Paper 2009/312

Jacobi Quartic Curves Revisited

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


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 $a = \pm1/2$.

Available format(s)
Publication info
Published elsewhere. Unknown where it was published
Efficient elliptic curve arithmeticscalar multiplicationJacobi model of elliptic curves.
Contact author(s)
h hisil @ isi qut edu au
2009-11-06: last of 3 revisions
2009-07-01: received
See all versions
Short URL
Creative Commons Attribution


      author = {Huseyin Hisil and Kenneth Koon-Ho Wong and Gary Carter and Ed Dawson},
      title = {Jacobi Quartic Curves Revisited},
      howpublished = {Cryptology ePrint Archive, Paper 2009/312},
      year = {2009},
      note = {\url{}},
      url = {}
Note: In order to protect the privacy of readers, does not use cookies or embedded third party content.