Algorithmic Number Theory

Volume 1838 of the series Lecture Notes in Computer Science pp 313-332

Counting Points on Hyperelliptic Curves over Finite Fields

  • Pierrick GaudryAffiliated withLIX, École Polytechnique
  • , Robert HarleyAffiliated withProjet Cristal, INRIA, Domaine de Voluceau – Rocquencourt

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We describe some algorithms for computing the cardinality of hyperelliptic curves and their Jacobians over finite fields. They include several methods for obtaining the result modulo small primes and prime powers, in particular an algorithm à la Schoof for genus 2 using Cantor’s division polynomials. These are combined with a birthday paradox algorithm to calculate the cardinality. Our methods are practical and we give actual results computed using our current implementation. The Jacobian groups we handle are larger than those previously reported in the literature.