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Paper 2006/174

Frobenius expansion and the Diffie Hellman problem

V. R. Sule

Abstract

This paper proposes investigation of special sessions of the Diffie Hellman (DH) key exchange scheme on elliptic curves for which the shared key can be computed by a polynomial time algorithm. Such sessions are called \emph{singular}. Existence of singular sessions are demonstrated using the Frobenius expansion and polynomial representation of public keys which lead to an expression for the shared key. When the Weil pairing can be computed on the elliptic curve along with a modified pairing defined by a distortion map efficiently, a sufficient condition is obtained for sessions to be singular which can be verified in polynomial time. Hence this condition identifies sessions whose singular nature can be determined in polynomial time. A single round three party key exchange scheme is proposed using singular sessions in which efficient computation of the shared key of a pair of users by the third party is a necessary requirement. This scheme is thus a positive application of singular sessions and offers a possible alternative to the need for using super singular curves on which pairings can be computed efficiently.

Note: PDF file frobenius&dhp.pdf

Metadata
Available format(s)
-- withdrawn --
Category
Public-key cryptography
Publication info
Published elsewhere. Not submitted for publication
Keywords
Diffie Hellman schemeFrobenius expansionTri-party key exchange.
Contact author(s)
vrs @ ee iitb ac in
History
2006-06-10: withdrawn
2006-05-22: received
See all versions
Short URL
https://ia.cr/2006/174
License
Creative Commons Attribution
CC BY
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