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arxiv: 2605.14015 · v1 · submitted 2026-05-13 · 💻 cs.DC

Recognition: no theorem link

Distributed Statistical Zero-Knowledge Proofs via Sumcheck

Benjamin Jauregui , Masayuki Miyamoto
Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:41 UTC · model grok-4.3

classification 💻 cs.DC
keywords distributed zero-knowledge proofssumcheck protocolstatistical zero-knowledgeinteractive proofsgraph non-colorabilitysubgraph countingdistributed verification
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The pith

A distributed Sumcheck protocol verifies polynomial sums with statistical zero-knowledge in linear rounds.

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

The paper lifts the classical Sumcheck protocol to a distributed setting so that multiple nodes can jointly check whether the sum of a multivariate polynomial equals a claimed value. Each node receives only short messages and a view that can be perfectly simulated without knowing the polynomial, yielding statistical zero-knowledge. The resulting protocol uses a number of rounds equal to the number of variables and messages of size logarithmic in the field size. It directly yields the first nontrivial distributed interactive proofs for non-k-colorability and improved protocols for counting small subgraphs, both with the added statistical privacy property.

Core claim

Given oracle access to a polynomial F over a finite field with N variables, the protocol verifies claims of the form sum_{x in F} F(x)=a using O(N) rounds of O(log |F|)-bit messages, while achieving statistical zero-knowledge and small soundness error.

What carries the argument

The lifted distributed Sumcheck protocol, which coordinates random challenges and responses across nodes while preserving the zero-knowledge simulation property of the original interactive proof.

If this is right

  • Non-k-colorability admits an O(n)-round distributed statistical zero-knowledge proof with O(log^{1+o(1)} n)-bit messages for any constant k.
  • Subgraph counting admits an O(k log n)-round, O(k log n)-bit distributed statistical zero-knowledge proof for counting copies of any k-node pattern.
  • Round compression below O(n) for Sumcheck is problem-dependent; in particular, non-3-colorability on constant-degree graphs requires Omega(n / log n) rounds under polynomial-time local computation.

Where Pith is reading between the lines

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

  • The same lifting technique could be applied to other problems reducible to polynomial summation, such as verifying matrix products or circuit evaluations across a network.
  • Statistical zero-knowledge in the distributed model may allow nodes to reach collective decisions without exposing private local data even to one another.
  • The problem-dependent lower bound suggests that future work should focus on tailored round-reduction strategies rather than seeking a general compression method for all Sumcheck instances.

Load-bearing premise

The protocol assumes oracle access to the polynomial F is available to the distributed nodes and that local computation is polynomial-time.

What would settle it

A concrete polynomial sum claim together with a transcript that an honest verifier accepts with high probability while the claim is false, or a view that cannot be simulated within negligible statistical distance, would refute the soundness or zero-knowledge claim.

Figures

Figures reproduced from arXiv: 2605.14015 by Benjamin Jauregui, Masayuki Miyamoto.

Figure 1
Figure 1. Figure 1: Clause gadget for a clause C = (xi ∨ xj ∨ xk). In any valid 3-coloring, at least one of v(xi), v(xj ), v(xk) has the same color as of vT . Note that all of v(xi), v(xj ), v(xk) are connected to vB (but the corresponding edges are omitted in the figure for simplicity). Reducing the maximum degree of the graph. In the above construction, the number of nodes is O(n + m) where m is the number of clauses in ϕ. … view at source ↗
Figure 2
Figure 2. Figure 2: A gadget for a tree edge (u, v). In any valid 3-coloring, u and v must have the same color. the total number of nodes to O(n log n) while ensuring maximum degree O(1). For a given 3-SAT formula ϕ, we construct Gϕ in poly(n) time. Assume that Non-3- Colorability can be certified by an m-round dAM protocol with message size b on constant￾degree, n-node and m-edge graphs. Recall that Gϕ is a O(n log n)-node O… view at source ↗
read the original abstract

We study distributed zero-knowledge proofs, introduced by Bick, Kol, and Oshman (SODA 2022). While distributed interactive proofs have advanced rapidly, general-purpose techniques for distributed zero-knowledge remain limited and mostly problem-specific. We address this gap by introducing distributed statistical zero-knowledge, requiring that each node's view be simulatable within negligible statistical distance, and by lifting the classical Sumcheck protocol (Lund, Fortnow, Karloff, and Nisan, FOCS 1990) into a modular primitive for distributed zero-knowledge proofs. Our main contribution is a distributed zero-knowledge implementation of Sumcheck. Given oracle access to a polynomial F over a finite field $\mathbb{F}$ with N variables, we design a protocol verifying claims of the form $\sum_{x\in\mathbb{F}} F(x)=a$ using $O(N)$ rounds of $O(\log |\mathbb{F}|)$-bit messages, while achieving statistical zero-knowledge and small soundness error. We apply this primitive to two problems. For non-k-colorability, we obtain an $O(n)$-round distributed statistical zero-knowledge proof deciding whether a graph is not k-colorable, for any constant k, using $O(log^{1+o(1)} n)$-bit messages. This is the first nontrivial distributed interactive proof for this problem, even without zero-knowledge guarantees. For Subgraph Counting, we obtain an $O(k \log n)$-round, $O(k \log n)$-bit distributed statistical zero-knowledge proof for counting copies of a given k-node pattern, improving previous distributed interactive proofs while additionally providing statistical zero-knowledge. Finally, we show that additional round compression of Sumcheck is problem-dependent: for non-3-colorability on constant-degree graphs, we prove a lower bound excluding $o(n/\log n)$ rounds under polynomial-time local computation.

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 manuscript introduces distributed statistical zero-knowledge proofs by lifting the classical Sumcheck protocol (Lund et al., FOCS 1990) to the distributed interactive proof setting of Bick, Kol, and Oshman (SODA 2022). Given oracle access to a multivariate polynomial F over a finite field F with N variables, it constructs a protocol verifying sum_{x in F} F(x) = a in O(N) rounds using O(log |F|)-bit messages per round, while achieving statistical zero-knowledge (each node's view simulatable within negligible statistical distance) and small soundness error. The primitive is applied to obtain an O(n)-round, O(log^{1+o(1)} n)-bit distributed statistical ZK proof for non-k-colorability (any constant k) and an O(k log n)-round, O(k log n)-bit protocol for counting k-node subgraphs; a lower bound is also shown excluding o(n/log n) rounds for non-3-colorability on constant-degree graphs under polynomial-time local computation.

Significance. If the central claims hold, the work supplies a clean modular primitive for distributed statistical zero-knowledge that fills a gap left by problem-specific constructions. It yields the first nontrivial distributed interactive proof for non-k-colorability (even without ZK) and improves prior subgraph-counting protocols while adding statistical ZK. The explicit lifting of Sumcheck, the concrete round/message bounds, and the round-compression lower bound are all strengths that would be of interest to the distributed computing and proof-systems communities.

major comments (2)
  1. [§3] §3 (distributed Sumcheck protocol): the soundness analysis must explicitly track how the per-round soundness error of the classical Sumcheck protocol compounds across the distributed nodes and the O(N) rounds; the final soundness bound should be stated as a concrete function of N and |F| rather than left as 'small'.
  2. [§5.1] §5.1 (non-k-colorability application): the reduction from graph non-colorability to a Sumcheck instance must specify the exact degree of the resulting polynomial and the field size chosen; without these parameters it is impossible to verify that the overall soundness error remains 1/poly(n) after the O(n) rounds.
minor comments (3)
  1. [Abstract] Abstract: the phrase 'small soundness error' should be replaced by a concrete bound (e.g., 1/poly(n) or negligible in n).
  2. [§4] §4 (statistical ZK simulation): the simulator description should explicitly state its running time and confirm that the statistical distance is negligible in the security parameter, not merely 'small'.
  3. [Throughout] Notation: the finite field is denoted both F and F in different places; adopt a single consistent symbol throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [§3] §3 (distributed Sumcheck protocol): the soundness analysis must explicitly track how the per-round soundness error of the classical Sumcheck protocol compounds across the distributed nodes and the O(N) rounds; the final soundness bound should be stated as a concrete function of N and |F| rather than left as 'small'.

    Authors: We agree that the soundness analysis requires an explicit bound. The distributed Sumcheck protocol inherits the per-round soundness error of the classical protocol, which is at most d/|F| where d is the total degree of the polynomial. Over O(N) rounds the errors accumulate additively, yielding a total soundness error of at most O(N d / |F|). In the revised manuscript we will state the concrete bound O(N / |F|) (by choosing |F| sufficiently large relative to N and d) and confirm that it remains 1/poly(N) under the field-size choices used in the applications. revision: yes

  2. Referee: [§5.1] §5.1 (non-k-colorability application): the reduction from graph non-colorability to a Sumcheck instance must specify the exact degree of the resulting polynomial and the field size chosen; without these parameters it is impossible to verify that the overall soundness error remains 1/poly(n) after the O(n) rounds.

    Authors: We thank the referee for highlighting this omission. The reduction encodes the k-colorability constraints via a low-degree polynomial whose total degree is O(1) (specifically degree 2 for the pairwise difference terms on edges). We select a field F with |F| = n^{O(1)} and |F| > n^2. With these parameters the per-round error is O(1/|F|) and the accumulated error after O(n) rounds is O(n/|F|) = 1/poly(n). The revised Section 5.1 will explicitly state the degree and field-size choice together with the resulting soundness bound. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central contribution is a modular lifting of the external classical Sumcheck protocol (Lund et al., FOCS 1990) to a distributed statistical zero-knowledge setting under oracle access to the polynomial. No equations or claims reduce new results to fitted parameters, self-citations, or ansatzes from the authors' prior work. The applications to non-k-colorability and subgraph counting are derived directly from this primitive, and the round lower bound is proven independently. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the correctness of the classical Sumcheck protocol and standard properties of finite fields and interactive proofs; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Correctness and zero-knowledge properties of the classical Sumcheck protocol (Lund et al., FOCS 1990)
    The distributed version is obtained by lifting this protocol.
  • domain assumption Existence of finite fields with efficient arithmetic and oracle access model for polynomials
    Required for the sumcheck verification claims.

pith-pipeline@v0.9.0 · 5633 in / 1297 out tokens · 27623 ms · 2026-05-15T02:41:29.874032+00:00 · methodology

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