article shows how by parallelising over a linear number of cores,its quartic time complexity becomes a cubic wallclock time complexity;and it presents the algorithmic choices of the FastECPP implementation inthe author's \cm\ software (https://www.multiprecision.org/cm/) which has been written with massive parallelisation over MPI in mind, and which has been used to establish a new primality record for the "repunit" $(10^{86453} - 1) / 9$

Schertz style class invariants for higher degree CM fields

Special values of Siegel modular functions for $\operatorname{Sp} (\mathbb{Z})$ generate class fields of CM fields. They also yield abelian varieties with a known endomorphism ring. Smaller alternative values of modular functions that lie in the same class fields (class invariants) thus help to speed up the computation of those mathematical objects. We show that modular functions for the subgroup $\Gamma^0 (N)\subseteq \operatorname{Sp}(\mathbb{Z})$ yield class invariants under some splitting conditions on $N$, generalising results due to Schertz from classical modular functions to Siegel modular functions. We show how to obtain all Galois conjugates of a class invariant by evaluating the same modular function in CM period matrices derived from an \emph{$N$-system}. Such a system consists of quadratic polynomials with coefficients in the real-quadratic subfield satisfying certain congruence conditions modulo $N$. We also examine conditions under which the minimal polynomial of a class invariant is real. Examples show that we may obtain class invariants that are much smaller than in previous constructions

The arithmetic of Jacobian groups of superelliptic cubics

International audienceWe present two algorithms for the arithmetic of cubic curves with a totally ramified prime at infinity. The first algorithm, inspired by Cantor's reduction for hyperelliptic curves, is easily implemented with a few lines of code, making use of a polynomial arithmetic package. We prove explicit reducedness criteria for superelliptic curves of genus 3 and 4, which show the correctness of the algorithm. The second approach, quite general in nature and applicable to further classes of curves, uses the FGLM algorithm for switching between Gröbner bases for different orderings. Carrying out the computations symbolically, we obtain explicit reduction formulae in terms of the input data

Discrete logarithms in curves over finite fields

International audienceA survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields

Brain-Derived Neurotrophic Factor (Val66Met) and Serotonin Transporter (5-HTTLPR) Polymorphisms Modulate Plasticity in Inhibitory Control Performance Over Time but Independent of Inhibitory Control Training

Several studies reported training-induced improvements in executive function tasks and also observed transfer to untrained tasks. However, the results are mixed and there is large interindividual variability within and across studies. Given that training-related performance changes would require modification, growth or differentiation at the cellular and synaptic level in the brain, research on critical moderators of brain plasticity potentially explaining such changes is needed. In the present study, a pre-post-follow-up design (N=122) and a three-weeks training of two response inhibition tasks (Go/NoGo and Stop-Signal) was employed and genetic variation (Val66Met) in the brain-derived neurotrophic factor (BDNF) promoting differentiation and activity-dependent synaptic plasticity was examined. Because Serotonin (5-HT) signaling and the interplay of BDNF and 5-HT are known to critically mediate brain plasticity, genetic variation in the 5-HT transporter (5-HTTLPR) was also addressed. The overall results show that the kind of training (i.e., adaptive vs. non-adaptive) did not evoke genotype-dependent differences. However, in the Go/NoGo task, better inhibition performance (lower commission errors) were observed for BDNF Val/Val genotype carriers compared to Met-allele ones supporting similar findings from other cognitive tasks. Additionally, a gene-gene interaction suggests a more impulsive response pattern (faster responses accompanied by higher commission error rates) in homozygous l-allele carriers relative to those with the s-allele of 5-HTTLPR. This, however, is true only in the presence of the Met-allele of BDNF, while the Val/Val genotype seems to compensate for such non-adaptive responding. Intriguingly, similar results were obtained for the Stop-Signal task. Here, differences emerged at post-testing, while no differences were observed at T1. In sum, although no genotype-dependent differences between the relevant training groups emerged suggesting no changes in the trained inhibition function, the observed genotype-dependent performance changes from pre- to post measurement may reflect rapid learning or memory effects linked to BDNF and 5-HTTLPR. In line with ample evidence on BDNF and BDNF-5-HT system interactions to induce (rapid) plasticity especially in hippocampal regions and in response to environmental demands, the findings may reflect genotype-dependent differences in the acquisition and consolidation of task-relevant information, thereby facilitating a more adaptive responding to task-specific requirements