TWC: Medium: CRYPTOGRAPHIC APPLICATIONS OF CAPACITY THEORY

该项目的主要目标是建立现代密码系统分析的数学基础。密码学是用于保护互联网通信安全的核心工具。 安全可靠的通信和数据存储对国家安全和世界经济的运作至关重要。 最近引人注目的研究成果使新型密码学的发展成为可能，令人兴奋的新的潜在应用，并希望从长远来看更有力地保证密码安全。 该项目旨在通过将数学中的新方法与密码工具相结合来增加对这些新构造的信心。几乎所有的公钥密码学都依赖于解决各种数论问题的假设难度来保证其安全性。密码学的最新发展，如全同态加密，候选多线性映射和有效的后量子格基密码学，产生了大量新的代数和数论密码硬度假设。 这些问题中有许多是新的，在很大程度上没有被计算数论界研究过。 该项目使用来自算术几何的工具，如容量理论和Arakelov理论，以更严格地理解用于确保密码安全的基本密码分析技术的理论和计算限制。 这个项目将改进密码分析中使用的数值和理论工具，并促进对许多实际密码系统安全性的更好理解。

The primary goal of this project is to develop a mathematical foundation underlying the analysis of modern cryptosystems. Cryptography is a core tool used to secure communications over the Internet. Secure and trustworthy communications and data storage are essential to national security and to the functioning of the world economy. Recent spectacular research results have enabled the development of new types of cryptography, exciting new potential applications, and hopes for stronger guarantees of cryptographic security in the long term. This project aims to increase confidence in these new constructions by unifying new methods from mathematics with cryptographic tools.Nearly all of public-key cryptography relies for its security on the assumed difficulty of solving various number theoretic problems. Recent developments in cryptography such as fully homomorphic encryption, candidate multilinear maps, and efficient post-quantum lattice-based cryptography have produced a multitude of new algebraicand number-theoretic cryptographic hardness assumptions. Many of these problems are new and largely unstudied by the computational number theory community. This project uses tools from arithmetic geometry such as capacity theory and Arakelov theory to develop a more rigorous understanding of the theoretical and computational limits to the fundamental cryptanalytic techniques used to assure cryptographic security. This project will improve the numerical and theoretical tools used in cryptanalysis, and promote a better understanding of the security of many practical cryptosystems.