Visualisation of Modern Key Exchange Schemes for more than two Parties in CrypTool and their Security Analysis

Key exchange is a prerequisite of most types of secure communication. Its purpose is to allow an arbitrarily large number of users n to communicate securely over an insecure line (the Internet). The Diffie-Hellman key exchange protocol is the most famous protocol that achieves key exchange for the case of two users only; this protocol was originally implemented only in finite fields of prime order. It is the purpose of this thesis to describe several other settings in which cryptography may be implemented: namely elliptic curves and pairings on elliptic curves. The translation of the Diffie-Hellman protocol to elliptic curves is explained and the two settings are compared. For the pairing setting, the Tate and Weil pairings are presented, and a basic overview of pairing-friendly curves with embedding degrees k = 2 and k = 12 is given. As a basic arrangement for three-partite key exchange in a pairing setting, a protocol by Joux is described and analysed. Finally, for arbitrary values of n, two versions of the BD I (named after its inventors Burmester and Desmedt) key exchange protocol and the BD II key exchange protocol are presented in finite fields and in the pairing setting. New additions to the protocols already described in the literature are: a slight extension for the BD II protocol in the pairing setting to include cases in which users might have only one child instead of two on one of its branches, and a turn-based BD I protocol for the pairing setting. For each of the protocols in the pairing setting, an explicit method has been given of how to modify the protocol for elliptic curves that do not support distortion maps. It was also the intention of this master project to implement multi-partite key exchange in JCrypTool – a didactic tool that demonstrates various cryptographic and security-related topics. Some details of this implementation are also presented in this thesis.