Séminaire de Cryptographie

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Alan Lauder

Computing the order of the group of rational points on the Jacobian of a hyperelliptic curve in characteristic 2

I will describe an algorithm for computing the zeta function of an arbitrary hyperelliptic curve in characteristic 2. This is a generalisation of an earlier method of myself and Wan, which tackled a restricted class of curves. The algorithm reduces the problem to that of computing the L-function of an additive character sum over an open subset of the projective line. This latter task can be achieved using the Dwork-Reich trace formula, Dwork's analytic construction of an additive character, and a method for `cohomological reduction' similar to the `Hermite reduction' algorithm used in the symbolic integration of rational functions. The talk is based upon joint work with Daqing Wan. See http://web.comlab.ox.ac.uk/oucl/work/alan.lauder/ for a version of the earlier paper, which has now appeared in LMS JCM, and also two other related papers.