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

Artyom Kuninets, Ekaterina Malygina

TL;DR

This work delivers explicit constructions of bases for Riemann–Roch spaces $ scrL(G)$ on elliptic curves for arbitrary divisors, enabling efficient algebraic-geometry code design beyond single-point divisors. It introduces an explicit basis for multipoint divisors using functions of the form $ rac{Y+A_{i,s}(X)}{(X- alpha_i)^s}$ and complementary $g_i$ polynomials, with rigorous proofs of independence and dimension $ deg G$, including a one-point case and char restrictions. The results pave the way for Goppa-like elliptic codes and, notably, quasi-cyclic subfield subcodes (QC-SSDE) that substantially reduce McEliece public-key sizes, while accounting for cryptanalytic considerations. Overall, the paper advances both the theory of Riemann–Roch spaces on elliptic curves and practical code-based cryptography for post-quantum security.

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.

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

TL;DR

This work delivers explicit constructions of bases for Riemann–Roch spaces on elliptic curves for arbitrary divisors, enabling efficient algebraic-geometry code design beyond single-point divisors. It introduces an explicit basis for multipoint divisors using functions of the form and complementary polynomials, with rigorous proofs of independence and dimension , including a one-point case and char restrictions. The results pave the way for Goppa-like elliptic codes and, notably, quasi-cyclic subfield subcodes (QC-SSDE) that substantially reduce McEliece public-key sizes, while accounting for cryptanalytic considerations. Overall, the paper advances both the theory of Riemann–Roch spaces on elliptic curves and practical code-based cryptography for post-quantum security.

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.

Paper Structure

This paper contains 5 sections, 3 theorems, 30 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Explicit bases of Riemann--Roch spaces for multipoint divisors on elliptic curves
  4. Cryptographic application
  5. Conclusion

Key Result

Lemma 2

Let $\mathcal{E}/\mathbb{F}_{p^m}$ be an elliptic curve, and $P = (\alpha, \beta) \in \mathcal{E}(\mathbb{F}_{p^m})\backslash P_\infty$. The basis of the Riemann -- Roch space associated with the divisor $G = kP$, where $k \in \mathbb{N}_{\geq 2}$, is defined as follows: $2 \leq s \leq k$ and $A_s(X) \in \mathbb{F}_{p^m}(\mathcal{E})$ is a function of degree $\deg(A_s) \leq s-1$, satisfying:

Theorems & Definitions (6)

  • Definition 1: Algebraic geometry code
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Lemma 4: Stichtenoth09