Informal analysis of authenticated key establishment (ake) protocols was commonly accepted as the valid argument for their security in the past. Although it can provide some confidence in protocol correctness, experience has shown time and again that ake protocols are likely to contain flaws even after an informal analysis is completed. Therefore, it has become increasingly common to expect a formal analysis, and preferably a mathematical proof, of any published ake protocol in order to obtain increased confidence in its security. In this research we use an appropriate model for analysing ake protocols based on its features and properties. The model allows us to design ake protocols modularly and reuse existing protocol components. We provide a detailed description of its formalisation, operations and usage. This description also includes ways of extracting new protocol components from existing ake protocols. Following the description of the model, we propose a new unauthenticated key establishment protocol for two-party communications. By composing this protocol with authentication protocols, we can construct several new secure ake protocols. These new protocols are compared with existing protocols for their computational efficiency. The comparison shows that our new proven secure protocols are as efficient as the existing protocols with an informal security analysis. We then propose a three-party key establishment protocol which involves a trusted server and two users. We also propose a non-interactive authentication protocol and discuss it and a variant of it. These components are used to construct a secure three-party ake protocol that supports a privacy framework. This framework allows users to remain anonymous while conducting electronic transactions with an independent service provider. A new password-based authentication protocol is proposed to address the problem of authentication using passwords. This protocol carries a proof of security and satisfies a slightly relaxed definition of security. We demonstrate its application by composing it with existing key establishment protocols. To maximise its use, we modified a two-party key establishment protocol to become three-party server based. By using the server for authentication, two users within a common network domain can establish a secure session key. Only a small number of ake protocols are demonstrated in this thesis. There exist many more provably secure ake protocols that can be constructed using the protocol components presented by applying the approach of "mix and match". That is, each new component results in a number of new ake protocols depending on the number of existing components.

Elliptic curves were first proposed as a basis for public key cryptography in the mid 1980's. They provide public key cryptosystems based on the difficulty of the elliptic curve discrete logarithm problem (ECDLP) , which is so called because of its similarity to the discrete logarithm problem (DLP) over the integers modulo a large prime. One benefit of elliptic curve cryptosystems (ECCs) is that they can use a much shorter key length than other public key cryptosystems to provide an equivalent level of security. For example, 160 bit ECCs are believed to provide about the same level of security as 1024 bit RSA. Also, the level of security provided by an ECC increases faster with key size than for integer based discrete logarithm (dl) or RSA cryptosystems. ECCs can also provide a faster implementation than RSA or dl systems, and use less bandwidth and power. These issues can be crucial in lightweight applications such as smart cards. In the last few years, ECCs have been included or proposed for inclusion in internationally recognized standards. Thus elliptic curve cryptography is set to become an integral part of lightweight applications in the immediate future. This thesis presents an analysis of several important issues for ECCs on lightweight devices. It begins with an introduction to elliptic curves and the algorithms required to implement an ECC. It then gives an analysis of the speed, code size and memory usage of various possible implementation options. Enough details are presented to enable an implementer to choose for implementation those algorithms which give the greatest speed whilst conforming to the code size and ram restrictions of a particular lightweight device. Recommendations are made for new functions to be included on coprocessors for lightweight devices to support ECC implementations Another issue of concern for implementers is the side-channel attacks that have recently been proposed. They obtain information about the cryptosystem by measuring side-channel information such as power consumption and processing time and the information is then used to break implementations that have not incorporated appropriate defences. A new method of defence to protect an implementation from the simple power analysis (spa) method of attack is presented in this thesis. It requires 44% fewer additions and 11% more doublings than the commonly recommended defence of performing a point addition in every loop of the binary scalar multiplication algorithm. The algorithm forms a contribution to the current range of possible spa defences which has a good speed but low memory usage. Another topic of paramount importance to ECCs for lightweight applications is whether the security of fixed curves is equivalent to that of random curves. Because of the inability of lightweight devices to generate secure random curves, fixed curves are used in such devices. These curves provide the additional advantage of requiring less bandwidth, code size and processing time. However, it is intuitively obvious that a large precomputation to aid in the breaking of the elliptic curve discrete logarithm problem (ECDLP) can be made for a fixed curve which would be unavailable for a random curve. Therefore, it would appear that fixed curves are less secure than random curves, but quantifying the loss of security is much more difficult. The thesis performs an examination of fixed curve security taking this observation into account, and includes a definition of equivalent security and an analysis of a variation of Pollard's rho method where computations from solutions of previous ECDLPs can be used to solve subsequent ECDLPs on the same curve. A lower bound on the expected time to solve such ECDLPs using this method is presented, as well as an approximation of the expected time remaining to solve an ECDLP when a given size of precomputation is available. It is concluded that adding a total of 11 bits to the size of a fixed curve provides an equivalent level of security compared to random curves. The final part of the thesis deals with proofs of security of key exchange protocols in the Canetti-Krawczyk proof model. This model has been used since it offers the advantage of a modular proof with reusable components. Firstly a password-based authentication mechanism and its security proof are discussed, followed by an analysis of the use of the authentication mechanism in key exchange protocols. The Canetti-Krawczyk model is then used to examine secure tripartite (three party) key exchange protocols. Tripartite key exchange protocols are particularly suited to ECCs because of the availability of bilinear mappings on elliptic curves, which allow more efficient tripartite key exchange protocols.