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arxiv: 2603.14009 · v3 · submitted 2026-03-14 · 💻 cs.CR

On secret sharing from extended norm-trace curves

Olav Geil This is my paper

Pith reviewed 2026-05-15 11:28 UTC · model grok-4.3

classification 💻 cs.CR
keywords ramp secret sharingextended norm-trace curvesalgebraic geometric codesrelative generalized Hamming weightsGoppa boundsecond security layerone-point codes
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The pith

Ramp secret sharing schemes from one-point codes on extended norm-trace curves deliver good parameters together with a second layer of security.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines ramp secret sharing schemes built from one-point algebraic geometric codes defined on extended norm-trace curves. It establishes that these schemes not only meet strong parameter requirements but also include an additional security feature. The analysis depends on computing relative generalized Hamming weights to confirm both the primary and secondary security properties. The work further shows that the estimation technique employed aligns with an established enhanced bound rather than offering a distinct alternative.

Core claim

Schemes defined from one-point algebraic geometric codes from extended norm-trace curves have good parameters and possess a second layer of security along the lines of prior work. The relative generalized Hamming weights of these codes guarantee the ramp parameters and the extra security property. The footprint-like approach to estimating the weights is shown to be a clever application of the enhanced Goppa bound rather than a competing method.

What carries the argument

Relative generalized Hamming weights of one-point algebraic geometric codes from extended norm-trace curves, used to bound both ramp parameters and the second security layer in secret sharing.

Load-bearing premise

The relative generalized Hamming weights of the codes from extended norm-trace curves are sufficient to guarantee both the claimed ramp parameters and the second security layer.

What would settle it

A concrete calculation for a specific extended norm-trace curve showing that its relative generalized Hamming weights fall below the thresholds needed for the stated ramp parameters and second security layer would disprove the central claim.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper claims that ramp secret sharing schemes constructed from one-point algebraic geometric codes on extended norm-trace curves achieve good parameters and possess an additional second layer of security in the sense of [11]. It reinterprets the footprint approach from [4] as an application of the enhanced Goppa bound in [12] rather than a competing method, and uses the resulting relative generalized Hamming weight estimates to support both the ramp parameters and the extra security property.

Significance. If the relative GHW bounds are shown to meet the explicit thresholds of [11] in the relevant regimes, the constructions would supply new families of ramp secret sharing schemes with a verifiable second security layer, extending the asymmetric quantum code applications in [4] and clarifying the relationship between the footprint method and the enhanced Goppa bound.

major comments (2)
  1. [Abstract] Abstract and the central claim: the assertion that the schemes possess a second layer of security along the lines of [11] is load-bearing, yet the manuscript supplies no explicit inequalities or numerical checks mapping the computed relative generalized Hamming weights to the security thresholds required by [11]. Without these verifications, it is unclear whether the weights are sufficient in all claimed parameter ranges.
  2. [Main results (assumed §3–4)] The weakest assumption identified in the reader's note is not discharged: the paper treats the relative GHW estimates (via the enhanced Goppa bound) as automatically delivering both the ramp parameters and the second security layer, but does not exhibit the precise gap conditions or threshold comparisons against [11] that would confirm this for the extended norm-trace curves.
minor comments (3)
  1. [Abstract] Abstract: 'posses' should be 'possess'.
  2. [Examples section] The manuscript should include a short table or explicit list of the relative GHW values for the example curves and the corresponding security thresholds from [11] to make the second-layer claim immediately verifiable.
  3. [Preliminaries] Notation for the enhanced Goppa bound and the footprint quantities should be aligned with [12] to avoid any appearance of a competing method.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of the second security layer. We agree that the manuscript would be strengthened by adding direct comparisons between our relative generalized Hamming weight bounds and the thresholds of [11]. We address each major comment below and will incorporate the requested material in the revision.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the central claim: the assertion that the schemes possess a second layer of security along the lines of [11] is load-bearing, yet the manuscript supplies no explicit inequalities or numerical checks mapping the computed relative generalized Hamming weights to the security thresholds required by [11]. Without these verifications, it is unclear whether the weights are sufficient in all claimed parameter ranges.

    Authors: We accept that the abstract claim requires supporting explicit inequalities and numerical checks. In the revised manuscript we will insert a new paragraph immediately after the statement of the main theorem that derives the minimal gap conditions on the relative weights implied by the enhanced Goppa bound and verifies that these gaps meet the security thresholds of [11] for the families of extended norm-trace curves under consideration. Representative numerical tables for small q and m will also be added. revision: yes

  2. Referee: [Main results (assumed §3–4)] The weakest assumption identified in the reader's note is not discharged: the paper treats the relative GHW estimates (via the enhanced Goppa bound) as automatically delivering both the ramp parameters and the second security layer, but does not exhibit the precise gap conditions or threshold comparisons against [11] that would confirm this for the extended norm-trace curves.

    Authors: We agree that the manuscript currently leaves the precise gap conditions implicit. We will revise the main results section (and the corresponding proofs) to state the exact inequalities required by [11] and to show, using the footprint-derived weight estimates, that these inequalities hold for the one-point codes on extended norm-trace curves. This will make the discharge of the assumption fully explicit rather than automatic. revision: yes

Circularity Check

0 steps flagged

Minor self-citation in bound reinterpretation; core claims rest on independent external results

full rationale

The derivation applies relative GHW estimates from [4] to obtain ramp parameters and invokes the second-layer security property directly from [11]. The paper's clarification that the footprint method equals the enhanced Goppa bound of [12] is presented as an observation rather than a load-bearing step in the secret-sharing construction itself. No equation or parameter is fitted to the target claim and then renamed as a prediction, and no uniqueness theorem or ansatz is imported solely via self-citation. The central assertions therefore remain dependent on externally computed weights and thresholds rather than reducing to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters or invented entities; the analysis rests on standard properties of algebraic-geometry codes and the enhanced Goppa bound from prior work.

axioms (1)
  • domain assumption One-point algebraic geometric codes from extended norm-trace curves admit relative generalized Hamming weights that support the claimed ramp secret-sharing parameters.
    Invoked when asserting good parameters and second-layer security.

pith-pipeline@v0.9.0 · 5476 in / 1152 out tokens · 45773 ms · 2026-05-15T11:28:29.756377+00:00 · methodology

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Reference graph

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