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On secret sharing from extended norm-trace curves

Olav Geil

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.

On secret sharing from extended norm-trace curves

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.
Paper Structure (6 sections, 4 theorems, 51 equations)

This paper contains 6 sections, 4 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Linear ramp secret sharing
  3. Relative parameters of one-point algebraic geometric codes and their relatives
  4. Bounds on relative parameters of codes from the extended norm-trace curve
  5. Schemes with a second layer of security
  6. Concluding remarks

Key Result

theorem 1

Let $A=\{ i_1 < \cdots < i_m\} \subseteq \{1, \ldots , n\}$. Assume $c_{i_1}^\prime , \ldots , c_{i_m}^\prime$ are simultaneously realizable shares in those positions, see (eqsnabel1). The amount of possible secrets $\vec{s}$ corresponding to such shares equals $q^s$ with where $\bar{A}=\{1, \ldots , n\} \backslash A$.

Theorems & Definitions (9)

  • definition 1
  • theorem 1
  • theorem 2
  • proof
  • lemma 1
  • proof
  • corollary 1
  • remark 1
  • remark 2