**Jacobi Quartic Curves Revisited**

*Huseyin Hisil and Kenneth Koon-Ho Wong and Gary Carter and Ed Dawson*

**Abstract: **This paper provides new results about efficient arithmetic on Jacobi quartic form elliptic curves, $y^2 = d x^4 + 2 a x^2 + 1$. With recent proposals, the arithmetic on Jacobi quartic curves became solidly faster than that of Weierstrass curves. These proposals use up to 7 coordinates to represent a single point. However, fast scalar multiplication algorithms based on windowing techniques, precompute and store several points which require more space than what it takes with 3 coordinates. Also note that some of these proposals require $d = 1$ for full speed. Unfortunately, elliptic curves having 2-times-a-prime number of points, cannot be written in Jacobi quartic form if $d = 1$. Even worse the contemporary formulae may fail to output correct coordinates for some inputs. This paper provides improved speeds using fewer coordinates without causing the above mentioned problems. For instance, our proposed point doubling algorithm takes only 2 multiplications, 5 squarings, and no multiplication with curve constants when $d$ is arbitrary and $a = \pm1/2$.

**Category / Keywords: **Efficient elliptic curve arithmetic, scalar multiplication, Jacobi model of elliptic curves.

**Date: **received 26 Jun 2009, last revised 5 Nov 2009

**Contact author: **h hisil at isi qut edu au

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Version: **20091106:010827 (All versions of this report)

**Short URL: **ia.cr/2009/312

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