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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 subquadratic time. This algorithm applies in particular to the sequential stage of Coppersmith's block Wiedemann algorithm. Experiments showed that our metho ..."
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Cited by 26 (4 self)
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This paper describes a new algorithm for computing linear generators (vector generating polynomials) for matrix sequences, running in subquadratic 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 elds we were interested in were nite elds of large characteristic.
Fast computation of linear generators for matrix sequences and application to the block Wiedemann algorithm
 PROC. ISSAC '2001
, 2001
"... In this paper we describe how the halfgcd algorithm can be adapted in order to speed up the sequential stage of Coppersmith's block Wiedemann algorithm for solving large sparse linear systems over any finite field. This very stage solves a subproblem than can be seen as the computation of a l ..."
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Cited by 17 (2 self)
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In this paper we describe how the halfgcd algorithm can be adapted in order to speed up the sequential stage of Coppersmith's block Wiedemann algorithm for solving large sparse linear systems over any finite field. This very stage solves a subproblem than can be seen as the computation of a linear generator for a matrix sequence. Our primary realm of interest is the field $\GF{q}$ for large prime power $q$. For the solution of a $N\times N$ system, the complexity of this sequential part drops from $O(N²)$ to $O(\mathsf{M}(N)\log N)$ where $\mathsf{M}(d)$ is the cost for multiplying two polynomials of degree $d$. We discuss the implications of this improvement for the overall cost of the block Wiedemann algorithm and how its parameters should be chosen for best efficiency.
Function Field Sieve in Characteristic Three
, 2004
"... In this paper we investigate the e#ciency of the function field sieve to compute discrete logarithms in the finite fields F3 n . Motivated by attacks on identity based encryption systems using supersingular elliptic curves, we pay special attention to the case where n is composite. This allows ..."
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Cited by 15 (4 self)
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In this paper we investigate the e#ciency of the function field sieve to compute discrete logarithms in the finite fields F3 n . Motivated by attacks on identity based encryption systems using supersingular elliptic curves, we pay special attention to the case where n is composite. This allows
Tests and Constructions of Irreducible Polynomials over Finite Fields
 In Foundations of Computational Mathematics
, 1997
"... In this paper we focus on tests and constructions of irreducible polynomials over finite fields. We revisit Rabin's [1980] algorithm providing a variant of it that improves Rabin's cost estimate by a log n factor. We give a precise analysis of the probability that a random polynomial of de ..."
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Cited by 12 (4 self)
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In this paper we focus on tests and constructions of irreducible polynomials over finite fields. We revisit Rabin's [1980] algorithm providing a variant of it that improves Rabin's cost estimate by a log n factor. We give a precise analysis of the probability that a random polynomial of degree n contains no irreducible factors of degree less than O(log n). This probability is naturally related to BenOr's [1981] algorithm for testing irreducibility of polynomials over finite fields. We also compute the probability of a polynomial being irreducible when it has no irreducible factors of low degree. This probability is useful in the analysis of various algorithms for factoring polynomials over finite fields.
Computation of Discrete Logarithms in ...
, 607
"... We describe in this article how we have been able to extend the record for computations of discrete logarithms in characteristic 2 from the previous record over F 2 503 to a newer mark of F 2 607 , using Coppersmith's algorithm. This has been made possible by several practical improvements ..."
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Cited by 10 (0 self)
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We describe in this article how we have been able to extend the record for computations of discrete logarithms in characteristic 2 from the previous record over F 2 503 to a newer mark of F 2 607 , using Coppersmith's algorithm. This has been made possible by several practical improvements to the algorithm. Although the computations have been carried out on fairly standard hardware, our opinion is that we are nearing the current limits of the manageable sizes for this algorithm, and that going substantially further will require deeper improvements to the method.