On secret sharing from extended norm-trace curves

2026-03-14

AI Summary

This work investigates ramp secret sharing schemes derived from one-point algebraic geometry codes based on the extended norm-trace curve. By analyzing their higher-order relative generalized Hamming weights, the study presents the first construction of a ramp scheme with dual security mechanisms using this curve and demonstrates its favorable parameters. A central contribution is the unification of existing “footprint-type” methods under the framework of the enhanced Goppa bound, revealing them as special cases thereof. This insight resolves longstanding comparative issues in weight estimation and clarifies the advantages of the proposed scheme in terms of both security and efficiency.

Technology Category

• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Search and Optimization: Distributed Search

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

In [4] Camps-Moreno et al. treated (relative) generalized Hamming weights of codes from extended norm-trace curves and they gave examples of resulting good asymmetric quantum error-correcting codes employing information on the relative distances. In the present paper we study ramp secret sharing schemes which are objects that require an analysis of higher relative weights and we show that not only do schemes defined from one-point algebraic geometric codes from extended norm-trace curves have good parameters, they also posses a second layer of security along the lines of [11]. It is left undecided in [4, page 2889] if the ``footprint-like approach'' as employed by Camps-Moreno herein is strictly better for codes related to extended norm-trace codes than the general approach for treating one-point algebraic geometric codes and their likes as presented in [12]. We demonstrate that the method used in [4] to estimate (relative) generalized Hamming weights of codes from extended norm-trace curves can be viewed as a clever application of the enhanced Goppa bound in [12] rather than a competing approach.

Problem

• secret sharing
• extended norm-trace curves
• ramp schemes
• algebraic geometric codes
• relative generalized Hamming weights

Innovation

• extended norm-trace curves
• ramp secret sharing
• algebraic geometric codes
• relative generalized Hamming weights
• enhanced Goppa bound