180 research outputs found
Computing isogenies between Jacobian of curves of genus 2 and 3
We present a quasi-linear algorithm to compute isogenies between Jacobians of
curves of genus 2 and 3 starting from the equation of the curve and a maximal
isotropic subgroup of the l-torsion, for l an odd prime number, generalizing
the V\'elu's formula of genus 1. This work is based from the paper "Computing
functions on Jacobians and their quotients" of Jean-Marc Couveignes and Tony
Ezome. We improve their genus 2 case algorithm, generalize it for genus 3
hyperelliptic curves and introduce a way to deal with the genus 3
non-hyperelliptic case, using algebraic theta functions.Comment: 34 page
Group law computations on Jacobians of hyperelliptic curves
We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form
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