## Cryptology ePrint Archive: Report 2004/070

Easy decision-Diffie-Hellman groups

Steven D Galbraith and Victor Rotger

Abstract: 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 $D=-8$.

Category / Keywords: public-key cryptography / pairings, elliptic curves, decision-Diffie-Hellman problem

Date: received 2 Mar 2004, last revised 31 Aug 2004

Contact author: Steven Galbraith at rhul ac uk

Available format(s): Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

Note: This paper has now appeared in LMS J. Comput. Math. 7 (2004) 201--218

See http://www.lms.ac.uk/jcm/7/lms2004-010

Short URL: ia.cr/2004/070

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