Cycles, Randomness, Discrete Logarithms, and Key Exchange

Abstract

With symmetric cryptography, it is necessary for the two parties who wish to communicate to have access to a common key so that one party can encrypt a message and the other party can decrypt the message. This would limit the ability of two parties who have not communicated in the past to engage in the kind of secure communication necessary for electronic commerce, for example. In this chapter we describe how number-theoretic constructs that create seemingly-random sequences of integers can be used to allow two parties to exchange information that would allow them to agree upon a common cryptographic key, even if no other communication has taken place between them. This exchange of key information can be done by exponentiation modulo a large prime number, in a manner similar to that of RSA encryption, or using elliptic curve groups in the same fashion. We will also cover the basics of the index calculus method that can be used, although with difficulty, to attack this kind of key exchange.