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Cryptographic protocols depend on the hardness of some computational
problems for their security. Joux briefly summarized known relations
between assumptions related bilinear map in a sense that if one
problem can be solved easily, then another problem can be solved
within a polynomial time \cite{Joux02}.
In this paper, we investigate additional relations between them.
Firstly, we show that the computational Diffie-Hellman assumption
implies the bilinear Diffie-Hellman assumption or the general
inversion assumption. Secondly, we show that a cryptographic useful
self-bilinear map does not exist. If a self-bilinear map exists, it
might be used as a building block for several cryptographic
applications such as a multilinear map. As a corollary, we show that
a fixed inversion of a bilinear map with homomorphic property is
impossible. Finally, we remark that a self-bilinear map proposed in
\cite{Lee04} is not essentially self-bilinear.
