Post-quantum cryptography with isogeny graphs of abelian varieties

具有阿贝尔簇同源图的后量子密码学

• 批准号: 2571327
• 金额: --
• 依托单位: University of Bristol
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2571327
• 关键词: Post; quantum; cryptography; isogeny; graphs

Quantum computers threaten to break most of the cryptography we currently use to protect our information security systems over an insecure channel such as the internet. In a quantum computer, performing operations comes from a quantum physical notion that works differently from a classical computer setting, and it gives an exponential speed-up for certain computations. The question of when a large-scale quantum computer will be built is not known and it is hard to estimate the exact time. Although it was not clear that large quantum computers are physically possible in the past, many scientists nowadays believe that it is just a significant engineering challenge. Currently, many researchers have been working to create quantum-resistant cryptographic systems by the time they are needed due to the sudden realization of the possible near arrival of a general quantum computer within the cryptographic community. If large quantum computers become practical one day, all widely used methods of asymmetric cryptography in use will be essentially broken. While the most optimistic believers of quantum computers suggest that such computers are years away to be constructed, maybe decades, it also takes years, maybe decades, to develop, test, and de- ploy new quantum-resistant schemes. To construct such quantum-resistant cryptographic systems, we need a new class of hard mathematical problems which seem to be unbreakable even by a quantum computer. Isogeny-based cryptography is a specific type of post-quantum cryptography that uses certain special maps between abelian varieties, mostly between elliptic curves, over finite fields. The reasons why isogeny-based cryptography grabbed the attention of the cryptographic community are the use of relatively short keys and the most sophisticated and rich mathematical structure among the other proposals for post-quantum candidates. Thus, it paves the way for many interesting questions to cryptographers and number theorists. As one of my research interests, I would like to construct new crypto- graphic protocols by using abelian varieties and their properties. Abelian varieties are the objects combining the fields of geometry and arithmetic. Due to their real-world applications in cryptography, abelian varieties have led to researchers to work on the computational and the arithmetic proper- ties of them. The basic examples of abelian varieties are elliptic curves and Jacobian varieties of hyperelliptic curves, and morphisms between abelian varieties concerning both the geometric and the arithmetic structures are called isogenies

量子计算机有可能打破我们目前用于通过互联网等不安全渠道保护信息安全系统的大多数加密技术。在量子计算机中，执行操作来自量子物理概念，与经典计算机设置不同，它为某些计算提供了指数级的加速。大规模量子计算机何时建成的问题尚不清楚，很难估计确切的时间。尽管过去还不清楚大型量子计算机在物理上是否可行，但现在许多科学家认为这只是一个重大的工程挑战。目前，许多研究人员一直致力于在需要的时候创建抗量子加密系统，因为突然意识到通用量子计算机可能即将到来。如果有一天大型量子计算机成为现实，那么所有广泛使用的非对称加密方法都将被彻底打破。虽然量子计算机最乐观的信徒认为，这样的计算机需要数年，也许几十年才能建成，但开发、测试和部署新的抗量子方案也需要数年，也许几十年。为了构建这种抗量子密码系统，我们需要一类新的数学难题，即使是量子计算机也似乎无法破解。基于等基因的密码学是一种特殊类型的后量子密码学，它使用有限域上阿贝尔变体（主要是椭圆曲线）之间的某些特殊映射。基于等基因的密码学之所以引起密码学界的注意，是因为在其他的后量子候选方案中，使用了相对较短的密钥和最复杂、最丰富的数学结构。因此，它为密码学家和数论学家提出了许多有趣的问题。作为我的研究兴趣之一，我想利用阿贝尔变体及其性质来构建新的加密协议。阿贝尔变分是几何和算术领域相结合的对象。由于它们在密码学中的实际应用，阿贝尔变体导致研究人员对它们的计算和算术性质进行了研究。阿贝尔变体的基本例子是椭圆曲线和超椭圆曲线的雅可比变体，涉及几何结构和算术结构的阿贝尔变体之间的态射称为等基因