TWC: Small: Automorphic Forms and Harmonic Analysis Methods in Lattice Cryptology

密码学是网络安全的一个基本组成部分，无论是在设计安全的应用程序，以及了解他们真正的安全性。 传统上，密码系统的数学基础是基于涉及整数的困难问题（最著名的是，将两个未知素数的乘积分解为这些素数因子的明显困难）。 最近，几种全新的密码学类型已经被提出来使用格的数学性质。 例如，同态加密提供了操纵加密数据而不泄露其内容的可能性。 该项目资助的研究旨在应用新的数学技术来理解晶格的困难特性，从而理解这些拟议的系统-这些特性被视为使它们实际上快速的关键。 提案中的项目涉及使用自守形式的工具（如Borel-Harish-Chandra约化理论，Eisenstein级数和基场的变化）来研究某些格族中短向量的具体问题。 它还包括对基于格的非阿贝尔推广的密码系统的研究，以及Coppermith方法中BKZ-约化基的几何（包括对小指数RSA的Boneh-Durfee攻击）。

Cryptography is a fundamental part of cybersecurity, both in designing secure applications as well as understanding how truly secure they really are. Traditionally, the mathematical underpinnings of cryptosystems were based on difficult problems involving whole numbers (most famously, the apparent difficulty of factoring a product of two unknown primes back into those prime factors). More recently, several completely new types of cryptography have been proposed using the mathematical properties of lattices. For example, homomorphic encryption offers the possibility of manipulating encrypted data without revealing its contents. The research funded by the project seeks to apply new mathematical techniques to understand difficult properties of lattices and hence of these proposed systems -- properties that are seen as crucial for making them practically fast, in particular. The projects in the proposal involve using tools from automorphic forms (such as Borel-Harish-Chandra reduction theory, Eisenstein series, and change of base field) to examine concrete questions about short vectors in certain families of lattices. It also includes a study of cryptosystems based on nonabelian generalizations of lattices, and the geometry of BKZ-reduced bases in Coppersmith's method (including the Boneh-Durfee attack on small exponent RSA).