Group Law Computations on Jacobians of Hyperelliptic Curves
Let C be an arbitrary smooth algebraic curve of genus g over a large finite field K. We revisit fast addition algorithms in the Jacobian of C due to Khuri-Makdisi (math.NT/0409209, to appear in Mathematics of Computation). The algorithms, which reduce to linear algebra in vector spaces of dimension O(g) once |K| ≫ g and which asymptotically require O(g) field operations using fast linear algebra, are shown to perform efficiently even for certain low genus curves. Specifically, we provide explicit formulae for performing the group law on Jacobians of C3,4 curves of genus 3. We show that, typically, the addition of two distinct elements in the Jacobian of a C3,4 curve requires 117 multiplications and 2 inversions in K, and an element can be doubled using 129 multiplications and 2 inversions in K. This represents an improvement of approximately 20% over previous methods.