Graph Coloring-based Relativistic ZKPs

• The paper demonstrates a novel graph coloring-based RZKP framework where a prover reveals only local consistency of a proper 3-coloring while the verifier enforces security via spatial separation.
• It implements a commit–challenge–response protocol using both linear commitments and subset relativistic bit commitments, enabling efficient local verification of NP witnesses.
• Experimental setups show linear scalability and robustness against classical, quantum, and entangled adversaries, paving the way for practical identity verification applications.

Searching arXiv for recent and foundational papers on graph-coloring-based relativistic zero-knowledge proofs. Searching arXiv for "graph coloring relativistic zero-knowledge proof". Graph coloring-based relativistic zero-knowledge proofs are multi-prover proof systems in which a prover demonstrates knowledge of a proper graph coloring—typically a proper $3$-coloring—while revealing no witness information beyond local consistency facts, and while deriving soundness from relativistic non-communication rather than computational hardness assumptions. In this line of work, the canonical NP witness is a coloring $c$ of a graph $G$ such that adjacent vertices receive different colors; the verifier checks only challenged local constraints, while space-like separation prevents separated prover agents from coordinating after the challenge. The area has developed from practical relativistic proofs for $\mathsf{3COL}$ with minimal communication (Crépeau et al., 2019), to experimental protocols with unconditional quantum security based on subset relativistic bit commitments and nonlocal games (Weng et al., 30 Jan 2025), and to identity-verification systems that emphasize deployability, message authentication, distance bounding, and security against entangled malicious provers (Ma et al., 18 Jul 2025).

1. NP witness structure and the role of graph coloring

The central NP statement is graph $3$-colorability. One formulation used in the literature is

$$\mathsf{3COL}=\{G=(V,E)\;:\;\exists\, c:V\to \mathbb{F}_3 \text{ such that } (i,j)\in E \Rightarrow c_i\neq c_j\},$$

with colors represented as trits in $\mathbb{F}_3=\{0,1,2\}$ (Crépeau et al., 2019). The same witness relation is also written as a proper $3$-coloring $c:V(G)\to \{1,2,3\}$ satisfying

$$c(u)\neq c(v)\qquad \text{for every edge } (u,v)\in E$$

(Ma et al., 18 Jul 2025).

Graph coloring is used because it is a natural NP witness relation: a valid witness is efficiently checkable by inspecting only local edge constraints, and the verifier can test correctness by asking to reveal information associated with one edge rather than the full witness. This locality makes graph coloring particularly compatible with zero-knowledge: the prover can commit to a randomly permuted coloring and reveal only the two endpoint colors on a challenged edge, so the verifier learns a local consistency statement without learning the global coloring (Ma et al., 18 Jul 2025).

The literature explicitly situates these protocols in the lineage of Blum’s graph-coloring proof idea and the relativistic NP-proof literature, including practical relativistic zero-knowledge for NP. This establishes graph coloring not merely as an illustrative language, but as the concrete combinatorial basis through which relativistic zero-knowledge for NP is engineered and experimentally realized (Ma et al., 18 Jul 2025).

2. Relativistic architecture and protocol families

The basic architecture is a commit–challenge–response protocol executed by spatially separated agents. In the two-prover setting, the prover is split into two separated agents, and the verifier side is likewise split into two stations so that the challenge and response timings enforce a non-communication window. One prover-side agent commits to masked coloring information, the verifier challenges an edge or related local query, and the separated agents must answer from local information only. The verifier then checks that the challenged adjacent vertices have distinct colors (Ma et al., 18 Jul 2025).

Two main graph-coloring-based RZKP design lines appear in the recent literature. One uses linear commitments over $c$0 with “implicit unveiling”; another uses subset relativistic bit commitments strengthened by a quantum nonlocality game. A later identity-verification line retains the graph-coloring witness relation while emphasizing engineering refinements and entanglement-robust variants.

Paper	Configuration and core primitive	Stated distinguishing properties
"Practical Relativistic Zero-Knowledge for NP" (Crépeau et al., 2019)	Two local provers or three entangled provers; linear commitment $c$1	One-round, single-shot, non-adaptive; provers reply two trits each
"Experimental relativistic zero-knowledge proofs with unconditional security" (Weng et al., 30 Jan 2025)	Two verifier-prover pairs; subset relativistic bit commitments plus $c$2	Unconditionally secure graph three-coloring protocol; linear relation between rounds and number of edges
"Quantum-Safe Identity Verification using Relativistic Zero-Knowledge Proof Systems" (Ma et al., 18 Jul 2025)	Two-prover identity-verification protocol, modified two-prover protocol, and three-prover extension	Distance constraint relaxed from $c$3 m to $c$4 m; stability and scalability enhancements; security against entangled malicious provers

In the linear-commitment line, each vertex $c$5 has a color $c$6 and a random mask $c$7, and the commitment takes the form

$c$8

for verifier-chosen randomness $c$9 (Crépeau et al., 2019). If the same vertex is queried twice with different randomizers, the color can be recovered from the two linear equations; over $G$0,

$G$1

and more generally

$G$2

(Crépeau et al., 2019). This is the paper’s “implicit unveiling” mechanism.

In the unconditional-security line, the commitment is algebraic but parameterized over a larger finite field. The prover prepares a random permutation of the coloring and random values $G$3 with $G$4; after receiving random nonzero field elements $G$5, prover $G$6 returns

$G$7

where $G$8 is the color of vertex $G$9. A second verifier then chooses a random edge $\mathsf{3COL}$0, and the second prover reveals the masks corresponding to that edge (Weng et al., 30 Jan 2025).

3. Security properties and adversary models

Completeness is straightforward across the graph-coloring RZKP literature. If the provers know a valid coloring and answer consistently, the verifier’s local edge checks always succeed. The practical relativistic NP protocols state perfect completeness, and the unconditional-security experimental protocol likewise states perfect completeness (Crépeau et al., 2019, Weng et al., 30 Jan 2025).

Soundness is physically enforced by the relativistic assumption that separated prover agents are space-like separated during the critical response window. In the identity-verification formulation, this is the standard relativistic cryptographic assumption: the provers cannot communicate faster than light between challenge and response, so they cannot adaptively fabricate a globally consistent answer after learning the challenge (Ma et al., 18 Jul 2025). In the experimental unconditional-security implementation, the separation threshold is explicitly

$\mathsf{3COL}$1

where $\mathsf{3COL}$2 is the distance between provers and $\mathsf{3COL}$3 is the speed of light (Weng et al., 30 Jan 2025).

The literature gives different quantitative soundness claims for different protocols and threat models. For the two-prover perfect-ZK protocol against classical or local provers,

$\mathsf{3COL}$4

(Crépeau et al., 2019). For the three-prover variant designed to handle entangled provers,

$\mathsf{3COL}$5

(Crépeau et al., 2019). In the unconditional-security experimental protocol, the commitment-size condition is

$\mathsf{3COL}$6

and for soundness $\mathsf{3COL}$7 the protocol uses

$\mathsf{3COL}$8

rounds, yielding linear scaling in the number of edges (Weng et al., 30 Jan 2025). In the identity-verification line, a modified two-prover protocol aimed at long-term security against entangled malicious provers is analyzed with the explicit upper bound

$\mathsf{3COL}$9

(Ma et al., 18 Jul 2025).

Zero-knowledge is obtained by restricting what the verifier learns from each challenge. In the graph-coloring framework, the verifier sees only a local consistency check and only after a random permutation of color labels, so the transcript does not identify the secret witness (Ma et al., 18 Jul 2025). The three-prover protocol in the practical NP line proves a particularly strong structural statement: the verifier can learn only one isolated vertex color, the colors of an edge, or the colors of a triangle; when two vertices are revealed they are adjacent, and when three are revealed they form a triangle, so the revealed colors are uniformly random subject only to the necessary distinctness constraints. The paper gives a classical black-box simulator and concludes

$3$0

hence perfect zero-knowledge against quantum verifiers (Crépeau et al., 2019). The unconditional-security experimental protocol likewise claims quantum zero-knowledge without quantum rewinding, with security resting on perfectly hiding commitments, relativistic causality, non-signaling, and quantum nonlocality bounds (Weng et al., 30 Jan 2025).

4. Entanglement, nonlocality, and prover multiplicity

A defining issue for graph-coloring-based RZKPs is that relativistic non-communication alone does not settle the behavior of entangled adversaries. The practical relativistic NP paper therefore distinguishes local or classical provers from entangled provers and uses a three-prover modification, together with a result of Kempe–Kobayashi–Matsumoto–Toner–Vidick, to transfer a classical soundness gap in a symmetric one-round two-prover game into a nontrivial bound on the entangled value of a three-prover game (Crépeau et al., 2019). This establishes a clean architectural point: a third prover is introduced not for convenience, but to alter the adversarial model in a way that supports entangled-soundness analysis.

The unconditional-security experimental paper takes a different route. It states that relativistic bit commitments are not automatically secure against quantumly correlated cheating strategies and addresses this by integrating a non-local CHSH game, specifically an $3$1-fold parallelized version,

$3$2

to certify the sum-binding property of the subset relativistic bit commitment even against quantumly entangled provers (Weng et al., 30 Jan 2025). In this design, special relativity prevents communication in time, while quantum nonlocality analysis limits the extent to which entanglement can substitute for communication.

The identity-verification paper makes entanglement-resilience a central long-term objective. It states both a modified two-prover protocol with a soundness upper bound against entangled malicious provers and a three-prover configuration demonstrating security of such relativistic protocols against entangled malicious provers (Ma et al., 18 Jul 2025). This suggests a stable division within the field: some protocols retain the two-prover geometry but modify the internal checks to constrain entangled cheating, while others change the prover multiplicity itself.

A recurring misconception is that all graph-coloring-based relativistic protocols inherit the same quantum-security guarantees. The literature does not support that view. The stated soundness guarantees are protocol-specific and depend on whether the model assumes local provers, entangled provers, subset relativistic commitments with nonlocal-game certification, or an explicitly modified multi-prover architecture (Crépeau et al., 2019, Weng et al., 30 Jan 2025, Ma et al., 18 Jul 2025).

5. Experimental realization and engineering trajectory

The engineering trajectory is explicit. The practical relativistic NP paper emphasizes one-round protocols with minimal equipment: each run transmits two vertex identifiers per prover, two trits from verifier to prover, and two trits back from prover to verifier. It states that the provers “only require them to reply two trits each,” and argues that the required minimal separation grows only logarithmically with $3$3 for fixed communication speed, rather than quadratically as in earlier Hamiltonian-cycle-based relativistic protocols (Crépeau et al., 2019).

The experimental unconditional-security paper provides a detailed physical implementation. The setup uses two spatially separated verifier-prover pairs, FPGAs on both prover sides, computers and FPGA cards on verifier sides, and GPS-based synchronization via UTC. The two sites are about

$3$4

apart, corresponding to

$3$5

and the GPS time synchronization server has absolute accuracy within

$3$6

(Weng et al., 30 Jan 2025). The measured timing differences satisfy the relativistic conditions

$3$7

with reported means $3$8 ns and $3$9 ns, and maxima $$\mathsf{3COL}=\{G=(V,E)\;:\;\exists\, c:V\to \mathbb{F}_3 \text{ such that } (i,j)\in E \Rightarrow c_i\neq c_j\},$$0 ns and $$\mathsf{3COL}=\{G=(V,E)\;:\;\exists\, c:V\to \mathbb{F}_3 \text{ such that } (i,j)\in E \Rightarrow c_i\neq c_j\},$$1 ns respectively (Weng et al., 30 Jan 2025). The prover commitment computation finishes within one FPGA clock cycle,

$$\mathsf{3COL}=\{G=(V,E)\;:\;\exists\, c:V\to \mathbb{F}_3 \text{ such that } (i,j)\in E \Rightarrow c_i\neq c_j\},$$2

at $$\mathsf{3COL}=\{G=(V,E)\;:\;\exists\, c:V\to \mathbb{F}_3 \text{ such that } (i,j)\in E \Rightarrow c_i\neq c_j\},$$3 MHz, while communication delay dominates at about $$\mathsf{3COL}=\{G=(V,E)\;:\;\exists\, c:V\to \mathbb{F}_3 \text{ such that } (i,j)\in E \Rightarrow c_i\neq c_j\},$$4 ns (Weng et al., 30 Jan 2025).

The same paper reports the principal scalability claim of that protocol family: a linear relationship between interactive rounds and the number of edges, compared with prior quantum-secure relativistic ZKPs said to scale as

$$\mathsf{3COL}=\{G=(V,E)\;:\;\exists\, c:V\to \mathbb{F}_3 \text{ such that } (i,j)\in E \Rightarrow c_i\neq c_j\},$$5

For a graph with $$\mathsf{3COL}=\{G=(V,E)\;:\;\exists\, c:V\to \mathbb{F}_3 \text{ such that } (i,j)\in E \Rightarrow c_i\neq c_j\},$$6 and $$\mathsf{3COL}=\{G=(V,E)\;:\;\exists\, c:V\to \mathbb{F}_3 \text{ such that } (i,j)\in E \Rightarrow c_i\neq c_j\},$$7, the reported storage requirement is $$\mathsf{3COL}=\{G=(V,E)\;:\;\exists\, c:V\to \mathbb{F}_3 \text{ such that } (i,j)\in E \Rightarrow c_i\neq c_j\},$$8 MB for the new protocol versus about $$\mathsf{3COL}=\{G=(V,E)\;:\;\exists\, c:V\to \mathbb{F}_3 \text{ such that } (i,j)\in E \Rightarrow c_i\neq c_j\},$$9 GB for the earlier quantum-secure relativistic protocol at the same soundness target $\mathbb{F}_3=\{0,1,2\}$0, which the paper describes as a reduction by thirteen orders of magnitude (Weng et al., 30 Jan 2025). With a $\mathbb{F}_3=\{0,1,2\}$1 trigger interval between rounds, the protocol completes in about $\mathbb{F}_3=\{0,1,2\}$2 seconds; under the same interval, the earlier quantum-secure protocol would take about $\mathbb{F}_3=\{0,1,2\}$3 years (Weng et al., 30 Jan 2025).

The identity-verification paper continues this engineering trend by relaxing the relativistic distance constraint from $\mathbb{F}_3=\{0,1,2\}$4 meters to $\mathbb{F}_3=\{0,1,2\}$5 meters and by emphasizing stability and scalability improvements, with dedicated sections on hardware, data processing and transmission, message authentication, and distance bounding (Ma et al., 18 Jul 2025). The paper presents these refinements as relevant for near-term use cases rather than as purely theoretical optimizations.

6. Identity verification, related methodologies, and scope boundaries

Identity verification is the application context that currently organizes much of the recent graph-coloring-based RZKP work. The problem is framed as confirming a claimed identity in sectors such as finance, healthcare, and online services, while avoiding credential theft through phishing or skimming attacks. In this setting, a secret witness—a valid graph coloring—functions as the credential, and the prover demonstrates knowledge of that witness without disclosing it (Ma et al., 18 Jul 2025). The overall architecture includes message authentication and distance bounding, underscoring that the system is intended as an experimentally realized authentication protocol rather than a purely abstract proof system (Ma et al., 18 Jul 2025).

At the same time, the boundaries of the topic are important. Distributed statistical zero-knowledge proofs for graph non-colorability via Sumcheck are graph-coloring-related, but they are not relativistic protocols. That literature defines per-node statistical zero-knowledge, proves

$\mathbb{F}_3=\{0,1,2\}$6

and uses algebraic arithmetization rather than witness-hiding under spacetime constraints (Jauregui et al., 13 May 2026). It is therefore conceptually adjacent but methodologically distinct.

Similarly, the graph-homomorphism and synchronous-game literature shows that a general synchronous non-local game can be reduced to a $\mathbb{F}_3=\{0,1,2\}$7-coloring game on a graph of size at most

$\mathbb{F}_3=\{0,1,2\}$8

and proves that transformed coloring strategies can be made perfect zero knowledge for an honest verifier in the tracial models $\mathbb{F}_3=\{0,1,2\}$9 (Harris, 2023). This establishes a deep connection between graph-coloring games, quantum graph parameters, and zero-knowledge answer distributions, but it is not itself a relativistic proof system.

Within relativistic cryptography proper, graph coloring remains the canonical witness language because it supports local checking, color-label randomization, and physically enforced non-communication. The recent literature shows three convergent directions: practical single-round constructions with minimal communication (Crépeau et al., 2019), experimental unconditional quantum security via relativistic commitments and nonlocal games (Weng et al., 30 Jan 2025), and deployable identity-verification systems that combine graph-coloring witnesses with authentication infrastructure and entanglement-aware security analysis (Ma et al., 18 Jul 2025).