[Resource Topic] 2004/070: Easy decision-Diffie-Hellman groups

Welcome to the resource topic for 2004/070

Easy decision-Diffie-Hellman groups

Authors: Steven D Galbraith, Victor Rotger


It is already known that the Weil and Tate pairings
can be used to solve many decision-Diffie-Hellman (DDH)
problems on elliptic curves.
A natural question is whether all DDH problems are
easy on supersingular curves.
To answer this question it is necessary to have
suitable distortion maps.
Verheul states that such maps exist,
and this paper gives methods to construct them.
The paper therefore shows that all DDH problems on
supersingular elliptic curves are easy.
We also discuss the issue of which DDH problems on ordinary
curves are easy.

A related contribution is a discussion of distortion maps
which are not isomorphisms. We give explicit
distortion maps for elliptic curves with complex
multiplication of discriminants D=-7 and D=-8.

ePrint: https://eprint.iacr.org/2004/070

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