Jordi Pujolàs

Explicit 2-power torsion of genus 2 curves over finite fields

We give an efficient explicit algorithm to find the structure and generators of the maximal 2-subgroup of the Jacobian of a genus 2 curve over a finite field of odd characteristic. We use the 2-torsion points as seeds to successively perform a chain of halvings to find divisors of increasing 2-power order. The halving loop requires a solution to certain degree 16 polynomials over the base field, and the termination of the algorithm is based on the description of the graph structure of the maximal 2-subgroup. The structure of our algorithm is the natural extension of the even characteristic case.

Halving for the 2-Sylow subgroup of genus 2 curves over binary fields

We give a deterministic polynomial time algorithm to find the structure of the 2-Sylow subgroup of the Jacobian of a genus 2 curve over a finite field of characteristic 2. Our procedure starts with the points of order 2 and then performs a chain of successive halvings while such an operation makes sense. The stopping condition is triggered when certain polynomials fail to have roots in the base field, as previously shown by I. Kitamura, M. Katagi and T. Takagi. The structure of our algorithm is similar to the already known case of genus 1 and odd characteristic.

Trisection for non-supersingular genus 2 curves in characteristic 2

We study division by 3 in Jacobians of genus 2 curves over binary fields with a 2-torsion subgroup of rank 1 or 2. We characterize the 3-torsion divisors and provide, for every , a formula for the coordinates of the divisors in the set .

Bisection and squares in genus 2

We show how to compute the pre-images of multiplication-by-2 in Jacobians of genus 2 curves C : y 2 = f ( x ) over F q with q odd. We characterize D = u ( x ) , v ( x ) ? 2 Jac ( C ) ( F q ) in terms of the quadratic character of u ( x ) at the roots of f ( x ) in imaginary models, and in terms of the quadratic character of the quotients of u ( x ) at pairs of roots of f ( x ) in real models. Our method reduces the problem to the computation of at most 5 square roots over the splitting field of f ( x ) plus the solution of a system of linear equations.

Halving in some genus 2 curves over binary fields

We give a method to compute the halving of a divisor in the Jacobian variety of a genus 2 curve in characteristic 2. We present explicit halving formulae for each possible divisor class for some curves (those with h(x)=x), detailing all the process to obtain them. We improve the fastest known formulae for some divisor classes in the studied type of curves.