Spatial Isolation Implies Zero Knowledge Even in a Quantum World

Abstract

Zero knowledge plays a central role in cryptography and complexity. The seminal work of Ben-Or et al. (STOC 1988) shows that zero knowledge can be achieved unconditionally for any language in NEXP, as long as one is willing to make a suitable physical assumption: if the provers are spatially isolated, then they can be assumed to be playing independent strategies.Quantum mechanics, however, tells us that this assumption is unrealistic, because spatially-isolated provers could share a quantum entangled state and realize a non-local correlated strategy. The MIP * model captures this setting.In this work we study the following question: does spatial isolation still suffice to unconditionally achieve zero knowledge even in the presence of quantum entanglement?We answer this question in the affirmative: we prove that every language in NEXP has a 2-prover zero knowledge interactive proof that is sound against entangled provers; that is, NEXP ⊆ ZK-MIP * .Our proof consists of constructing a zero knowledge interactive PCP with a strong algebraic structure, and then lifting it to the MIP * model. This lifting relies on a new framework that builds on recent advances in low-degree testing against entangled strategies, and clearly separates classical and quantum tools.Our main technical contribution is the development of new algebraic techniques for obtaining unconditional zero knowledge; this includes a zero knowledge variant of the celebrated sumcheck protocol, a key building block in many probabilistic proof systems. A core component of our sumcheck protocol is a new algebraic commitment scheme, whose analysis relies on algebraic complexity theory. Keywords: zero knowledge; multi-prover interactive proofs; quantum entangled strategies; interactive PCPs; sumcheck protocol; algebraic complexity † This work was supported in part by the UC Berkeley Center for Long-Term Cybersecurity. A part of the earlier technical report [CFS17] was merged with this work. ContentsClaim 6.1 (Approximate consistency to trace distance [Vid11; Vid16]). Let |Ψ be a permutation-invariant entangled state on r ≥ 2 registers, and let {A z }, {B z } be single-register measurements with outcomes in the same set. Then, z A In the rest of this section we prove Lemma 8.1. Specifically, in Section 8.1 we begin with a classical preprocessing step (a query reduction); in Section 8.2 we present our transformation; in Section 8.3 we prove soundness against entangled provers; and in Section 8.4 we prove preservation of zero knowledge. The conceptual contribution of Lemma 8.1 is that it provides an abstraction of techniques in [IV12; Vid16].Remark 8.2 (on preserving round complexity). If we do not wish to preserve zero knowledge, then the round complexity of the MIP * that is obtained in Lemma 8.1 can be reduced by 1 (see discussion at the end of Section 8.2). In addition, if the original low-degree IPCP makes a single, uniformly distributed query to its PCP oracle, then the preprocessing step is not required, and we can save an additional round. In particular,...

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