Algorithms and cryptography in global fields

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The last few decades have seen an unprecedented increase in the use of computing and communication**technology. Coupled with this development is a crucial need for information protection: confidential data must be guarded against access or modification by unauthorized parties, users of online banking need to be authenticated to ensure that they are who they claim they are, and the origin of e-mails must be verifiable to ascertain if they are legitimate. Sadly, the public has become accustomed to news flashes about virus-infected computers, hacking, defaced web sites, cases of identity theft, and even potential threats of internet terrorism.****Cryptography is an effective vehicle for preserving the confidentiality, integrity and authenticity of digital information. The security of many modern cryptosystems relies on certain mathematical problems that experts believe to be extremely difficult to solve. The idea underlying the design of such a system is that an adversary would have to solve an instance of one of these very hard problems in order to crack the scheme. It is therefore of great interest, both theoretical and practical, to thoroughly study these types of problems and devise new cryptosystems that make use of them. To this end, I propose to investigate a type of mathematical structure called a global field. Certain global fields provide an excellent framework for efficient and very secure cryptosystems; this includes the celebrated elliptic curve cryptosystems which have been deployed in a range of consumer electronics products such as BluRay technology and the BlackBerry. My research into fast arithmetic in global fields will make the corresponding cryptographic schemes more efficient. Moreover, my exploration into the difficulty of certain hard mathematical problems in global fields will lead to a better understanding of the security of such schemes. The outcomes of the proposed work will be a collection of well understood algorithms for number theoretic computations as well as a suite of secure and efficient cryptographic protocols that can be used to protect electronic communications.********