Algorithms in number theory and cryptography

在过去的几十年里，密码学，数据安全的研究，已经成为我们社会的一个重要问题。随着我们的个人信息越来越多地通过互联网访问，保护和验证数据的需求比以往任何时候都更加迫切。现代公钥密码学中许多方法的安全性依赖于数学中某些计算问题的假定难度，特别是数论。这些方案的设置方式假定攻击者破坏给定系统的唯一方法是解决此类问题的一个实例。目前，很少有数论问题被广泛接受作为安全公钥协议的基础。此外，这些问题大多具有很强的理论关系。在许多情况下，能够有效地解决一个问题意味着能够有效地解决另一个问题。换句话说，一个引人注目的数学发现可能会使许多当前的公钥加密技术变得不安全。

In the past few decades, cryptography, the study of data security, has emerged as an important concern in our society. As our personal information becomes increasingly accessible through the Internet, the need to protect and authenticate data is more imperative than ever. The security of many methods in modern public-key cryptography relies on the supposed difficulty of certain computational problems in mathematics and, in particular, number theory. These schemes are set up in such a way that presumably the only way for an attacker to break the given system is to solve an instance of one such problem. Currently, very few number-theoretic problems are widely accepted as foundations for secure public-key protocols. Furthermore, most of these problems have strong theoretical relationships. In many cases, being able to solve one efficiently implies the ability to solve another efficiently as well. In other words, one dramatic mathematical discovery could render many current public-key cryptographic techniques insecure.