A Cryptographic Application of Weil Descent

@inproceedings{Galbraith1999ACA,
  title={A Cryptographic Application of Weil Descent},
  author={Steven D. Galbraith and Nigel P. Smart},
  booktitle={IMACC},
  year={1999}
}
This paper gives some details about howWeil descent can be used to solve the discrete logarithm problem on elliptic curves which are defined over finite fields of small characteristic. The original ideas were first introduced into cryptography by Frey. We discuss whether these ideas are a threat to existing public key systems based on elliptic curves. 
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  • Mathematics, Computer Science
  • 2005
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    EUROCRYPT
  • 2001
TLDR
The method of Weil descent for solving the ECDLP is compared against the standard method of parallelised Pollard rho and it is shown that composite degree extensions of degree divisible by four should be avoided.
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Weil restriction of scalars can be used to construct curves suitable for cryptography whose Jacobian has known group order, but only a small proportion of the set of all curves can be constructed in this way.
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  • S. Arita
  • Mathematics, Computer Science
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  • 2000
The paper shows that some of elliptic curves over finite fields of characteristic three of composite degree are attacked by a more effective algorithm than Pollard's ρ method. For such an elliptic
Weil descent attack for Kummer extensions
TLDR
The Weil descent attack of Gaudry, Hess and Smart can be adapted to work for some hyperelliptic curves defined over fields of odd characteristic, and it is shown that those are the only families of nonsingular curves defining Kummer extensions for which this method will work.
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