CAREER: Geometric Methods in Cryptography

随着越来越多的人使用计算机网络交换机密数据和执行商业交易，公钥密码学正迅速成为当今电子世界中最关键的工具之一。 使用密码学可以执行许多重要的任务，从电子投票到数字合同签署，在公共网络上安全的虚拟会议等等。 所有这些应用程序最终都依赖于底层加密原语的安全性（即构建所有其他更复杂的加密应用程序的基本构建块）。 这项研究涉及数学领域的计算问题的研究，称为数字几何，可以用来设计新的密码原语，并测试旧的，并评估其安全性。研究人员研究点格的复杂性。 这些是可以被描述为规则n维网格的交点集合的几何对象。 这项研究涉及硬格问题的识别，并寻找更好的算法来解决格问题，承认有效的解决方案。 硬格问题随后被用来设计新的密码函数，而新的格算法被用来设计新的密码分析攻击现有的密码原语。

As more and more people use computer networks to exchange confidential data and perform business transactions, public key cryptography is rapidly becoming one of the most critical tools in today's electronic world. Using cryptography it is possible to perform many important tasks ranging from electronic voting, to digital contract signing, secure virtual conferences on public networks and many more. All these applications ultimately rely on the security of the underlying cryptographic primitives (i.e. the fundamental building blocks using which all other more complex cryptographic applications are built). This research involves the study of computational problems from an area of mathematics called geometry of numbers that can be used both to design new cryptographic primitives, and to test old ones and assess their security.The investigators study the complexity of point lattices. These are geometric objects that can be described as the set of intersection points of a regular n-dimensional grid. This research involves both the identification of hard lattice problems, and the search for better algorithms to solve lattice problems that admit efficient solution. Hard lattice problems are subsequently used to design new cryptographic functions, while new lattice algorithms are used to design new cryptanalytic attacks against existing cryptographic primitives.