Elliptic Curves Suitable for Pairing Based Cryptography

For pairing based cryptography we need elliptic curves defined over finite fields whose group order is divisible by some prime with where k is relatively small. In Barreto et al. and Dupont et al. [Proceedings of the Third Workshop on Security in Communication Networks (SCN 2002), LNCS, 2576, 2003; Building curves with arbitrary small Mov degree over finite fields, Preprint, 2002], algorithms for the construction of ordinary elliptic curves over prime fields with arbitrary embedding degree k are given. Unfortunately, p is of size .We give a method to generate ordinary elliptic curves over prime fields with p significantly less than which also works for arbitrary k. For a fixed embedding degree k, the new algorithm yields curves with where or depending on k. For special values of k even better results are obtained.We present several examples. In particular, we found some curves where is a prime of small Hamming weight resp. with a small addition chain.AMS classification: 14H52, 14G50     

Non-Smooth Optimization for Robust Control of Infinite-Dimensional Systems

We use a non-smooth trust-region method for H _∞-control of infinite-dimensional systems. Our method applies in particular to distributed and boundary control of partial differential equations. It is computationally attractive as it avoids the use of system reduction or identification. For illustration the method is applied to control a reaction-convection-diffusion system, a Van de Vusse reactor, and to a cavity flow control problem.     

Sulfoxide as a ligand in iron(II) porphyrinates: S- or O-bound?

In order to study the bonding of sulfoxides to iron(II) porphyrinates, an equilibrium study of Fe(TPP) with tetramethylenesulfoxide (TMSO) has been performed. UV-vis spectra at different concentrations of TMSO have shown distinct character belonging to three species: four-coordinate Fe(II)(TPP), five-coordinate [Fe(II)(TPP)(TMSO)], and six-coordinate [Fe(II)(TPP)(TMSO)2]. The isosbestic points for the low TMSO concentrations suggest that the equilibrium constant K1 is much larger than K2. Analysis of spectral data by the nonlinear least-squares program SQUAD gives K1 = 267 and K2 approximately 1. Even though the five-coordinate species is the dominant species under the synthetic conditions, only the six-coordinate species was crystallized and characterized by an X-ray diffraction study. [Fe(TPP)(TMSO)2] (C52H44Fe-N4O2S2): monoclinic, P2(1)/c, a = 11.2580(3) A, b = 15.9262(5) A, c = 12.3930(4) A, beta = 116.246(1) degrees , V = 1992.95(10) A3, Z = 2. X-ray crystallography demonstrates the complex is a low-spin bis-TMSO ligated species. The average Fe-Np distance is 1.999(4) A. The most important feature is that TMSO is coordinated to iron(II) by the sulfur donors, not oxygen. The Fe-S distance is 2.2220(3) A.     

Cyclohexylphosphonic acid

The title compound, C₆H₁₃O₃P, displays a crystallographic mirror plane. Bond lengths in the phosphonic acid moiety are P—O = 1.5557 (13) Å and P=O = 1.5089 (18) Å. The mol­ecules are linked via intermolecular hydrogen bonding to form a one‐dimensional chain of fused rings. There are no significant contacts between planes.     

The sodium salt of a tris­(tridentate anion)­gadolinium(III) complex: penta­sodium bis­[chelidamato(3−)][chelidamato(2−)]­gadolinate(III) hexa­deca­hydrate

The sodium salt of a complex anion formed between gadolinium(III) and three variously deprotonated chelidamic acid (4‐hydroxy­pyridine‐2,6‐di­carboxyl­ic acid) ligand moi­eties, assigned as Na₅[Gd(C₇H₂NO₅)₂(C₇H₃NO₅)]·16H₂O, i.e. pentasodium (4‐hydroxy­pyridine‐2,6‐di­carboxyl­ate)­bis(4‐oxido­pyridine‐2,6‐di­carboxyl­ate)­gadolinium(III) hexadecahydrate, forms as colourless monoclinic crystals upon vapour diffusion of ethanol into its aqueous solution. The ligand moieties, assigned as two trianionic and one dianionic chelidamate species, are all tridentate in the complex anion of tricapped trigonal prismatic donor‐atom geometry. The geometry of the ligands and that of the primary coordination sphere is very similar to that of the analogous anionic tris­(ligand)–rare earth complexes of the pyridine‐2,6‐di­carboxyl­ate (dipicolinate) dianion.