Improved Lower Bound on DHP: Towards the Equivalence of DHP and DLP for Important Elliptic Curves Used for Implementation
This work addresses a foundational cryptographic security issue for users of elliptic curve cryptography, though it is incremental as it builds on prior reduction algorithms.
The paper tackles the problem of establishing a tighter lower bound on the Diffie-Hellman problem (DHP) for elliptic curves used in practice, achieving the tightest estimates to date and moving closer to proving the equivalence of DHP and the discrete logarithm problem (DLP) for these curves.
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.