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arxiv: 2605.22426 · v1 · pith:VKD4UPBVnew · submitted 2026-05-21 · 💻 cs.IT · cs.DC· math.IT

Monotone Erasure Codes

Vivien Bammert , Annalisa Cimatti , Orestis Alpos , Giuliano Losa , Christian Cachin This is my paper

Pith reviewed 2026-05-22 04:46 UTC · model grok-4.3

classification 💻 cs.IT cs.DCmath.IT
keywords monotone erasure codesaccess structureserasure codingdistributed storageverifiable information dispersallinear codestrust assumptionsblockchain protocols
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The pith

Monotone erasure codes can be constructed for any access structure given by a monotone Boolean formula, supporting arbitrary trust assumptions in storage systems.

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

The paper establishes that erasure codes can be generalized beyond threshold assumptions to respect any monotone access structure on nodes, defined via Boolean formulas. This matters because modern systems like blockchains often use complex trust models instead of simple majority failures. The authors provide an efficient algorithm to build linear monotone erasure codes for arbitrary access structures and a specialized efficient method for partitioned structures that minimizes storage. They further demonstrate how these codes yield a communication-efficient generalized asynchronous verifiable information dispersal primitive, useful for reliable broadcast and consensus.

Core claim

For any access structure specified by a monotone Boolean formula, a monotone erasure code exists that recovers the data if and only if the set of nodes satisfies the formula. Linear versions of these codes can be constructed efficiently using linear algebra over finite fields for any such structure, and for partitioned access structures this construction achieves minimal storage overhead. These codes in turn support a generalized AVID primitive with reduced communication costs compared to threshold-based versions.

What carries the argument

Monotone erasure codes, which encode data such that it is recoverable precisely when the participating nodes form an authorized set under the given monotone access structure.

If this is right

  • Any monotone access structure admits an efficient construction of a corresponding linear monotone erasure code.
  • Partitioned access structures allow constructions with minimal storage overhead.
  • Monotone erasure codes enable a generalized AVID that is communication-efficient.
  • This supports more flexible reliable broadcast and consensus protocols under non-threshold failure assumptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Implementations could adapt existing erasure code libraries to handle Boolean formula-based access structures for practical deployment.
  • Testing recovery guarantees on sample access structures would validate the linear algebra approach.
  • The generalization might extend to other coding primitives beyond AVID in distributed systems.

Load-bearing premise

That standard linear algebra operations over a finite field can always produce a valid encoding meeting the exact recovery guarantees required by the monotone Boolean formula access structure.

What would settle it

A counterexample access structure defined by a monotone formula where either no linear encoding satisfies the recovery condition or the proposed algorithm fails to generate one that does.

Figures

Figures reproduced from arXiv: 2605.22426 by Annalisa Cimatti, Christian Cachin, Giuliano Losa, Orestis Alpos, Vivien Bammert.

Figure 1
Figure 1. Figure 1: Access tree representing A and 𝜂. △ 4 Linear Monotone Erasure Codes We now consider monotone erasure codes for data represented by symbols that are elements of a finite field F𝑞 . This means the file is a vector 𝑓 ∈ F 𝑘 𝑞 of length 𝑘. Moreover, we take the fragments (𝑔1, . . . , 𝑔𝑛) from the set G = G1 × · · · × G𝑛, where G𝑖 = F 𝑚𝑖 𝑞 ∪ {⊥} or G𝑖 = {⊥}. Then 𝑚𝑖 ∈ N denotes the size of 𝑝𝑖’s fragment, and we … view at source ↗
Figure 2
Figure 2. Figure 2: Access tree 𝑇 of A Let 𝑇𝑖 be depth-one subtrees of 𝑇 for 𝑖 ∈ [3] with root labeled 𝑡𝑖 with 𝑟𝑖 children. We aim to construct an encoding matrix 𝑀 ∈ F 𝑘×𝑚 𝑞 for a sufficient large 𝑞 and a labeling function 𝜙 for A such that the nodes of each access set yield at least 𝑘 linearly independent columns of 𝑀. To achieve this, the matrix 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Access tree of Example 4. 7.1 Uniform assignment Let A be an 𝐿-level partitioned access structure with tree 𝑇 and partition 𝐵1, . . . , 𝐵𝛿. We propose here a fragments assignment, called uniform assignment, that is efficient to compute but not always optimal in terms of overhead. In more detail, the uniform assignment provides parameters (𝑚, 𝑘) for the base code such that the resulting monotone erasure cod… view at source ↗
Figure 4
Figure 4. Figure 4: Uniform assignment of base fragments. Example 4 demonstrates that in some cases it is beneficial to disregard nodes from P, i.e., to assign them zero base fragments. Based on this, in Section 7.2, we provide a method that builds an optimal assignment for any partitioned access structure. 7.2 Optimal fragment assignment algorithm We present here an algorithm that finds an optimal solution to LPP for any par… view at source ↗
Figure 5
Figure 5. Figure 5: Access tree 𝑇 of A △ 7.3 Analysis We now show that applying Algorithm 2 to any partitioned access structure A provides an optimal linear monotone erasure code for A. To do that, we first rephrase the problem we want to solve, namely, LPP, in a recursive way, which makes the proof easier. Recall that, given an access structure A = {𝐴1, . . . , 𝐴𝜔}, LPP is defined as follows: min 𝑦∈R𝑛 ∑︁ 𝑖∈ [𝑛] 𝑦𝑖 subject to… view at source ↗
read the original abstract

Erasure codes are a critical component in reliable storage systems today, and many blockchain systems use consensus protocols that involve erasure codes to reduce their communication cost. Existing erasure codes rely on a threshold failure assumption, but recent blockchain systems have departed from this simple model and use generalized failure assumptions. This paper introduces monotone erasure codes that respect arbitrary trust assumptions on a set of nodes. The paper first describes a method for constructing a monotone erasure code from any access structure given by a monotone Boolean formula. Next, the relevant notion of a linear monotone erasure code is introduced, which works on vectors over a finite field and where the encoding is a linear operation. We then focus on constructing linear monotone erasure codes: We give an efficient algorithm to construct linear monotone erasure codes for any access structure, and we show how to efficiently construct linear monotone erasure codes for the special case of partitioned access structures with minimal storage overhead. Last but not least, this work also shows how to use monotone erasure codes to obtain a communication-efficient, generalized version of the well-known asynchronous verifiable information dispersal (AVID) primitive, which is a key building block for developing efficient reliable broadcast and consensus protocols.

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

1 major / 1 minor

Summary. The paper introduces monotone erasure codes that respect arbitrary trust assumptions on a set of nodes, given by monotone Boolean formulas. It describes a method to construct such codes from any access structure, defines linear monotone erasure codes over finite fields, provides an efficient algorithm to construct linear versions for arbitrary access structures, gives a specialized efficient construction for partitioned access structures with minimal storage overhead, and shows how to apply monotone erasure codes to obtain a communication-efficient generalized version of asynchronous verifiable information dispersal (AVID).

Significance. If the constructions and algorithm are shown to be correct, the work could enable erasure coding under generalized (non-threshold) failure assumptions, which is relevant for blockchain and distributed consensus protocols that already use erasure codes to reduce communication. The explicit algorithm for arbitrary access structures and the AVID generalization are potential strengths, provided they include correctness arguments and field-size bounds.

major comments (1)
  1. [algorithm for any access structure] The algorithm for constructing linear monotone erasure codes for any access structure (described after the definition of linear monotone erasure codes): the recovery guarantee requires that every authorized set induces a submatrix of full column rank while unauthorized sets do not, yet the description relies on standard linear-algebra operations over a finite field without stating a field-size lower bound or a deterministic method (such as explicit Vandermonde or random selection from a sufficiently large field) to ensure the required independence relations for arbitrary formulas. This is load-bearing for the central claim that the algorithm produces valid encodings meeting the recovery condition.
minor comments (1)
  1. [Abstract] The abstract and introduction would benefit from a small concrete example (e.g., a 3-node access structure with its encoding matrix) to illustrate how the linear construction satisfies the monotone recovery property.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our manuscript. We address the major comment point by point below and outline the revisions we will make to strengthen the presentation of the algorithm for linear monotone erasure codes.

read point-by-point responses

  1. Referee: The algorithm for constructing linear monotone erasure codes for any access structure (described after the definition of linear monotone erasure codes): the recovery guarantee requires that every authorized set induces a submatrix of full column rank while unauthorized sets do not, yet the description relies on standard linear-algebra operations over a finite field without stating a field-size lower bound or a deterministic method (such as explicit Vandermonde or random selection from a sufficiently large field) to ensure the required independence relations for arbitrary formulas. This is load-bearing for the central claim that the algorithm produces valid encodings meeting the recovery condition.

    Authors: We agree with the referee that an explicit field-size bound and a clear method for ensuring the required linear independence are necessary to make the correctness of the algorithm fully rigorous. In the revised manuscript, we will add a dedicated paragraph immediately following the algorithm description that specifies the construction operates over a finite field F_q with q > n (where n is the number of nodes), chosen large enough to guarantee the existence of suitable coefficients. We will explain that the algorithm assigns random elements from F_q to the encoding matrix entries corresponding to each variable in the monotone Boolean formula, and that by the Schwartz-Zippel lemma (or a union bound over all minimal authorized and maximal unauthorized sets), the probability that any required rank condition fails is at most O(1/q). We will also note that a deterministic construction is possible by enumerating a sufficiently large field and selecting a Vandermonde-like matrix that satisfies the rank conditions for the given access structure, at the cost of a larger but still polynomial field size. A short proof sketch will be included showing why authorized sets yield full column rank and unauthorized sets do not, thereby supporting the central claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit constructive algorithm is self-contained.

full rationale

The paper presents a direct algorithmic construction of linear monotone erasure codes from an arbitrary monotone Boolean formula representing the access structure, together with a specialized efficient construction for partitioned access structures that minimizes storage overhead. These steps rely on standard linear-algebra operations over a finite field to enforce the required recovery properties for authorized sets and non-recovery for unauthorized sets. No load-bearing step reduces by the paper's own equations to a fitted parameter, a self-referential definition, or a chain of self-citations whose validity is presupposed by the present work. The central claims are therefore independent of any circular reduction and rest on the external correctness of the described linear encoding procedure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on standard linear algebra over finite fields and the definition of monotone access structures; no free parameters are introduced in the abstract, no new physical entities are postulated, and the only axioms are background results from coding theory and monotone Boolean functions.

axioms (2)
  • domain assumption Linear operations over a finite field produce a valid encoding whose recovery properties match the given monotone access structure.
    Invoked when defining linear monotone erasure codes and the efficient algorithm for any access structure.
  • domain assumption Monotone Boolean formulas correctly capture the authorized sets for recovery in the target trust model.
    Stated in the opening description of monotone erasure codes respecting arbitrary trust assumptions.
invented entities (1)
  • monotone erasure code no independent evidence
    purpose: Erasure code whose recovery threshold respects an arbitrary monotone access structure rather than a fixed threshold.
    The new primitive introduced in the paper; no independent evidence outside the construction is provided in the abstract.

pith-pipeline@v0.9.0 · 5743 in / 1568 out tokens · 44179 ms · 2026-05-22T04:46:40.349781+00:00 · methodology

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