Efficient Algorithms for gcd and Cubic Residuosity in the Ring of Eisenstein Integers

We present simple and efficient algorithms for computing gcd and cubic residuosity in the ring of Eisenstein integers, Z[zeta] , i.e. the integers extended with zeta , a complex primitive third root of unity. The algorithms are similar and may be seen as generalisations of the binary integer gcd and derived Jacobi symbol algorithms. Our algorithms take time O(n^2) for n bit input. This is an improvement from the known results based on the Euclidian algorithm, and taking time O(n· M(n)), where M(n) denotes the complexity of multiplying n bit integers. The new algorithms have applications in practical primality tests and the implementation of cryptographic protocols. The technique underlying our algorithms can be used to obtain equally fast algorithms for gcd and quartic residuosity in the ring of Gaussian integers, Z[i]

Quantum Algorithms for Some Hidden Shift Problems

Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of "unknown shift" problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure

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

A hierarchically blocked Jacobi SVD algorithm for single and multiple graphics processing units

We present a hierarchically blocked one-sided Jacobi algorithm for the singular value decomposition (SVD), targeting both single and multiple graphics processing units (GPUs). The blocking structure reflects the levels of GPU's memory hierarchy. The algorithm may outperform MAGMA's dgesvd, while retaining high relative accuracy. To this end, we developed a family of parallel pivot strategies on GPU's shared address space, but applicable also to inter-GPU communication. Unlike common hybrid approaches, our algorithm in a single GPU setting needs a CPU for the controlling purposes only, while utilizing GPU's resources to the fullest extent permitted by the hardware. When required by the problem size, the algorithm, in principle, scales to an arbitrary number of GPU nodes. The scalability is demonstrated by more than twofold speedup for sufficiently large matrices on a Tesla S2050 system with four GPUs vs. a single Fermi card.Comment: Accepted for publication in SIAM Journal on Scientific Computin

A GPU-based hyperbolic SVD algorithm

A one-sided Jacobi hyperbolic singular value decomposition (HSVD) algorithm, using a massively parallel graphics processing unit (GPU), is developed. The algorithm also serves as the final stage of solving a symmetric indefinite eigenvalue problem. Numerical testing demonstrates the gains in speed and accuracy over sequential and MPI-parallelized variants of similar Jacobi-type HSVD algorithms. Finally, possibilities of hybrid CPU--GPU parallelism are discussed.Comment: Accepted for publication in BIT Numerical Mathematic

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