Isogeny-based cryptography: A comprehensive review on advancements, analysis of attacks, and future directions

Abstract

Recent developments in quantum computing present significant challenges to traditional cryptographic approaches, especially those relying on integer factorization and discrete logarithms. To counter these challenges, a variety of post-quantum cryptographic (PQC) methods have been proposed, including cryptographic systems based on lattices, hashing, codes, multivariate equations, and isogenies. Each of these is based on complex mathematical problems that are challenging to solve even with quantum computers. Among them, isogeny-based cryptography has garnered considerable attention for its small key sizes, making it a leading candidate for developing efficient post-quantum cryptographic primitives. This article provides an in-depth review of isogeny-based cryptographic techniques, emphasizing key exchange, zero-knowledge proofs (ZKP), and signature protocols. It begins by elucidating the foundational concepts of elliptic curves, isogenies, and the primary protocols utilizing them. The discussion then shifts to the challenges that current schemes face and examines potential research directions aimed at improving security, scalability, and functionality.