Paper 2002/107

Efficient Arithmetic on Hyperelliptic Curves

Tanja Lange


Using the Frobenius endomorphism the operation of computing scalar-mulitples in the Jacobian of a hyperelliptic curve is sped-up considerably. The kind of curves considered are Kobiltz i.e. subfield curves, defined over a small finite field which are then considered over a large extension field. We deal with computation of the group order over various extension fields, algorithms to obtain the mentioned speed-up, and experimental results concerning both issues. Additionally an alternative set-up is treated which uses arihtmetic in the finite field only and allows shorter code for similar security. Furthermore explicit formulae to perform the arithmetic in the ideal class group explicitely are derived and can thus be used for implementation in hardware; in software they are also faster than the generic Cantor algorithm. As a second group suitable for cryptographic applications the trace-zero-variety is considered. Here we investigate the group operation and deal with security issues.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. PhD Thesis, Preprint 4-2002, Universität-Gesamthochschule Essen
hyperelliptic curvesscalar multiplicationarithmeticimplementationclass number
Contact author(s)
lange @ itsc ruhr-uni-bochum de
2003-12-15: last of 2 revisions
2002-08-04: received
See all versions
Short URL
Creative Commons Attribution


      author = {Tanja Lange},
      title = {Efficient Arithmetic on Hyperelliptic Curves},
      howpublished = {Cryptology ePrint Archive, Paper 2002/107},
      year = {2002},
      note = {\url{}},
      url = {}
Note: In order to protect the privacy of readers, does not use cookies or embedded third party content.