On secret sharing from extended norm-trace curves

TL;DR

This paper analyzes ramp secret sharing schemes based on algebraic geometric codes from extended norm-trace curves, highlighting their security features and comparing estimation methods for code weights.

Contribution

It demonstrates that codes from extended norm-trace curves have strong parameters and a second security layer, and clarifies the estimation approach for their weights.

Findings

Codes from extended norm-trace curves have good parameters for secret sharing.

The method used is an application of the enhanced Goppa bound, not a competing approach.

The schemes possess a second layer of security.

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.