  In 2004, Muzereau et al. showed how to use a reduction algorithm of the
discrete logarithm problem to Diffie-Hellman problem in order to estimate lower
bound on Diffie-Hellman problem on elliptic curves. They presented their
estimates for various elliptic curves that are used in practical applications.
In this paper, we show that a much tighter lower bound for Diffie-Hellman
problem on those curves can be achieved, if one uses the multiplicative group
of a finite field as an auxiliary group. Moreover, improved lower bound
estimates on Diffie-Hellman problem for various recommended curves are also
given which are the tightest; thus, leading us towards the equivalence of
Diffie-Hellman problem and the discrete logarithm problem for these recommended
elliptic curves.
