, Volume 15, Issue 1, pp 1946
First online:
Constructive and destructive facets of Weil descent on elliptic curves
 P. GaudryAffiliated withLIX, École Polytechnique
 , F. HessAffiliated withSchool of Mathematics and Statistics F07, University of Sydney
 , N. P. SmartAffiliated withComputer Science Department, University of Bristol
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In this paper we look in detail at the curves which arise in the method of Galbraith and Smart for producing curves in the Weil restriction of an elliptic curve over a finite field of characteristic 2 of composite degree. We explain how this method can be used to construct hyperelliptic cryptosystems which could be as secure as cryptosystems based on the original elliptic curve. On the other hand, we show that the same technique may provide a way of attacking the original elliptic curve cryptosystem using recent advances in the study of the discrete logarithm problem on hyperelliptic curves.
We examine the resulting higher genus curves in some detail and propose an additional check on elliptic curve systems defined over fields of characteristic 2 so as to make them immune from the methods in this paper.
Key words
Function fields Divisor class group Cryptography Elliptic curves Title
 Constructive and destructive facets of Weil descent on elliptic curves
 Journal

Journal of Cryptology
Volume 15, Issue 1 , pp 1946
 Cover Date
 200203
 DOI
 10.1007/s001450010011x
 Print ISSN
 09332790
 Online ISSN
 14321378
 Publisher
 Springer New York
 Additional Links
 Topics
 Keywords

 Function fields
 Divisor class group
 Cryptography
 Elliptic curves
 Industry Sectors
 Authors

 P. Gaudry ^{(1)}
 F. Hess ^{(2)}
 N. P. Smart ^{(3)}
 Author Affiliations

 1. LIX, École Polytechnique, 91128, Palaiseau, France
 2. School of Mathematics and Statistics F07, University of Sydney, 2006, Sydney, NSW, Australia
 3. Computer Science Department, University of Bristol, Woodland Road, BS8 1UB, Bristol, England