Isogeny-based cryptography and its applications

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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.