Post-quantum cryptography with isogeny graphs of abelian varieties

量子计算机有可能破坏我们目前使用的大部分密码学，这些密码学用于通过互联网等不安全的渠道保护我们的信息安全系统。在量子计算机中，执行操作来自量子物理概念，其工作方式与经典计算机设置不同，并且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，也很难估计确切的时间。虽然在过去，大型量子计算机在物理上是否可行尚不清楚，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时创建抗量子密码系统，这是因为密码学界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变得实用，所有广泛使用的非对称密码学方法都将被打破。虽然量子计算机最乐观的信徒认为，这种计算机还需要几年甚至几十年的时间才能建成，但开发、测试和部署新的抗量子方案也需要几年甚至几十年的时间。为了构建这样的量子抵抗密码系统，我们需要一类新的硬数学问题，即使是量子计算机也似乎无法破解。 基于同构的密码学是一种特定类型的后量子密码学，它使用有限域上阿贝尔簇之间的某些特殊映射，主要是椭圆曲线之间的映射。基于同源性的密码学之所以引起密码学界的注意，是因为它使用了相对较短的密钥，以及在其他后量子候选方案中最复杂、最丰富的数学结构。因此，它为密码学家和数论家提出许多有趣的问题铺平了道路。 作为我的研究兴趣之一，我想利用阿贝尔簇及其性质构造新的密码协议。阿贝尔簇是结合几何和算术领域的对象。由于阿贝尔簇在密码学中的实际应用，研究人员开始研究阿贝尔簇的计算和算术性质。阿贝尔簇的基本例子是椭圆曲线和超椭圆曲线的雅可比簇，阿贝尔簇之间关于几何和算术结构的态射称为同构

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