Number Theory and Applications

该提案中的数学应用之一是设计和实现保密系统-密码学。我们生活在一个电子通信占主导地位的时代，确保隐私的需要呈指数级增长，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消息以及将个人医疗记录归档到大型数据库方面。密码学在电子商务、普遍的互联网应用、大型数据库的安全、高性能计算以及(潜在的)量子计算中发挥着至关重要的作用，仅举几例。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合成整数的因式分解，以及椭圆曲线密码中使用的离散对数问题)，因此计算数理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，还为确保没有人篡改信息提供了方法。它还允许我们确定是谁发送了数据。今天实用的密码学应用渗透到我们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和通信技术(信通技术)更加重要的核心。这是通过扩大信通技术业务基础设施、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，这本书于2005年5月出版，长达700页。最终目标是在本书所给内容的基础上，为加密知识做出贡献，这最终可能对整个加拿大的安全有利。

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.