## Abstract

We consider interactive proofs for social graphs, where the verifier has only oracle access to the graph and can query for the i^{th} neighbor of a vertex v, given i and v. In this model, we construct a doubly-efficient public-coin two-message interactive protocol for estimating the size of the graph to within a multiplicative factor ε >0. The verifier performs O(1/ε^{2} · τ_{mix}· Δ) queries to the graph, where τ_{mix} is the mixing time of the graph and Δ is the average degree of the graph. The prover runs in quasi-linear time in the number of nodes in the graph. Furthermore, we develop a framework for computing the quantiles of essentially any (reasonable) function f of vertices/edges of the graph. Using this framework, we can estimate many health measures of social graphs such as the clustering coefficients and the average degree, where the verifier performs only a small number of queries to the graph. Using the Fiat-Shamir paradigm, we are able to transform the above protocols to a non-interactive argument in the random oracle model. The result is that social media companies (e.g., Facebook, Twitter, etc.) can publish, once and for all, a short proof for the size or health of their social network. This proof can be publicly verified by any single user using a small number of queries to the graph.

Original language | English |
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Title of host publication | Advances in Cryptology - CRYPTO 2020 - 40th Annual International Cryptology Conference, Proceedings |

Editors | Daniele Micciancio, Thomas Ristenpart |

Publisher | Springer |

Pages | 574-601 |

Number of pages | 28 |

ISBN (Print) | 9783030568764 |

DOIs | |

State | Published - 2020 |

Externally published | Yes |

Event | 40th Annual International Cryptology Conference, CRYPTO 2020 - Santa Barbara, United States Duration: 17 Aug 2020 → 21 Aug 2020 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 12172 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 40th Annual International Cryptology Conference, CRYPTO 2020 |
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Country/Territory | United States |

City | Santa Barbara |

Period | 17/08/20 → 21/08/20 |

### Bibliographical note

Funding Information:This work was done (in part) while the second and third authors were visiting the Simons Institute for the Theory of Computing. Eylon Yogev is funded by the ISF grants 484/18, 1789/19, Len Blavatnik and the Blavatnik Foundation, and The Blavatnik Interdisciplinary Cyber Research Center at Tel Aviv University.

Funding Information:

Acknowledgments. This work was done (in part) while the second and third authors were visiting the Simons Institute for the Theory of Computing. Eylon Yogev is funded by the ISF grants 484/18, 1789/19, Len Blavatnik and the Blavatnik Foundation, and The Blavatnik Interdisciplinary Cyber Research Center at Tel Aviv University.

Publisher Copyright:

© International Association for Cryptologic Research 2020.

## Keywords

- Interactive proofs
- Social graphs
- Succinct arguments