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arxiv: 2606.27996 · v1 · pith:PST7LR45new · submitted 2026-06-26 · 🪐 quant-ph · cs.CR

Quantum Multi-Party Threshold Private Set Intersection with Explicit Cardinality Testing

Zixian Gong , Kun Tian , Yi Zhang , Fengxia Liu This is my paper

Pith reviewed 2026-06-29 04:19 UTC · model grok-4.3

classification 🪐 quant-ph cs.CR
keywords threshold private set intersectionquantum multiparty protocolcardinality testingphoton rotationshidden-label measurementsoblivious linear evaluationgarbled circuits
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The pith

A rotation-based quantum protocol allows multiple parties to test if their private set intersection meets a threshold without revealing its details.

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

This paper develops a quantum multiparty threshold private set intersection protocol that performs explicit cardinality testing. Single-photon sequences undergo participant data rotations, masking by the third party, and aggregate rotations to produce hidden-label measurement vectors. The third party measures these but cannot interpret their meaning, enabling an OLE-based inner product and garbled circuit to output solely the indicator of whether the intersection size is at least the threshold. Only then is the intersection reconstructed if needed. The design improves on prior protocols by preventing the third party from directly accessing semantic results, with proofs of correctness and security plus Qiskit simulations.

Core claim

The protocol develops a rotation-based quantum construction in which single-photon sequences are sequentially processed through participant-side data rotations, TP--participant masking rotations, and correlated aggregate rotations. This design produces hidden-label measurement vectors: TP can complete the final measurement, but cannot interpret the semantic meaning of the outcomes. Based on these hidden measurements, we further realize the threshold decision through an oblivious linear evaluation (OLE)-based inner product procedure and a lightweight garbled circuit, revealing only 1[|∩_i X_i| ≥ τ] before conditional intersection reconstruction. We prove the correctness and security of the pr

What carries the argument

Rotation-based quantum construction producing hidden-label measurement vectors via sequential participant data rotations, TP masking rotations, and aggregate rotations.

If this is right

  • The protocol outputs only the boolean 1[|intersection| >= τ].
  • The third party performs final measurements without semantic interpretation capability.
  • Conditional reconstruction of the intersection follows only if the threshold is met.
  • Correctness and security are proven, with feasibility shown in Qiskit simulations.

Where Pith is reading between the lines

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

  • The hidden measurement approach could extend to other quantum multiparty protocols requiring controlled revelation.
  • Practical testing on physical quantum hardware beyond simulators would validate real-world applicability.
  • This method of decoupling measurement from interpretation may apply to broader quantum privacy-preserving computations.

Load-bearing premise

The third party honestly applies the masking rotations without colluding or deriving semantic meaning from the hidden-label vectors, and the OLE and garbled circuit components are secure.

What would settle it

Observing that the third party recovers semantic information about the sets or their intersection from the measurement vectors despite following the protocol would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.27996 by Fengxia Liu, Kun Tian, Yi Zhang, Zixian Gong.

Figure 1
Figure 1. Figure 1: Ideal functionalities for OLE and VOLE. 3.1. Oblivious Inner Product (OIP) from OLE OLE and Vector OLE Functionalities. OLE can be viewed as the linear case of Oblivious Polynomial Evaluation (OPE) [NP99] that enables a receiver to compute a linear com￾bination of the sender’s inputs. Vector OLE (VOLE) gen￾eralizes this primitive to vectors, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Ideal functionality for oblivious inner product (OIP). [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: OIP Constructed from OLE. Consistency-share generation. TP and the participant side invoke  𝑝 𝖮𝖨𝖯 to obtain additive shares of ⟨𝑧 𝖲 , 𝑚 ⊙ (1 − 𝜌)⟩, ⟨𝑧 𝖮, 𝑚 ⊙ 𝜌⟩, ⟨𝑧 𝖲 , 𝑚 ⊙ (1 − 𝜌)⟩, and ⟨𝑧 𝖮, 𝑚 ⊙ 𝜌⟩. Zixian Gong et al.: Preprint submitted to Elsevier Page 3 of 11 [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Masked Hidden-label Cardinality Testing Protocol. [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Flowchart. Protocol 1 Quantum Multi-party Threshold PSI with Cardinality Testing Input: Private sets 𝑋𝑖 ⊆  , threshold 𝜏, anchor sets + , − , repetition number 𝓁. Output: Reveal ⋂𝑛 𝑖=1 𝑋𝑖 iff its cardinality is at least 𝜏. Phase 1: Setup. 1: Let ̃ =  ∪ + ∪ − and 𝑀 = |̃|. 2: Participants run TP-free QCKA to obtain 𝐾𝖯 and derive 𝑘 ∈ ℤ∗ 𝑀 , 𝐛 = (𝑏0 , …, 𝑏𝑀−1) ∈ {0, 1}𝑀 , 𝚯 = (Θ0 , …, Θ𝑀−1) with Θ𝑡 = 𝑏… view at source ↗
Figure 7
Figure 7. Figure 7: Quantum Circuit of toy model. t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7 Hidden position t 0.0 0.2 0.4 0.6 0.8 1.0 Probability 0.988 0.984 0.013 0.984 0.257 0.016 0.742 0.256 0.012 0.016 0.987 0.016 0.743 0.984 0.258 0.744 Noisy same probability Noisy opposite probability [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Noisy Probabilities. and opposite measurement relations are 𝑝 𝖲 𝑡 = cos2 ( 𝑟𝑡𝜋∕𝑛 + 𝑏𝑡𝜋 2 ) , 𝑝𝖮 𝑡 = sin2 ( 𝑟𝑡𝜋∕𝑛 + 𝑏𝑡𝜋 2 ) . To evaluate the robustness of the quantum phase, we perform noisy simulation by adding depolarizing noise with rate 0.2%, phase-damping noise with rate 0.4%, and read￾out error with rate 0.5% which can be realized through Qiskit_aer.noise. These channels are chosen to reflect rep￾res… view at source ↗
read the original abstract

Threshold private set intersection (TPSI) allows parties to reveal their intersection only when its cardinality reaches a prescribed threshold. Existing quantum TPSI protocols typically rely on a third party (TP) to interpret the final results, which deviates from the cardinality-testing paradigm of TPSI. In this paper, we propose a quantum multiparty TPSI protocol with explicit cardinality testing. Our protocol develops a rotation-based quantum construction in which single-photon sequences are sequentially processed through participant-side data rotations, TP--participant masking rotations, and correlated aggregate rotations. This design produces hidden-label measurement vectors: TP can complete the final measurement, but cannot interpret the semantic meaning of the outcomes. Based on these hidden measurements, we further realize the threshold decision through an oblivious linear evaluation (OLE)-based inner product procedure and a lightweight garbled circuit, revealing only \(\mathbf 1[|\bigcap_i X_i|\ge \tau]\) before conditional intersection reconstruction. We prove the correctness and security of the proposed protocol, and further validate its feasibility through quantum-circuit simulations implemented on the IBM \textsf{Qiskit} platform.

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 / 0 minor

Summary. The paper proposes a quantum multi-party threshold private set intersection (TPSI) protocol. It uses a rotation-based construction on single-photon sequences involving participant data rotations, TP-participant masking rotations, and correlated aggregate rotations to generate hidden-label measurement vectors. A third party (TP) performs the final measurement but cannot interpret semantic meaning. Threshold decision is then handled via an OLE-based inner product and lightweight garbled circuit, revealing only the indicator 1[|∩_i X_i| ≥ τ] before conditional reconstruction. The authors claim proofs of correctness and security, plus feasibility validation via Qiskit simulations.

Significance. If the security reduction holds, the work would advance quantum secure multiparty computation by enabling explicit cardinality testing in TPSI without excess leakage to the TP, addressing a limitation in prior quantum protocols. The quantum hiding mechanism combined with classical primitives (OLE, garbled circuits) offers a hybrid approach that could support privacy applications where only threshold information is needed.

major comments (1)
  1. [Security analysis section] Security analysis (the section containing the proof of security against TP): the central claim that TP cannot interpret semantic meaning from the hidden-label measurement vectors requires an explicit reduction showing that the final density operator (after aggregate rotations) is independent of the intersection labels from TP's perspective. The abstract asserts such a proof exists, but without the concrete calculation demonstrating that the masking angles fully randomize the basis relative to TP's knowledge, the subsequent OLE inner-product and garbled-circuit steps cannot be guaranteed to enforce the claimed leakage bound.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment on the security analysis. We address the point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Security analysis section] Security analysis (the section containing the proof of security against TP): the central claim that TP cannot interpret semantic meaning from the hidden-label measurement vectors requires an explicit reduction showing that the final density operator (after aggregate rotations) is independent of the intersection labels from TP's perspective. The abstract asserts such a proof exists, but without the concrete calculation demonstrating that the masking angles fully randomize the basis relative to TP's knowledge, the subsequent OLE inner-product and garbled-circuit steps cannot be guaranteed to enforce the claimed leakage bound.

    Authors: We agree that the security analysis would benefit from an explicit reduction. The current proof sketch in the security analysis section establishes that the aggregate rotations produce a maximally mixed state from the TP's viewpoint by averaging over the random masking angles chosen by the participants, but we acknowledge that a fully expanded calculation tracing out the dependence on the intersection labels is not written out in sufficient detail. In the revised version we will add this concrete calculation, showing that the final density operator ho_TP is independent of the labels (i.e., ho_TP = (1/2)^n I for n photons) and therefore that the subsequent OLE inner-product and garbled-circuit steps inherit the claimed leakage bound. revision: yes

Circularity Check

0 steps flagged

No circularity; protocol construction and security claims are independent of self-referential definitions or fitted inputs

full rationale

The abstract and description present a rotation-based quantum protocol producing hidden-label vectors, followed by OLE inner-product and garbled-circuit steps for the threshold indicator. No equations, self-citations, or parameter-fitting steps are exhibited that would reduce the claimed correctness, security, or hidden-label property to a tautology or to the paper's own inputs by construction. The security assumptions (TP honesty, OLE/garbled-circuit security) are stated as external and load-bearing but are not derived from the protocol equations themselves. This is the normal case of a self-contained construction claim without detectable circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger reflects high-level protocol elements; no explicit free parameters or invented entities are identifiable. The work rests on standard domain assumptions rather than new postulates.

axioms (1)
  • domain assumption Standard assumptions of quantum mechanics for single-photon rotations and measurements, plus security of OLE and garbled circuit primitives.
    These underpin the correctness, security, and hidden-label claims in the abstract.

pith-pipeline@v0.9.1-grok · 5722 in / 1440 out tokens · 65696 ms · 2026-06-29T04:19:34.898530+00:00 · methodology

discussion (0)

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