There has been a rapid increase interest in computational number theory ever since the invention of public-key cryptography. Various attempts to solve the underlying hard problems behind public-key cryptosystems has led to interesting problems in computational number theory. One such problem, called the cubic sieve congruence problem, arises in the context of the cubic sieve method for solving the discrete logarithm problem in prime fields. The cubic sieve method requires a nontrivial solution to the Cubic Sieve Congruence (CSC)x3 y2z (mod p), where p is a given prime. A nontrivial solution must satisfy x3 y2z (mod p), x3 ≠ y2z, 1≤ x, y, z < pα , where α is a given real number ⅓ < α ≤ ½. The CSC problem is to find an efficient algorithm to obtain a nontrivial solution to CSC. This thesis is concerned with the CSC problem. Recently, the parametrization x y2z (mod p) and y υ3z (mod p) of CSC was introduced. We give a deterministic polynomial-time (O(ln3p) bit-operations) algorithm to determine, for a given υ, a nontrivial solution to CSC, if one exists. Previously it took Õ(pα) time to do this. We relate the CSC problem to the gap problem of fractional part sequences. We also show in the α = ½ case that for a certain class of primes the CSC problem can be solved deterministically Õ(p⅓) time compared to the previous best of Õ(p½). It is empirically observed that about one out of three primes are covered by this class, up to 109

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.