# Jacobian Coordinates on Genus 2 Curves

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## Abstract

This paper presents a new projective coordinate system and new explicit algorithms which together boost the speed of arithmetic in the divisor class group of genus 2 curves. The proposed formulas generalize the use of Jacobian coordinates on elliptic curves, and their application improves the speed of performing cryptographic scalar multiplications in Jacobians of genus 2 curves over prime fields by an approximate factor of 1.25x. For example, on a single core of an Intel Core i7-3770 (Ivy Bridge), we show that replacing the previous best formulas with our new set improves the cost of generic scalar multiplications from 239,000 to 192,000 cycles and drops the cost of specialized GLV-style scalar multiplications from 155,000 to 123,000 cycles.

## Keywords

Genus 2 Hyperelliptic curves Explicit formulas Jacobian coordinates Scalar multiplication## Notes

### Acknowledgments

We thank Joppe Bos, Michael Naehrig, Benjamin Smith, and Osmanbey Uzunkol for their useful comments on an early draft of this work. We also thank the anonymous Asiacrypt 2014 referees for their valuable comments, and Patrick Longa for independently benchmarking our code on different processors.

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