**Diffie-Hellman Problems and Bilinear Maps**

*Jung Hee Cheon and Dong Hoon Lee*

**Abstract: **We investigate relations among the discrete logarithm (DL)
problem, the Diffie-Hellman (DH) problem and the bilinear
Diffie-Hellman (BDH) problem when we have an efficient computable
non-degenerate bilinear map $e:G\times G \rightarrow H$. Under a
certain assumption on the order of $G$, we show that the DH
problem on $H$ implies the DH problem on $G$, and both of them are
equivalent to the BDH problem when $e$ is {\it weak-invertible}.
Moreover, we show that given the bilinear map $e$ an injective
homomorphism $f:H\rightarrow G$ enables us to solve the DH problem
on $G$ efficiently, which implies the non-existence a {\it
self-bilinear} map $e:G\times G \rightarrow G$ when the DH problem
on $G$ is hard. Finally we introduce a sequence of bilinear maps
and its applications.

**Category / Keywords: **public-key cryptography / Diffie-Hellman, Bilinear Diffie-Hellman, Bilinear map

**Date: **received 11 Aug 2002

**Contact author: **jhcheon at icu ac kr

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

**Version: **20020812:205617 (All versions of this report)

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