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

Counting points in medium characteristic using Kedlaya's algorithm

Recently, many new results have been found concerning algorithms for counting points on curves over finite fields of characteristic p, mostly due to the use of p-adic liftings. The complexity of these new methods is exponential in p therefore they behave well when p is small, the ideal case being p=2. When applicable, these new methods are usually faster than those based on SEA algorithms, and are more easily extended to non-elliptic curves. We investigate more precisely this dependance on the characteristic, and in particular, we show that after a few modifications using fast algorithms for radix-conversion, Kedlaya's algorithm works in time almost linear in p. As a consequence, this algorithm can also be applied to medium values of p. We give an example of a cryptographic size genus 3 hyperelliptic curve over a finite field of characteristic 251

Secondary crystalline phases identification in Cu2ZnSnSe4 thin films: contributions from Raman scattering and photoluminescence

In this work, we present the Raman peak positions of the quaternary pure selenide compound Cu2ZnSnSe4 (CZTSe) and related secondary phases that were grown and studied under the same conditions. A vast discussion about the position of the X-ray diffraction (XRD) reflections of these compounds is presented. It is known that by using XRD only, CZTSe can be identified but nothing can be said about the presence of some secondary phases. Thin films of CZTSe, Cu2SnSe3, ZnSe, SnSe, SnSe2, MoSe2 and a-Se were grown, which allowed their investigation by Raman spectroscopy (RS). Here we present all the Raman spectra of these phases and discuss the similarities with the spectra of CZTSe. The effective analysis depth for the common back-scattering geometry commonly used in RS measurements, as well as the laser penetration depth for photoluminescence (PL) were estimated for different wavelength values. The observed asymmetric PL band on a CZTSe film is compatible with the presence of CZTSe single-phase and is discussed in the scope of the fluctuating potentials’ model. The estimated bandgap energy is close to the values obtained from absorption measurements. In general, the phase identification of CZTSe benefits from the contributions of RS and PL along with the XRD discussion.info:eu-repo/semantics/publishedVersio

Extracting Bits from Coordinates of a Point of an Elliptic Curve

In the classic Di#e-Hellman protocol based on a generic group G, Alice and Bob agree on a common secret KAB (master secret) which is indistinguishable from another element of G but not from a random bits-string of the same length. In this paper, we present a new deterministic method to extract bits from KAB when G is an elliptic curve defined over a quadratic extension of a finite field. In the last section, we show that it is also possible to extract a few bits when G is an elliptic curve defined over a prime field

Extracting

bits from coordinates of a point of an elliptic curv