Cryptographic primitives and error control coding

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The first part of the proposal is concerned with the investigation of the fundamental primitives of cryptography of homomorphic encryption and hash functions. Homomorphic encryption systems find wide application in numerous important cryptographic protocols. New homomorphic encryption systems will be developed with a view to making their application to certain secure multiparty computation protocols more efficient. Hash functions are an essential part of many commonly used cryptographic protocols such as digital signatures. Most currently used such functions are fast but lack a security proof. The few currently known hash functions that have some form of security proof tend to rely on sophisticated mathematics and are very slow. The proposed research will investigate possible techniques to formulate a new function that has some form of security proof and improves on the speed. For the second part of the proposal, further properties of random coding techniques for the erasure channel will be considered. Such systems have proved very useful for multicast internet downloads where a symbol erasure corresponds to a packet loss. Systems that have linear complexity (in the number of information symbols) encoding and decoding techniques, for the binary case, are available although have limitations in terms of implementation. Nonbinary systems will be investigated that have the potential to improve performance and efficiency in terms of speeds of convergence and simplicity of implementation.