# American Institute of Mathematical Sciences

February  2010, 4(1): 23-47. doi: 10.3934/amc.2010.4.23

## Efficient halving for genus 3 curves over binary fields

 1 Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands 2 Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile

Received  March 2009 Revised  September 2009 Published  February 2010

In this article, we deal with fast arithmetic in the Picard group of hyperelliptic curves of genus 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 the optimal performance curves, we provide explicit halving and doubling formulas; not only for the most frequent case but also for all possible special cases that may occur when performing arithmetic on the proposed curves. In this situation, we show that halving offers equivalent performance to that of doubling when computing scalar multiples (by means of an halve-and-add algorithm) in the divisor class group.
For the other types of curves where halving may give performance gains (when the group order is twice an odd number), we give explicit halving formulas which outperform the corresponding doubling formulas by about 10 to 20 field multiplications per 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.
Citation: Peter Birkner, Nicolas Thériault. Efficient halving for genus 3 curves over binary fields. Advances in Mathematics of Communications, 2010, 4 (1) : 23-47. doi: 10.3934/amc.2010.4.23
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