SaTC: CORE: Small: Number-theoretic aspects of lattice cryptology

该奖项支持密码学数学基础的研究。 自20世纪70年代后期以来，基于某些数学问题的难度，已经开发了广泛使用的密码系统。 新的应用，以及对现有密码系统的攻击的改进，要求更好地理解这些基本的数学问题。 特别是，开发量子计算机（以及它们能够执行的新技术）的潜力使许多现有的密码系统，如众所周知的Rivest-Shamir-Adleman（RSA）系统和椭圆曲线密码学，面临可能的长期风险。 出于这个原因，研究后量子替代方案很重要，例如基于格子的密码学（这是该项目的重点）。 格还提供了其他潜在的好处，包括新类型的功能，例如对加密数据执行某些操作的能力（这将加强云安全性）。该项目集中在最近提出的许多加密方案中使用的特殊类型格（例如理想格或整数格）的加密强度。 研究人员和合作者打算研究这些特殊晶格在更大的自然家族中的分布，看看它们是否由目前未知的几何特征区分，这些几何特征可能使攻击成为可能。 他们计划使用自守形式和代数数论的技术。 尾界（如高斯质量的Banaszczyk的尾界）将被研究为更一般的区域和适合于这些格的函数。 从不同的计算角度，该项目将探索为这些特殊类型的格量身定制的格约简算法，并分析如何生成这些格的挑战性基础。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award supports research in the mathematical underpinnings of cryptography. Since the late 1970s, widely-used cryptosystems have been developed based on the perceived difficulty of certain mathematical problems. New applications, as well as improvements in attacks on existing cryptosystems, call for a better understanding of these underlying mathematical problems. In particular, the potential to develop quantum computers (and new techniques they would be able to execute) puts many existing cryptosystems, such as the well known Rivest-Shamir-Adleman (RSA) system and Elliptic Curve Cryptography, at possible long-term risk. For this reason, it is important to study post-quantum alternatives, such as lattice-based cryptology (which is the focus of this project). Lattices also offer other potential benefits, including new types of functionalities such as the ability to perform some operations on encrypted data (which would strengthen cloud security).The project centers on the cryptographic strength of special types of lattices (such as ideal lattices or integer lattices) that are used in numerous recent proposed cryptographic schemes. The investigator and collaborators intend to study the distribution of these special lattices amongst larger natural families, to see whether or not they are distinguished by presently-unknown geometric features that might make attacks possible. They plan to use techniques from automorphic forms and algebraic number theory. Tail bounds (such as Banaszczyk's tail bound for Gaussian mass) will be studied for more general regions and functions appropriate to these lattices. From a different, computational perspective, the project will explore lattice-reduction algorithms tailored for these special types of lattices, and analyze how challenging bases of these lattices can be generated.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.