A binary algorithm with low divergence for modular inversion on SIMD architectures

When performing modular arithmetic the most computationally expensive operation is the modular inversion of an integer. Its cost might be a problem for cryptanalytical applications (like the transformation from projective to affine coordinates within a Pollard rho algorithm implementation to solve the discrete logarithm problem on an elliptic curve) where performances constitute a key aspect. Good platforms for such operations are single-instruction multiple-data architectures, like graphic processing units (GPUs) because of their extremely competitive performance/price ratio. Unfortunately, when a single thread computes a single inversion, the whole computation on GPUs can be significantly slowed down in the presence of divergent threads. In this paper we describe a new algorithm to compute modular inversion on GPUs based on Stein’s Binary GCD. By exploiting the De Bruijn sequences and the Montgomery arithmetic, our version of Stein’s algorithm better fits GPUs, since it reduces the divergence among threads of the original algorithm. The paper includes a brief report on tests of our algorithm in six prime fields with characteristics of size ranging from 109 to 359 bits and in the two prime fields associated to the Mersenne primes \(2^{521} - 1\) and \(2^{607} - 1\).     

A new synthetic entry to 3-carboxamido-4-carboxylic acid derivatives of isoxazole and pyrazole

Ethyl 2-amino-3-methoxycarbonyl-4-oxo-2-pentenoate (3) reacts with hydroxylamine or hydrazines to give isoxazole and pyrazole ortho-dicarboxylic acid esters 4 and 5, respectively. Partial hydrolysis of diesters 4 and 5 afforded the corresponding dicarboxylic acid monoesters 6 and 7. Amidation of the intermediate acid chlorides 8,9 followed by hydrolysis of 4-methylesters 10, 11 gave the title compounds 1 and 2, respectively.     

Double resonance with Landesman-Lazer conditions for planar systems of ordinary differential equations

We prove the existence of periodic solutions for first order planar systems at resonance. The nonlinearity is indeed allowed to interact with two positively homogeneous Hamiltonians, both at resonance, and some kind of Landesman-Lazer conditions are assumed at both sides. We are thus able to obtain, as particular cases, the existence results proposed in the pioneering papers by Lazer and Leach (1969) [27], and by Frederickson and Lazer (1969) [18]. Our theorem also applies in the case of asymptotically piecewise linear systems, and in particular generalizes Fabry's results in Fabry (1995) [10], for scalar equations with double resonance with respect to the Dancer-Fu?ik spectrum.     

Diffusion reproduction processes

Interesting physics emerges from studying a population of reproducing individuals. Each can be regarded as a random walker, but it can either duplicate or die. Novel features of the collective behavior are quite surprising: if individuals reproduce or die freely, the life expectation is proportional to the size of the population, and if it is kept constant, the center of mass moves in space as if it were a single walker (i.e., the diffusion constant is independent of population's size). Biology-inspired interactions are also considered.     

Directions of Coaxiality between Pure Strain and Stress in Linear Elasticity

For a linear anisotropic hyperelastic body subjected to a state of pure strain it is shown that there are at least three directions such that the stress and strain tensors are coaxial.