Lubicz : A quasi quadratic time algorithm for hyperelliptic curve point counting. Available online at http://www.medicis.polytechnique.fr∼lercier/preprints/riemann.pdf

Abstract. We describe an algorithm to compute the cardinality of Jacobians of ordi-nary hyperelliptic curves of small genus over finite fields F2n with cost O(n 2+o(1)). This algorithm is derived from ideas due to Mestre. More precisely, we state the mathemat-ical background behind Mestre’s algorithm and develop from it a variant with quasi-quadratic time complexity. Among others, we present an algorithm to find roots of a system of generalized Artin-Schreier equations and give results that we obtain with an efficient implementation. Especially, we were able to obtain the cardinality of curves of genus one, two or three in finite fields of huge size.