Number Theory and Applications

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Among the applications of the mathematics in the proposal is to the design and implementation of secrecy systems --- cryptography. We are living in an age dominated by electronic communication and the need to ensure privacy has increased exponentially, particularly in the areas of confidentiality and Internet communication, transfer of electronic funds, sending of military or personal e-mail messages, and filing of personal medical records in large databases. Cryptography plays a vital role in e-commerce, general Internet usage, security of large databases, high-performance computing, and (potentially) quantum computing, to mention a few. This demonstrates that the importance of cryptography to academia, the military, business, and industrial enterprises is paramount. Since the basis for the security of cryptosystems involves the solving of difficult mathematical problems (for example, the factoring of large composite integers used in the RSA cryptosystem, and the discrete logarithm problem used in elliptic curve ciphers), then computational number theory fits in as an integral part of the security scene. Cryptography provides us with methods for not only keeping sensitive information secret, but also for ensuring that nobody tampers with the information. It also allows us to determine who sent the data. Today's utile applications of cryptography permeate our information-based world, rendering them crucial ingredients in our recipe for a secure society. In this new millennium, cryptography will continue to become an even more essential centerpiece of any and all information and communications technology (ICT). This is being accomplished via the expansion of ICT business infrastructure, research and development arenas, all of which are discussed in my eighth book, "CODES --- The Guide to Secrecy from Ancient to Modern Time", a 700-page tome published in May of 2005. The end-goal is to build upon what is given in this book and contribute to cryptographic knowledge, which may ultimately be of benefit to the security of Canada at large.