Algorithms and cryptographic protocols using elliptic curves

En els darrers anys, la criptografia amb corbes el.líptiques ha adquirit una importància creixent, fins a arribar a formar part en la actualitat de diferents estàndards industrials. Tot i que s'han dissenyat variants amb corbes el.líptiques de criptosistemes clàssics, com el RSA, el seu màxim interès rau en la seva aplicació en criptosistemes basats en el Problema del Logaritme Discret, com els de tipus ElGamal. En aquest cas, els criptosistemes el.líptics garanteixen la mateixa seguretat que els construïts sobre el grup multiplicatiu d'un cos finit primer, però amb longituds de clau molt menor. Mostrarem, doncs, les bones propietats d'aquests criptosistemes, així com els requeriments bàsics per a que una corba sigui criptogràficament útil, estretament relacionat amb la seva cardinalitat. Revisarem alguns mètodes que permetin descartar corbes no criptogràficament útils, així com altres que permetin obtenir corbes bones a partir d'una de donada. Finalment, descriurem algunes aplicacions, com són el seu ús en Targes Intel.ligents i sistemes RFID, per concloure amb alguns avenços recents en aquest camp.

Generalization of Vélu’s formulae for isogenies between elliptic curves

Given an elliptic curve E and a finite subgroup G, V ́lu’s formulae concern to a separable isogeny IG : E → E ′ with kernel G. In particular, for a point P ∈ E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P + G as the difference between the abscissa of IG (P ) and the first elementary symmetric polynomial on the abscissas of the nontrivial points of the kernel G. On the other hand, they express Weierstraß coefficients of E ′ as polynomials in the coefficients of E and two additional parameters: w0 = t and w1 = w. We generalize this by defining parameters wn for all n ≥ 0 and giving analogous formulae for all the elementary symmetric polynomials and the power sums on the abscissas of the points in P +G. Simultaneously, we obtain an efficient way of performing computations concerning the isogeny when G is a rational group.

Spontaneous Emergence of Chirality in the Limited Enantioselectivity Model: Autocatalytic Cycle Driven by an External Reagent

The emergence of chirality in enantioselective autocatalysis for compounds unable to transform according to the Frank-like reaction network is discussed with respect to the controversial limited enantioselectivity (LES) model composed of coupled enantioselective and non-enantioselective autocatalyses. The LES model cannot lead to spontaneous mirror symmetry breaking (SMSB) either in closed systems with a homogeneous temperature distribution or in closed systems with a stationary non-uniform temperature distribution. However, simulations of chemical kinetics in a two-compartment model demonstrate that SMSB may occur if both autocatalytic reactions are spatially separated at different temperatures in different compartments but coupled under the action of a continuous internal flow. In such conditions, the system can evolve, for certain reaction and system parameters, toward a chiral stationary state; that is, the system is able to reach a bifurcation point leading to SMSB. Numerical simulations in which reasonable chemical parameters have been used suggest that an adequate scenario for such a SMSB would be that of abyssal hydrothermal vents, by virtue of the typical temperature gradients found there and the role of inorganic solids mediating chemical reactions in an enzyme-like role.

Regio- and stereoselective microwave-assisted synthesis of 5-alkyl-4-alkenyl-4-phenyl-1,3-oxazolidin-2-ones

Chiral symmetrical alk-2-yne-1,4-diols have been stereoselectively transformed into 5-alkyl-4-alkenyl-4-phenyl-1,3-oxazolidin- 2-ones, which are precursors of quaternary α-amino β-hydroxy acids. The key step was the cyclization of the bis(tosylcarbamates) of 2- phenylalk-2-yne-1,4-diols, easily obtained from the starting chiral diols. These cyclizations were accomplished with complete regioselectivity and up to 92:8 dr in the presence of catalytic amounts of Ni(0) or Pd (II) derivatives under microwave heating.

Total synthesis of (-)-isoavenaciolide

An enantioselective approach to (-)-isoavenaciolide was achieved starting from 1- undecyn-3-ol. The synthesis relied upon the preparation of a chiral 4-silyloxy-2-alkenylborane by hydroboration of a protected 2,3-allenol and subsequent stereoselective addition to 2- thiophenecarboxaldehyde

Modelo de cuentas estelares de la Vía Láctea

Son muchos los estudios que se han realizado sobre la Vía Láctea. El Grupo de Investigación de Morfología y Dinámica de la Vía Láctea del Instituto de Astrofísica de Canarias (IAC) ha realizado numerosos trabajos acerca de la estructura galáctica, entre los cuales son de especial relevancia los relativos a la estructura del disco fino, el bulbo y la barra larga. Sin embargo, el grupo carece de un código propio que unifique en un modelo común el conocimiento actual de la estructura galáctica. En este proyecto de final de carrera se presenta la implementación de un código informático de un modelo de cuentas estelares de la Vía Láctea realizado en colaboración con investigadores del IAC.

Multigram synthesis and in vivo efficacy studies of a novel multitarget anti-Alzheimer's compound

We describe the multigram synthesis and in vivo efficacy studies of a donepezil‒huprine hybrid that has been found to display a promising in vitro multitarget profile of interest for the treatment of Alzheimer's disease (AD). Its synthesis features as the key step a novel multigram preparative chromatographic resolution of intermediate racemic huprine Y by chiral HPLC. Administration of this compound to transgenic CL4176 and CL2006 Caenorhabditis elegans strains expressing human Aβ42, here used as simplified animal models of AD, led to a significant protection from the toxicity induced by Aβ42. However, this protective effect was not accompanied, in CL2006 worms, by a reduction of amyloid deposits. Oral administration for 3 months to transgenic APPSL mice, a well-established animal model of AD, improved short-term memory, but did not alter brain levels of Aβ peptides nor cortical and hippocampal amyloid plaque load. Despite the clear protective and cognitive effects of AVCRI104P4, the lack of Aβ lowering effect in vivo might be related to its lower in vitro potency toward Aβ aggregation and formation as compared with its higher anticholinesterase activities. Further lead optimization in this series should thus focus on improving the anti-amyloid/anticholinesterase activity ratio.