Hyperelliptic Curves: Operations on the Jacobian over F_q

The techniques for point counting are due to P. Gaudry and R. Harley and have been implemented in Magma by P. Gaudry.

• Enumeration of all rational points (small fields), random point
• Order of a point via the Shanks or the Pollard-rho algorithms
• Counting points: Shanks and Pollard methods
• Counting points: An index calculus method for when the genus is large compared to the base field
• Counting points: Schoof algorithm for finding the order modulo small primes in the case of a genus 2 curve
• Structure of the abelian group of rational points
• Weil pairing of points