Efficient halving for genus 3 curves over binary fields

Abstract

In this article, we deal with fast arithmetic in the Picard group of hyperellipticcurves ofgenus 3 over binary fields. We investigate both the optimal performance curves,where $h(x)=1$, and the more general curves where the degree of $h(x)$ is 1, 2 or 3.For theoptimal performance curves, we provide explicit halving and doubling formulas; notonly for the most frequent case but also for all possible special cases that mayoccur when performing arithmetic on the proposed curves. In this situation, weshow that halving offers equivalent performance to that of doubling when computingscalar multiples (by means of an halve-and-add algorithm) in the divisor class group.  &nbspFor the other types of curves where halving may give performance gains (when thegroup order is twice an odd number), we give explicit halving formulas whichoutperform the corresponding doubling formulas by about 10 to 20 field multiplicationsper halving. These savings more than justify the use of halvings for these curves,making them significantly more efficient than previously thought.For halving on genus 3 curves there is no previous work published so far.