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Constructive and Destructive Facets of Weil Descent on Elliptic Curves

Gaudry, P.; Hess, F.; Smart, Nigel P.


Keyword(s): function fields; divisor class group; cryptography; elliptic curves

Abstract: 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 two 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 this 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 two so as to make them immune from the methods in this paper. Notes: Florian Hess, School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia. P. Gaudry, LIX, Ecole Polytechnique, 91128 Palaiseau, France

19 Pages

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