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It is generally regarded that there are two main potential advantages of public key techniques over symmetric cryptography. The first is that public key systems allow the straightforward definition of digital signatures, thereby enabling the service of non-repudiation which is so useful in commercial applications. The second is the simplification of key management, because there is no requirement for the online third party that is part of typical protocols based on symmetric cryptography. The first of these advantages is not really our concern in this book since non-repudiation is of limited value in authentication and key establishment. However, the second advantage has led to a great variety of new key establishment protocols since the invention of public key cryptography. In the modern distributed communications environments exemplified by the Internet, public-key-based protocols have become far more important than …