## Cryptology ePrint Archive: Report 2009/312

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

Short URL: ia.cr/2009/312

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