Isogeny-based cryptography and its applications

现代密码学对于在对手在场的情况下保护我们的在线通信至关重要。它允许我们以保密的方式传输消息，同时确保消息的真实性和未被篡改。密码学在现实世界中有大量的应用，包括即时消息、电子商务和数字货币。然而，量子计算的出现揭示了，如果建造大规模的量子计算机，它们可以破解当今广泛使用的许多公钥密码系统。这是因为，与经典计算机不同，量子计算机可以同时处理大量信息，这意味着它们可以有效地解决经典计算机无法在可行的时间内解决的问题。这引发了开发新密码学的竞赛，这种密码学可以使用经典计算机实现，但可以抵御来自量子计算机的攻击，从而产生了密码学的一个分支，称为后量子密码学(PQC)。我的研究将集中在最年轻的PQC类型，即基于同源的密码学。这种类型的密码术使用椭圆曲线之间的映射来构建协议，这些椭圆曲线具有某些属性，称为同源。这些方案的安全性基于以下问题：给定有限域上定义的两条椭圆曲线，找出一条椭圆曲线到另一条椭圆曲线的同源。由于这是一个年轻的研究领域，有许多研究方向需要探索。我感兴趣的两个主要研究方向是：使用基于同源的密码学来构造更复杂的密码协议(超越密钥交换和签名方案)；提高基于同源的密码原语的效率。我们注意到，在提高效率方面取得的任何进展都将直接有利于在追求先前研究方向的同时开发的任何建筑的实用性。

Modern cryptography is essential to protect our online communications in the presence of adversaries. It allows us to transmit messages in a confidential manner whilst ensuring they are authentic and have not been altered. Cryptography has a vast array of real-world applications, including instant messaging, electronic commerce, and digital currencies. The advent of quantum computation, however, has revealed that if large-scale quantum computers are built, they can break many of the public-key cryptosystems that are widely used today. This is because, unlike classical computers, quantum computers can process a large amount of information simultaneously, meaning they can efficiently solve problems that no classical computer can solve in a feasible amount of time. This sparked the race to develop new cryptography that can be implemented using classical computers but can withstand attacks from quantum ones, creating a branch of cryptography called post-quantum cryptography (PQC).My research will focus on the youngest type of PQC, namely isogeny-based cryptography. This type of cryptography builds protocols using maps between elliptic curves that have certain properties, called isogenies. The security of these schemes is based on the following problem: given two elliptic curves defined over a finite field, find an isogeny that maps one elliptic curve to the other. As it is a young field of research, there are many directions of research to explore. My two main research directions of interest are:Using isogeny-based cryptography to construct more complex cryptographic protocols (beyond key exchange and signature schemes);Improving the efficiency of isogeny-based cryptographic primitives. We note that any progress made in improving efficiency will directly benefit the practicality of any constructions developed whilst pursuing the previous research direction.