Traces of the Group Law on the Kummer Surface of a Curve of Genus 2 in Characteristic 2

Abstract

In the early 1990s, Flynn gave an explicit description of the Jacobian of a genus 2 hyperelliptic curve in order to perform efficient arithmetic on these objects. In this paper, we give a generalization of Flynn’s work when the ground field has characteristic 2. More precisely, we give an explicit description of the Kummer surface. We also give and explain how we found, using symbolic computations, explicit formulas for the structure of the group law on the Jacobian preserved on the Kummer surface. Finally, we use these new objects to give a very fast scalar multiplication algorithm for hyperelliptic curve cryptography in characteristic 2.