Cryptographic primitives and error control coding

该方案的第一部分是关于同态加密和散列函数密码学的基本原语的研究。同态加密系统在众多重要的密码协议中有着广泛的应用。将开发新的同态加密系统，以期使其在某些安全多方计算协议中的应用更加有效。散列函数是许多常用密码协议(例如数字签名)的重要组成部分。目前使用的大多数此类功能速度都很快，但缺乏安全证明。目前为数不多的具有某种形式安全性证明的散列函数往往依赖于复杂的数学，并且速度非常慢。建议的研究将调查可能的技术来制定一个新的函数，具有某种形式的安全证明，并提高速度。 对于建议的第二部分，将考虑用于擦除信道的随机编码技术的进一步特性。这种系统已被证明对于多播互联网下载非常有用，其中符号擦除对应于分组丢失。对于二进制情况，具有线性复杂性(在信息符号的数量方面)的编码和解码技术的系统是可用的，尽管在实现方面有限制。将研究在收敛速度和实现简单性方面具有提高性能和效率的潜力的非二进制系统。

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.