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

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(g 4 ) to O(g 3 ) 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 empirically. 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.