On the Worst Case of Three Algorithms for Computing the Jacobi Symbol

We study the worst-case behavior of three iterative algorithms- Eisenstein\u27s algorithm, Lebesgue\u27s algorithm, and the ordinary Jacobi symbol algorithm - for computing the Jacobi symbol. Each algorithm is similar in format to the Euclidean algorithm for computing gcd (u,v)

An O(M(n) log n) algorithm for the Jacobi symbol

The best known algorithm to compute the Jacobi symbol of two n-bit integers runs in time O(M(n) log n), using Sch\"onhage's fast continued fraction algorithm combined with an identity due to Gauss. We give a different O(M(n) log n) algorithm based on the binary recursive gcd algorithm of Stehl\'e and Zimmermann. Our implementation - which to our knowledge is the first to run in time O(M(n) log n) - is faster than GMP's quadratic implementation for inputs larger than about 10000 decimal digits.Comment: Submitted to ANTS IX (Nancy, July 2010

On the Design of Cryptographic Primitives

The main objective of this work is twofold. On the one hand, it gives a brief overview of the area of two-party cryptographic protocols. On the other hand, it proposes new schemes and guidelines for improving the practice of robust protocol design. In order to achieve such a double goal, a tour through the descriptions of the two main cryptographic primitives is carried out. Within this survey, some of the most representative algorithms based on the Theory of Finite Fields are provided and new general schemes and specific algorithms based on Graph Theory are proposed

Local Fourier Analysis of the Complex Shifted Laplacian preconditioner for Helmholtz problems

In this paper we solve the Helmholtz equation with multigrid preconditioned Krylov subspace methods. The class of Shifted Laplacian preconditioners are known to significantly speed-up Krylov convergence. However, these preconditioners have a parameter beta, a measure of the complex shift. Due to contradictory requirements for the multigrid and Krylov convergence, the choice of this shift parameter can be a bottleneck in applying the method. In this paper, we propose a wavenumber-dependent minimal complex shift parameter which is predicted by a rigorous k-grid Local Fourier Analysis (LFA) of the multigrid scheme. We claim that, given any (regionally constant) wavenumber, this minimal complex shift parameter provides the reader with a parameter choice that leads to efficient Krylov convergence. Numerical experiments in one and two spatial dimensions validate the theoretical results. It appears that the proposed complex shift is both the minimal requirement for a multigrid V-cycle to converge, as well as being near-optimal in terms of Krylov iteration count.Comment: 20 page

Mimo Systems Low complexity SVD Implementation Analysis

This paper analyses the implementation of the singular value decomposition (SVD) using approximation to the exact computation for MIMO systems in the case of modulation-mode and power assignment set-up. The study developed in the paper focuses on the use of low complexity algorithm with low computational load oriented to the use of devices with limited resources as FPGA, highlighting some of the advantages and drawbacks against more sophisticated devices. The implementation of the SVD is analyzed through the algorithms that efficiently perform the required computations, seeking for computationally efficient solutions that provide parallelism and low complexity. The CORDIC algorithm seems to be a good candidate for this task since it can efficiently compute the singular value decomposition. It is shown that this algorithm provides an efficient tool for SVD computation with appropriate accuracy and the computational complexity obtained and the required resources make it feasible to be implemented on an FPGA device. System performance degradation is analyzed compared with conventional and exact method for SVD obtaining some key conclusions