This paper describes a variant of the Diffie-Hellman protocol, generalized for three participants, and based on elliptic curves as the mathematical tool.

In order to devise this tripartite extension, the author uses the Weil and Tate pairings. These pairings were involved initially in cryptography, as cryptanalytic tools to reduce the complexity of the discrete logarithm problem on “weak” elliptic curves.

The tripartite Diffie-Hellman protocol uses only one round of communication (all such protocols known before have at least two rounds). Moreover, it can be transformed into a noninteractive public key crypto system.

The paper has six sections, starting with an introduction. The second section, “The Discrete Logarithm Problem on Weak Elliptic Curves,” includes a short presentation of MOV and FR reductions (based on the Weil and, respectively, the Tate pairing).

The third section presents “A Tripartite Diffie-Hellman Protocol,“ which involves three participants and only one pass of communication for constructing a common secret key, using two points of an elliptic curve. Two variants are proposed and analyzed.

Section 4 discusses a single point approach, where the previously presented protocol is improved, using only one point of the elliptic curve. Three different techniques are built, and their advantages and drawbacks are examined.

Section 5, on “Security Issues,” presents some remarks about security assumptions (the difficulty of the discrete logarithm problem, the computational Diffie-Hellman problem, and the decision Diffie-Hellman problem) and their relations.

The paper is a revised and updated version of another paper by Joux [1].