International Association for Cryptologic Research

International Association
for Cryptologic Research


Paper: Diffie-Hellman Problems and Bilinear Maps

Jung Hee Cheon
Dong Hoon Lee
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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.
  title={Diffie-Hellman Problems and Bilinear Maps},
  booktitle={IACR Eprint archive},
  keywords={public-key cryptography / Diffie-Hellman, Bilinear Diffie-Hellman, Bilinear map},
  note={ 11911 received 11 Aug 2002},
  author={Jung Hee Cheon and Dong Hoon Lee},