Number Theory and Applications

该提案中数学的应用之一是设计和实现保密系统——密码学。我们生活在一个以电子通信为主导的时代，确保隐私的需求呈指数级增长，特别是在保密和互联网通信、电子资金转移、发送军事或个人电子邮件以及在大型数据库中存档个人医疗记录等领域。密码学在电子商务、一般互联网使用、大型数据库的安全性、高性能计算和（潜在的）量子计算等方面发挥着至关重要的作用。这表明密码学对学术界、军事、商业和工业企业的重要性是至高无上的。由于密码系统安全性的基础涉及解决困难的数学问题（例如，RSA密码系统中使用的大组合整数的分解，以及椭圆曲线密码中使用的离散对数问题），因此计算数论适合作为安全场景的一个组成部分。密码学不仅为我们提供了保密敏感信息的方法，而且还确保没有人篡改信息。它还允许我们确定谁发送了数据。今天，密码学的实用应用渗透到我们这个以信息为基础的世界，使它们成为我们安全社会配方的关键成分。在这个新的千年里，密码学将继续成为任何和所有信息通信技术（ICT）的更重要的核心。这是通过扩大信息通信技术业务基础设施、研究和开发领域来实现的，所有这些都在我的第八本书《密码——从古代到现代的保密指南》中进行了讨论，这本700页的大部头书于2005年5月出版。最终目标是建立在本书中给出的内容之上，并为密码学知识做出贡献，这最终可能有利于加拿大的安全。

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.