[Paper Review] Entanglement-Resistant Two-Prover Interactive Proof Systems and Non-Adaptive Private Information Retrieval Systems

Richard Cleve, Dmitry Gavinsky+1 more|ArXiv.org|Jul 12, 2007

Cryptography and Data Security15 references21 citations

TL;DR

This paper presents a two-prover interactive proof system for NP that resists cheating via quantum entanglement, using a novel connection to non-adaptive private information retrieval (PIR) systems. The key contribution is an entanglement-resistant protocol with a constant completeness-soundness gap, achieving the strongest known expressive power for constant-bit, multi-prover quantum-robust proof systems.

ABSTRACT

We show that, for any language in NP, there is an entanglement-resistant constant-bit two-prover interactive proof system with a constant completeness vs. soundness gap. The previously proposed classical two-prover constant-bit interactive proof systems are known not to be entanglement-resistant. This is currently the strongest expressive power of any known constant-bit answer multi-prover interactive proof system that achieves a constant gap. Our result is based on an "oracularizing" property of certain private information retrieval systems, which may be of independent interest.

Motivation & Objective

• To construct a two-prover interactive proof system for NP that remains sound even when provers share quantum entanglement.
• To overcome the failure of classical oracularization techniques in the presence of entanglement, which can otherwise allow cheating with perfect soundness.
• To demonstrate that certain private information retrieval (PIR) systems possess a non-adaptive property robust against quantum entanglement, enabling secure oracularization.
• To establish that the expressive power of constant-bit, two-prover quantum-robust proof systems is at least NP, matching the strongest known result for such systems.
• To provide a new technique for oracularizing provers using PIR properties, distinct from prior methods that fail under entanglement.

Proposed method

• The protocol uses a classical PCP verifier as a subroutine to generate random indices i, j, k and a bit δ, which define a 3-bit query to a witness w.
• The verifier samples a random string s uniformly and computes t = s ⊕ e_i ⊕ e_j ⊕ e_k, sending s to Alice and t to Bob.
• Alice and Bob each respond with a single bit a and b, computed as a = w·s and b = w·t, respectively, under the classical strategy.
• The verifier accepts if and only if a ⊕ b = f_x(i, j, k, δ), where f_x is the predicate derived from the PCP for input x.
• The protocol's security relies on a transversal XOR game framework, where the optimal strategy for entangled provers is shown to be classical and non-adaptive.
• A key technical component is the use of PIR systems with a non-adaptive property: provers cannot coordinate answers to depend non-trivially on the queries, even with entanglement.

Experimental results

Research questions

• RQ1 Can a two-prover interactive proof system for NP be constructed that remains sound under arbitrary quantum entanglement between provers?
• RQ2 Do certain private information retrieval (PIR) systems possess a non-adaptive property that prevents entangled provers from exploiting correlations to cheat?
• RQ3 Is it possible to achieve a constant completeness-soundness gap in a constant-bit answer, multi-prover quantum-robust proof system, despite the failure of classical oracularization under entanglement?
• RQ4 What is the maximum expressive power of constant-bit, two-prover quantum-robust interactive proof systems, and can it reach NP?
• RQ5 Can the oracularization of provers be achieved via PIR-based protocols that are robust against quantum side-channel attacks?

Key findings

• The proposed protocol achieves a constant completeness-soundness gap for NP, with completeness at least 1−ε and soundness at most 1/2+ε for any ε>0.
• The protocol is entanglement-resistant: even with shared quantum entanglement, the optimal strategy for provers is classical and non-adaptive, yielding no advantage over classical strategies.
• The soundness bound of 1/2+ε is derived from the soundness of the underlying PCP procedure, which ensures that no non-trivial witness can satisfy more than half the constraints plus a small error.
• The non-adaptive property of the PIR-based protocol ensures that provers cannot coordinate answers to depend on the query structure in a way that would violate oracularization.
• The result establishes that the expressive power of constant-bit, two-prover quantum-robust proof systems is at least NP, representing the strongest known result in this setting.
• The transversal XOR game framework is generalized to show that entanglement provides no advantage in maximizing acceptance probability, even in the presence of quantum side channels.