Efficient Algorithms for Isogeny Computation on Hyperelliptic Curves: Their Applications in Post-Quantum Cryptography
Mohammed El Baraka, Siham Ezzouak
TL;DR
This work targets the efficiency bottleneck of isogeny computations in post-quantum cryptography by exploiting hyperelliptic curves. It introduces a novel genus $2$ isogeny algorithm that reduces the asymptotic cost from $O(g^4)$ to $O(g^3)$ field operations, with extensions to small-degree isogenies in higher genus and comprehensive empirical validation. The manuscript also provides a rigorous mathematical foundation (Jacobian arithmetic, kernel theory, Richelot isogenies), a detailed security analysis (including resistance to Shor’s and Grover’s algorithms), and practical recommendations for parameter selection and implementation on constrained devices. Overall, hyperelliptic isogeny-based cryptography emerges as a promising, efficient, and quantum-resistant alternative for post-quantum cryptographic protocols, supported by both理论ic and empirical evidence.
Abstract
We present e cient algorithms for computing isogenies between hyperelliptic curves, leveraging higher genus curves to enhance cryptographic protocols in the post-quantum context. Our algorithms reduce the computational complexity of isogeny computations from O(g4) to O(g3) operations for genus 2 curves, achieving significant ciency gains over traditional elliptic curve methods. Detailed pseudocode and comprehensive complexity analyses demonstrate these improvements both theoretically and empirically. Additionally, we provide a thorough security analysis, including proofs of resistance to quantum attacks such as Shor's and Grover's algorithms. Our findings establish hyperelliptic isogeny-based cryptography as a promising candidate for secure and e cient post-quantum cryptographic systems.