R. Flassenberg and S. Paulus. Sieving in function fields. Preprint, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is large in comparison to the size of the field of definition of the Jacobian. In this case there are conjectured subexponential methods. The first of these was due to Adleman, De Marrais and Huang which is based on the number field sieve factoring method. Paulus [17] and Flassenberg and Paulus [7] have implemented such a method for solving discrete logarithms in Jacobians of hyperelliptic curves. Flassenberg and Paulus did not, however, use the method of Adleman, De Marrais and Huang directly. Instead they made use of the fact that our hyperelliptic curves correspond to real quadratic ....

R. Flassenberg and S. Paulus. Sieving in function fields. Preprint, 1997.

....F q n =F q and obtain an Abelian variety A of dimension ng over F q . One then searches for a curve C lying on A in such a way that one can pull the divisors D 1 and D 2 back to Jac(C) F q ) and then solve the discrete logarithm problem there using one of the available algorithms (see [1] [4], 7] for solving such problems on high genus curves. Before giving flesh to this skeleton, we discuss the practical situation we have in mind. 2. The Cryptographic application The most relevant cases of Jacobians of curves for cryptography are when C is a hyperelliptic curve of genus 2 or 3 ....

....Section 7. It is not very easy to express the complexity of Weil descent in a meaningful sense (i.e. one which can be used to determine the security of a discrete logarithm problem) Part of the problem is a lack of experience with solving discrete logarithms on high genus curves (although see [4], 7] 11] Examples 2 and 3 (subsections 4.7 and 4.8) show that there are cases of curves of genus 3 over certain field extensions for which the discrete logarithm problem is significantly easier to solve than had been previously thought (although still exponential complexity) To completely ....

R. Flassenberg and S. Paulus, Sieving in function fields, Experimental Mathematics, 8, No. 4, 339--349 (1997)

....but that our concern is to simplify the analysis. For instance, in practice it should usually suffice to create about 2n relations instead of 40n and the relations should not be obtained randomly, but by sieving techniques. For a description of an implementation based on a sieving approach, see [10]. 3.2 Computing individual logarithms To relate D (1) and D (2) with the primes in B, we have to find B smooth divisors D (1) D (1) and D (2) D (2) i.e. divisors which can be decomposed into the primes in B. To do so, we again choose random vectors e 2 f0; E Gamma ....

Ralf Flassenberg and Sachar Paulus. Sieving in function fields. Preprint; available at ftp://ftp.informatik.tu-darmstadt.de/pub/TI/- TR/TI-97-13.rafla.ps.gz, 1997.

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R. Flassenberg and S. Paulus. Sieving in function fields. Preprint, 1997.

....use sparse techniques. In a practical algorithm this would become a major computational bottleneck. The method we propose will produce sparse matrices. 2. In function fields of degree greater than two it appears unlikely that an efficient sieving technique like that applied in degree two fields in [9] can be found. The method below does allow efficient sieving strategies to be employed. 3. The factor base for the Hafner McCurley style method is the set of all prime ideals of norm less than some bound. In our method we need only to take all prime ideals of inertia degree one less than some ....

R. Flassenberg and S. Paulus. Sieving in function fields. Preprint, 1997.

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