Riemann-Roch bases for arbitrary elliptic curve divisors and their application in cryptography

TL;DR

This paper develops explicit bases for Riemann-Roch spaces on elliptic curves, facilitating efficient code construction and cryptographic applications, notably reducing key sizes in post-quantum McEliece cryptosystems.

Contribution

It provides explicit constructions of bases for arbitrary divisors on elliptic curves, enhancing algebraic geometry code design and cryptographic scheme efficiency.

Findings

Enables efficient construction of algebraic geometry codes

Facilitates the creation of quasi-cyclic subfield subcodes

Reduces public key size in McEliece cryptosystem

Abstract

This paper presents explicit constructions of bases for Riemann-Roch spaces associated with arbitrary divisors on elliptic curves. In the context of algebraic geometry codes, the knowledge of an explicit basis for arbitrary divisors is especially valuable, as it enables efficient code construction. From a cryptographic point of view, codes associated with arbitrary divisors with many points are closer to Goppa codes, making them attractive for embedding in the McEliece cryptosystem. Using the results obtained in this work, it is also possible to efficiently construct quasi-cyclic subfield subcodes of elliptic codes. These codes enable a significant reduction in public key size for the McEliece cryptosystem and, consequently, represent promising candidates for integration into post-quantum code-based schemes.