136 research outputs found
Computational linear algebra over finite fields
We present here algorithms for efficient computation of linear algebra
problems over finite fields
Discrete logarithm computations over finite fields using Reed-Solomon codes
Cheng and Wan have related the decoding of Reed-Solomon codes to the
computation of discrete logarithms over finite fields, with the aim of proving
the hardness of their decoding. In this work, we experiment with solving the
discrete logarithm over GF(q^h) using Reed-Solomon decoding. For fixed h and q
going to infinity, we introduce an algorithm (RSDL) needing O (h! q^2)
operations over GF(q), operating on a q x q matrix with (h+2) q non-zero
coefficients. We give faster variants including an incremental version and
another one that uses auxiliary finite fields that need not be subfields of
GF(q^h); this variant is very practical for moderate values of q and h. We
include some numerical results of our first implementations
Integer Factorization with a Neuromorphic Sieve
The bound to factor large integers is dominated by the computational effort
to discover numbers that are smooth, typically performed by sieving a
polynomial sequence. On a von Neumann architecture, sieving has log-log
amortized time complexity to check each value for smoothness. This work
presents a neuromorphic sieve that achieves a constant time check for
smoothness by exploiting two characteristic properties of neuromorphic
architectures: constant time synaptic integration and massively parallel
computation. The approach is validated by modifying msieve, one of the fastest
publicly available integer factorization implementations, to use the IBM
Neurosynaptic System (NS1e) as a coprocessor for the sieving stage.Comment: Fixed typos in equation for modular roots (Section II, par. 6;
Section III, par. 2) and phase calculation (Section IV, par 2
Efficient Parallelization of Lanczos Type Algorithms
We propose a new parallelization technique for Lanczos type algorithms for solving large sparse linear systems over finite fields on mesh cluster architecture. The algorithm computation time scales as on P^{-1/2}P$
Subquadratic computation of vector generating polynomials and improvement of the block Wiedemann algorithm
This paper describes a new algorithm for computing linear generators (vector generating polynomials) for matrix sequences, running in sub-quadratic time. This algorithm applies in particular to the sequential stage of Coppersmith's block Wiedemann algorithm. Experiments showed that our method can be substituted in place of the quadratic one proposed by Coppersmith, yielding important speedups even for realistic matrix sizes. The base fields we were interested in were finite fields of large characteristic. As an example, we have been able to compute a linear generator for a sequence of 4*4 matrices of length 242 304 defined over GF(2^607) in less than two days on one 667MHz alpha ev67 cpu
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