A summary of recent and old results on the security of the Diffie-Hellman key exchange protocol in finite groups

Regarding fundamental protocols in cryptography, the Diffie-Hellman (Diffie and Hellman, 1976) public key exchange protocol is one of the oldest and most widely used in today’s applications. Consequently, many specific cryptographic implementations depend on its security. Typically, an underlying (finite dimensional) group is selected to provide candidates for the key. The study of the security of the exchange as depending on the structure of the underlying group is even today poorly understood, with the most common approaches relying on the security of the Discrete Logarithm problem or on the size of the group. Recent developments bring to attention that the relationship is not necessarily valid and that more research is needed that will relate the underlying structure of the group and the security of the Diffie-Hellman exchange. In this chapter we describe the problem in detail, we present the relationship with the previously studied Discrete Logarithm and Computational Diffie-Hellman problems, we expose the various concepts of security, and we introduce a new statistical concept specifically designed to serve the assessment of the security of the exchange.