### 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.

Original language | English (US) |
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Title of host publication | Proceedings - 59th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2018 |

Editors | Mikkel Thorup |

Publisher | IEEE Computer Society |

Pages | 755-765 |

Number of pages | 11 |

ISBN (Electronic) | 9781538642306 |

DOIs | |

State | Published - Nov 30 2018 |

Event | 59th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2018 - Paris, France Duration: Oct 7 2018 → Oct 9 2018 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2018-October |

ISSN (Print) | 0272-5428 |

### Other

Other | 59th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2018 |
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Country | France |

City | Paris |

Period | 10/7/18 → 10/9/18 |

### Keywords

- Algebraic complexity
- Interactive PCPs
- Multi-prover interactive proofs
- Quantum entangled strategies
- Sumcheck protocol
- Zero knowledge

### ASJC Scopus subject areas

- Computer Science(all)

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## Cite this

*Proceedings - 59th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2018*(pp. 755-765). [8555155] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS; Vol. 2018-October). IEEE Computer Society. https://doi.org/10.1109/FOCS.2018.00077