The Diffie–Hellman problem and generalization of Verheul’s theorem

Abstract

Bilinear pairings on elliptic curves have been of much interest in cryptography recently. Most of the protocols involving pairings rely on the hardness of the bilinear Diffie-Hellman problem. In contrast to the discrete log (or Diffie-Hellman) problem in a finite field, the difficulty of this problem has not yet been much studied. In 2001, Verheul [66] proved that on a certain class of curves, the discrete log and Diffie-Hellman problems are unlikely to be provably equivalent to the same problems in a corresponding finite field unless both Diffie-Hellman problems are easy. In this paper we generalize Verheul's theorem and discuss the implications on the security of pairing based systems. We also include a large table of distortion maps.

Elliptic curve cryptography: The serpentine course of a paradigm shift

“…It turns out that Verheul's theorem can be generalized (see [30,84]) to all supersingular curves and all finite fields. Thus, the construction of such a map would imply that the Diffie-Hellman problem is easy in all finite fields and all supersingular elliptic curves.…”