Volume 17, Number 2

Efficient Algorithms for Isogeny Computation on Hyperelliptic Curves: Their Applications in Post-Quantum Cryptography

  Authors

Mohammed El Baraka and Siham Ezzouak, Sidi Mohammed Ben Abdellah University, Morocco

  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 com- putations from O(g4) to O(g3) operations for genus 2 curves, achieving signi cant e ciency gains over traditional elliptic curve methods. Detailed pseudocode and comprehensive complexity analyses demonstrate these improvements both theoretically and em- pirically. Additionally, we provide a thorough security analysis, including proofs of resistance to quantum attacks such as Shor's and Grover's algorithms. Our ndings establish hyperelliptic isogeny-based cryptography as a promising candidate for secure and e cient post-quantum cryptographic systems.

  Keywords

isogenies; hyperelliptic curves; post-quantum cryptography; complexity reduction; e ciency gains; empirical evaluation; quantum resistance.