Improved Lower Bound on DHP: Towards the Equivalence of DHP and DLP for Important Elliptic Curves Used for Implementation

TL;DR

This paper improves the lower bounds on the Diffie-Hellman problem for important elliptic curves by leveraging the multiplicative group of finite fields, moving closer to establishing their equivalence with the discrete logarithm problem.

Contribution

It introduces a tighter lower bound for the Diffie-Hellman problem on elliptic curves using auxiliary groups, advancing understanding of their computational hardness.

Findings

Tighter lower bounds for Diffie-Hellman on elliptic curves.

Establishment of bounds close to the discrete logarithm problem.

Implication of potential equivalence between DHP and DLP for certain curves.

Abstract

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.