Anonymous Shamir's Secret Sharing via Reed-Solomon Codes Against Permutations, Insertions, and Deletions

Roni Con

REVIEW 2 major objections 4 minor 35 references

Anonymous Shamir's Secret Sharing via Reed-Solomon Codes Against Permutations, Insertions, and Deletions

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Reed-Solomon codes that withstand permutation-plus-deletion attacks yield fully anonymous secret sharing with perfect reconstruction.

desk verdict Solid existence proof for fully anonymous ramp secret sharing via permutation-insdel-robust RS codes; the main theorem holds up, though the explicit construction section is sketchy. read the letter →

arxiv 2412.17003 v2 pith:YOFPLOJY submitted 2024-12-22 cs.IT cs.CRmath.IT

Roni Con This is my paper · ORCID

classification cs.ITcs.CRmath.IT MSC 94B0594B3594A62

keywords Reed-Solomoncodesinsertionsanddeletionsanonymoussecretsharingpermutation-insdeladversaryrampfullyVandermondematricesSchwarz-Zippellemma

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a bridge between error-correcting codes and anonymous cryptography: it argues that a Reed-Solomon code robust against an adversary who permutes a codeword and then inserts or deletes symbols can be turned into a secret-sharing scheme in which unauthorized shares leak neither the secret nor the identities of their holders. The main result claims that for any k,n with 2k-1 <= n, over a sufficiently large prime field there exist such codes, and hence a fully anonymous (k-1,2k-1,n) ramp secret-sharing scheme with perfect reconstruction. This matters because earlier anonymous schemes either sacrificed perfect reconstruction or only hid identities statistically, and the paper's construction gives the first gap-threshold schemes that are fully anonymous in the strongest sense. The proof is algebraic: it shows that the evaluation points can be chosen so that, for any two distinct low-degree polynomials, no two length-(2k-1) subsequences of their evaluations can coincide after an arbitrary permutation.

What carries the argument

The load-bearing object is the V-matrix $V_{I,J}(X)$, the $(2k-1)\times(2k-1)$ matrix whose $s$-th row is $(1,X_{I_s},\ldots,X_{I_s}^{k-1},X_{J_s},\ldots,X_{J_s}^{k-1})$ for two distinct-element sequences $I,J\in[n]^{2k-1}$. A collision of two codewords under permutation-plus-insdel is exactly a nonzero vector in the right kernel of $V_{I,J}(\alpha)$ that does not have the form $(0,f_1,\ldots,f_{k-1},-f_1,\ldots,-f_{k-1})$. The proof decomposes $(I,J)$ into minimal equal sub-pairs (maximal disjoint index blocks on which the two sequences contain the same set of symbols) and proves, by induction with a uniquely realized monomial, that $\det V_{I,J}(X)$ is a nonzero polynomial whenever there are at most $k$ such blocks. For pairs with $k+1$ blocks, a second family of matrices $A_{I_1,\ldots,I_k}(X)$ with full-rank conditions forces any kernel vector to come from a single polynomial. The Schwarz-Zippel lemma then shows that random evaluation points over a large field satisfy both algebraic conditions simultaneously.

What would settle it

For a small k (say k=3) and n>=2k-1, symbolically expand $\det V_{I,J}(X)$ for every pair of distinct-element sequences $I,J$ of length $2k-1$ whose decomposition has at most $k$ minimal equal sub-pairs; if any of these determinants is identically zero as a polynomial, the central existence argument collapses.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.18: for integers k,n with 2k-1 <= n and any prime q >= 200 * (\binom{n}{2k-1} \cdot (2k)!)^2, there exists an RS_{n,k}(\alpha_1,\ldots,\alpha_n) code that is robust against the (n-2k+1)-permutation-insdel adversary. Equivalently, no two distinct codewords can be mapped to the same string by first applying an arbitrary permutation and then at most n-2k+1 insertions and deletions. Corollary 1.19 then obtains a fully anonymous (k-1,2k-1,n) ramp secret-sharing scheme over F_q: any k-1 shares are uniform and independent, so they hide the secret and the participants' identities, while any 2k-1 shares, presented in any order, determine the secret exactly. The author also shows that earlier explicit Reed-Solomon constructions for insertions and deletions already give such codes in the two-dimensional case with q=O($n^{3}$) and in general dimension with q=$n^{{O(k^2((2k)!)^2)}}$.

Load-bearing premise

The load-bearing step is the claim that a certain determinant polynomial stays nonzero for every pair of index sequences that share at most k small matching blocks; if that claim ever fails, the random evaluation points that the existence proof needs cannot be guaranteed.

Editorial extensions

If this is right

For every k,n with 2k-1 <= n and q satisfying the theorem's bound, a fully anonymous (k-1,2k-1,n) ramp secret-sharing scheme exists with perfect reconstruction and shares of size O(k log n) bits for constant k.
The two-dimensional explicit construction yields a fully anonymous (1,3,n) ramp scheme over q=O(n^3), so even the smallest non-trivial gap is covered explicitly.
The gap between k-1 and 2k-1 is forced by the half-Singleton bound: a reconstruction threshold below 2k-1 would require the underlying linear code to correct more than n-2k+1 insdel errors, which is impossible.
The general explicit construction gives a deterministic fully anonymous scheme for all k,n with 2k<=n over q=n^{O(k^2((2k)!)^2)}, though the bit-length of shares is polynomial in n only for very slowly growing k.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The algebraic kernel condition is what really matters, so codes other than Reed-Solomon whose evaluation matrices satisfy the same rank property would also yield fully anonymous schemes; the connection seems transferable.
Because the half-Singleton bound rules out linear-code routes to the exact threshold, closing the gap to k would require either nonlinear codes or a fundamentally different secret-sharing construction.
A concrete next step would be to seek evaluation sets with algebraic structure that allows reconstructing from any 2k-1 unordered evaluations in polynomial time; the paper's brute-force reconstruction is the main practical bottleneck.
For non-constant k the field-size bound is dominated by ((2k)!)^2, so the existential and explicit schemes are mainly of theoretical interest; practical use would need a different trade-off.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this paper delivers what it promises. The main theorem, existence of RS codes robust against the permutation-insdel adversary at the half-Singleton bound, is proven carefully and the proof holds up. The corollary, fully anonymous (k-1, 2k-1, n) ramp secret sharing with perfect reconstruction, follows cleanly from the code robustness. That answers the open question from BGI+24 for ramp thresholds, which is a real step.

What's new: the permutation-insdel model for RS codes is a genuine extension of the insdel-only setting; the observation that permutation robustness is exactly what you need for anonymous reconstruction is simple but right. The algebraic core is Proposition 3.10, an induction showing the V-matrix determinant does not vanish when the pair has at most k minimal equal sub-pairs. I went through that induction and it works. The deletion step in Case 1 is the delicate point, but the minimal equal sub-pair structure plus Claim 3.6 keeps the invariant intact. The Schwarz-Zippel applications then give random evaluation points with high probability, and the prime-selection argument (find a prime in [200C,400C] not dividing some integer coefficient) is correct, if slightly indirect.

The soft spots are real but not load-bearing. First, the explicit construction in Section 5.2 is a sketch: it says the proof of CST23 goes through with Proposition 3.11 swapped in, without re-deriving the details. For a reader who wants to actually use those evaluation points, that's frustrating. Second, Corollary 1.19 silently needs the alpha_i to be nonzero; Theorem 1.18 doesn't enforce that, but it's a trivial union-bound fix since you can avoid zero in the random draw. Third, reconstruction is brute-force; the author acknowledges this. The field size in the existential result is large but not surprising for this kind of argument.

Citation pattern is fine. The paper leans on CST23/CST24, but uses them as tools and proves the new algebraic conditions independently. No circularity, no fitted parameters. The main limitation is conceptual: the factor-of-two gap between privacy and reconstruction thresholds is likely inherent for the RS-based approach, and the author says so.

Who should read this: anyone working on insdel-correcting codes or anonymous secret sharing. It deserves a serious referee; the central claim is correct and the writeup is mostly clear, though Section 5.2 needs tightening. I'd send it to an editor for full review.

Referee Report

2 major / 4 minor

Summary. This paper studies Reed–Solomon codes under an adversarial model that first permutes the codeword and then performs insertions and deletions, and applies these codes to fully anonymous ramp secret sharing. The main technical work is an algebraic condition on evaluation points (Section 2), a determinant-nonvanishing lemma for V-matrices whose index pairs have at most k minimal equal sub-pairs (Proposition 3.10), and two Schwarz–Zippel arguments (Propositions 3.11 and 3.14) that yield evaluation points satisfying the required rank/determinant conditions. From this, the author derives Theorem 1.18, the existence of [n,k] RS codes robust against the (n-2k+1)-permutation-insdel adversary over fields of size q >= 200 * (binom(n,2k-1) * (2k)!)^2, and Corollary 1.19, a fully anonymous (k-1,2k-1,n) ramp secret-sharing scheme with perfect reconstruction. The paper also claims explicit schemes using the evaluation points of [CST24] and [CST23].

Significance. If the main theorem and corollary are established, this is a meaningful step: it gives the first gap-threshold secret-sharing schemes satisfying the strongest anonymity notion (uniform, independent shares for unauthorized sets) together with perfect anonymous reconstruction, thereby addressing an open question from [BGI+24] in the ramp regime. It also extends the insdel analysis of RS codes to a permutation-then-insdel adversary. The core determinant lemma, Proposition 3.10, is carefully argued by induction with a unique-monomial technique, and the algebraic conditions are explicit and falsifiable. The two-dimensional explicit construction over O(n^3) fields is a useful concrete instantiation. However, the present proof contains field-size gaps that need repair before the claims are fully established.

major comments (2)

[Section 4, proof of Theorem 1.18, with Proposition 3.14] The field-size inequality used to apply Propositions 3.11 and 3.14 is false for all parameters stated in Theorem 1.18. The proof claims q >= 200 * (binom(n,2k-1) * (2k)!)^2 >= 200 * max{ k^2 * binom(n,2k-1)^2 * (2k-1)!, k^3 * (en)^(2k-1) }. For n = 3 and k = 2 the left side is 115,200, while the right side (using the k^3 term that the proof of Proposition 3.14 actually derives) is 200 * max{24, 8 * (3e)^3} = 867,720. Hence the field size guaranteed by Theorem 1.18 does not satisfy the hypotheses of the propositions as stated, and the Schwarz-Zippel step is unjustified for these parameters. Moreover, Proposition 3.14's statement requires only q >= 200 * k^2 * (en)^(2k-1), but its proof establishes the stronger condition q >= 200 * k^3 * (en)^(2k-1); these two bounds are inconsistent. This is repairable by a sharper count of the determinants in Proposition 3.14 or by treating small n, k separately, but as written the main existence proof is incomplete.
[Section 1.3, Corollary 1.19] The proof begins 'Let alpha_1, ..., alpha_n be nonzero elements such that RS_{n,k}(alpha_1,...,alpha_n) is robust...', but Theorem 1.18 only guarantees distinct evaluation points; it does not guarantee that any of them is nonzero. If some alpha_i = 0, then the share of participant i equals the secret f(0) = s, so for k > 1 a single share is not uniform over F_q and the full-anonymity condition in Definition 1.13 fails. The gap is easily fixed by adding the n events alpha_i = 0 to the union bound in Proposition 3.11, but the current statement asserts nonzeroness without proof.

minor comments (4)

[Claim 3.6] The first sentence of the proof says 'Assume for contradiction that A1 ∩ A2 = ∅'; the intended contradiction hypothesis is A1 ∩ A2 ≠ ∅. As written the sentence is nonsensical.
[Proposition 3.10, cases 2 and 3] The displayed unique monomial is written as X_{I_i}^{2k-2} m, but in cases 2 and 3 the variables I_i and J_j are distinct, so the monomial should be X_{I_i}^{k-1} X_{J_j}^{k-1} m. The preceding text is consistent with the corrected expression.
[Section 5.1, Theorem 5.1] The determinant computation for the case in which I and J agree on two coordinates is only sketched; the formula det(V_{I,J}) = (alpha_{J3} - alpha_{I3})(alpha_{J2} - alpha_{J1}) should be derived explicitly, since the signed expression depends on the choice of the two agreeing coordinates.
[Remark 1.20] The remark correctly explains why the reconstruction threshold of this RS-based construction cannot go below 2k-1, but the opening phrase 'the gap ... is necessary and cannot be improved' could be misread as a general impossibility for all fully anonymous ramp schemes. The restriction to this construction should be stated more prominently.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main existence theorem is self-contained; self-citations are contextual, not load-bearing.

full rationale

Theorem 1.18 is proved from in-paper algebraic lemmas. Proposition 3.10 gives an induction showing det(V_{I,J}(X)) is a nonzero polynomial with a uniquely obtained ±1 monomial whenever (I,J) has at most k minimal equal sub-pairs, and Proposition 3.13 proves the A-matrix determinants are nonzero; Propositions 3.11 and 3.14 then apply Schwarz–Zippel after a coefficient-primorial argument. None of these steps fits parameters to data or renames a prior prediction. The secret-sharing corollary (Corollary 1.19) follows directly: share uniformity is the standard random-polynomial calculation Pr = q^{−ℓ} for ℓ ≤ k−1, and perfect anonymous reconstruction is exactly the injectivity condition of Claim 1.11, which is equivalent to the permutation-insdel robustness definition. Self-citations to [CST23]/[CST24] appear in the framework (Section 2.1) and in the explicit constructions (Section 5), and Proposition 5.5 borrows Claim 5.6 from [CST23] with an omitted proof; but the main existence result does not depend on those prior theorems, and the cited lemmas are external published results rather than the target claim. The paper itself flags the real limitations—brute-force reconstruction (Remark 1.14, Section 1.4) and the existential, non-explicit nature of the main theorem—which are correctness/efficiency issues, not circularity. No equation in the derivation reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof rests on standard algebraic tools and prior theorems from the literature. No free parameters are fitted to data; the only parameters are the explicit field-size bounds q = O((n choose 2k-1)^2 ((2k)!)^2) and the constructions from CST23 and CST24. No new physical or mathematical entities are introduced.

assumptions (5)

standard math Schwarz-Zippel-Demillo-Lipton lemma
Used in Propositions 3.11 and 3.14 to show a random evaluation vector satisfies the algebraic conditions with high probability.
domain assumption Half-Singleton bound for linear insdel codes (Theorem 1.16, CGHL23)
Used in Remark 1.20 to argue that the gap between k-1 and 2k-1 is necessary.
standard math Mason-Stothers theorem (Theorem 5.4, VW03)
Used in the proof of Proposition 5.5 to show the determinant of the A-matrix is nonzero for the explicit construction.
domain assumption Framework and results of CST23 (Proposition 2.2, Construction 27, Claims 28 and 29)
The paper builds on the V-matrix framework and the explicit construction; several proof details are deferred to CST23.
standard math LCS characterization of edit distance (Lemma 1.7)
Used in Claim 1.11 to relate permutation-insdel robustness to distinct-element sequence collisions.

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Cite this review

Pith. "Pith review of Anonymous Shamir's Secret Sharing via Reed-Solomon Codes Against Permutations, Insertions, and Deletions." pith.science (2026). https://pith.science/paper/YOFPLOJY

@misc{pith2026241217003,
  author       = {Pith},
  title        = {Pith review of: Anonymous Shamir's Secret Sharing via Reed-Solomon Codes Against Permutations, Insertions, and Deletions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YOFPLOJY}},
  note         = {Machine review of arXiv:2412.17003}
}

abstract

In this work, we study the performance of Reed-Solomon codes against an adversary that first permutes the symbols of the codeword and then performs insertions and deletions. This adversarial model is motivated by the recent interest in fully anonymous secret-sharing schemes [EBG+24],[BGI+24]. A fully anonymous secret-sharing scheme has two key properties: (1) the identities of the participants are not revealed before the secret is reconstructed, and (2) the shares of any unauthorized set of participants are uniform and independent. In particular, the shares of any unauthorized subset reveal no information about the identity of the participants who hold them. In this work, we first make the following observation: Reed-Solomon codes that are robust against an adversary that permutes the codeword and then deletes symbols from the permuted codeword can be used to construct ramp threshold secret-sharing schemes that are fully anonymous. Then, we show that over large enough fields of size, there are $[n,k]$ Reed-Solomon codes that are robust against an adversary that arbitrary permutes the codeword and then performs $n-2k+1$ insertions and deletions to the permuted codeword. This implies the existence of a $(k-1, 2k-1, n)$ ramp secret sharing scheme that is fully anonymous. That is, any $k-1$ shares reveal nothing about the secret, and, moreover, this set of shares reveals no information about the identities of the players who hold them. On the other hand, any $2k-1$ shares can reconstruct the secret without revealing their identities. We also provide explicit constructions of such schemes based on previous works on Reed-Solomon codes correcting insertions and deletions. The constructions in this paper give the first gap threshold secret-sharing schemes that satisfy the strongest notion of anonymity together with perfect reconstruction.

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