International Association for Cryptologic Research

International Association
for Cryptologic Research


Paper: Extending the GHS Weil Descent Attack

Steven D. Galbraith
F. Hess
Nigel P. Smart
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Abstract: In this paper we extend the Weil descent attack due to Gaudry, Hess and Smart (GHS) to a much larger class of elliptic curves. This extended attack still only works for fields of composite degree over $\F_2$. The principle behind the extended attack is to use isogenies to find a new elliptic curve for which the GHS attack is effective. The discrete logarithm problem on the target curve can be transformed into a discrete logarithm problem on the new isogenous curve. One contribution of the paper is to give an improvement to an algorithm of Galbraith for constructing isogenies between elliptic curves, and this is of independent interest in elliptic curve cryptography. We conclude that fields of the form $\F_{q^7}$ should be considered weaker from a cryptographic standpoint than other fields. In addition we show that a larger proportion than previously thought of elliptic curves over $\F_{2^{155}}$ should be considered weak.
  title={Extending the GHS Weil Descent Attack},
  booktitle={IACR Eprint archive},
  keywords={public-key cryptography / elliptic curve cryptosystems},
  note={ 11509 received 6 Jul 2001},
  author={Steven D. Galbraith and F. Hess and Nigel P. Smart},