Many cryptographic systems use exponentiation in an appropriate finite group as the critical part of the method of encryption. The security of any such system depends on the mathematical intractability of the computational problem of inverting the operation of exponentiation in that group, a problem known as the discrete-log problem. The difficulty or intractability of the discrete-log problem may – and will – depend on the specfic group. The discrete-log problem is dificult in some groups and is easy in other groups. The belief in the intractability of this computational problem in many groups is based on anecdotal evidence rather than on mathematical proof. This means that a public-key cryptosystem based on exponentiation always entails the risk that the underlying inverse computational problem is actually easy. Indeed, that computational problem may already have been solved by a secluded and secretive cryptanalyst.

Diffie-Hellman key exchange

The Diffie–Hellman key exchange (or, more descriptively, the Diffie–Hellman key agreement or key creation) is a method by which two parties with no prior communication establish a secret key by messages sent over a clear public two-way channel. No key is actually exchanged despite the terminology. More correctly, a key is created by both parties working together. The two parties, between them, establish a common secret key even though all of their communication takes place on a public channel. It is believed that no eavesdropper is able to determine the secret key even though the eavesdropper has access to all communications.