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

Discrete logarithms in curves over finite fields

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

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

Schertz style class invariants for quartic CM fields

A class invariant is a CM value of a modular function that lies in a certain unram-ified class field. We show that Siegel modular functions over $Q$ for $Γ^0 (N) ⊆ Sp_4 (Z)$yield class invariants under some splitting conditions on N. Small class invariants speed up constructions in explicit class field theory and public-key cryptography. Our results generalise results of Schertz's from elliptic curves to abelian varieties and from classical modular functions to Siegel modular functions

Short addition sequences for theta functions

International audienceThe main step in numerical evaluation of classical Sl2 (Z) modular forms and elliptic functions is to compute the sum of the first N nonzero terms in the sparse q-series belonging to the Dedekind eta function or the Jacobi theta constants. We construct short addition sequences to perform this task using N + o(N) multiplications. Our constructions rely on the representability of specific quadratic progressions of integers as sums of smaller numbers of the same kind. For example, we show that every generalised pentagonal number c 5 can be written as c = 2a + b where a, b are smaller generalised pentagonal numbers. We also give a baby-step giant-step algorithm that uses O(N/ log r N) multiplications for any r > 0, beating the lower bound of N multiplications required when computing the terms explicitly. These results lead to speed-ups in practice

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

Capture Hi-C identifies the chromatin interactome of colorectal cancer risk loci.

Multiple regulatory elements distant from their targets on the linear genome can influence the expression of a single gene through chromatin looping. Chromosome conformation capture implemented in Hi-C allows for genome-wide agnostic characterization of chromatin contacts. However, detection of functional enhancer-promoter interactions is precluded by its effective resolution that is determined by both restriction fragmentation and sensitivity of the experiment. Here we develop a capture Hi-C (cHi-C) approach to allow an agnostic characterization of these physical interactions on a genome-wide scale. Single-nucleotide polymorphisms associated with complex diseases often reside within regulatory elements and exert effects through long-range regulation of gene expression. Applying this cHi-C approach to 14 colorectal cancer risk loci allows us to identify key long-range chromatin interactions in cis and trans involving these loci