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:y2=f(x) over Fq with q odd. We characterize D=[u(x),v(x)]∈2Jac(C)(Fq) 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.