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

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.

💡 Research Summary

The paper addresses a long‑standing gap in algebraic‑geometry (AG) coding theory: the lack of explicit bases for Riemann‑Roch spaces associated with arbitrary divisors on elliptic curves. While bases for the special case (G=kP_{\infty}) are well known, practical code construction and cryptographic applications require bases for multi‑point divisors of the form (G=\sum_{i=1}^{z}k_iP_i). The authors provide constructive algorithms, rigorous proofs of correctness, and a detailed analysis of the resulting code families.

Mathematical contributions.

1. Single‑point divisor basis (Lemma 2). For a point (P=(\alpha,\beta)) on an elliptic curve (E) over (\mathbb{F}_{q}) (characteristic (\neq3)), they define functions
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