## Abstract

Hypergraphical sources are a natural class of sources for secret key generation, within which different subsets of terminals sharing secrets are allowed to discuss publicly in order to agree upon a global secret key. While their secrecy capacity, i.e., the maximum rate of a secret key that can be agreed upon by the entire set of terminals, is well-understood, what remains open is the maximum rate of a secret key that can be generated when there is a restriction on the overall rate of public discussion allowed. In this paper, we obtain a family of explicitly computable upper bounds on the number of bits of secret key that can be generated per bit of public discussion. These upper bounds are derived using a lamination technique based on the submodularity of the entropy function. In particular, a specific instance of these upper bounds, called the edge-partition bound, is shown to be tight for the pairwise independent network model, a special case of the hypergraphical source when the hypergraph is a graph. The secret key generation scheme achieving this upper bound is the tree-packing protocol of Nitinawarat et al., thereby resolving in the affirmative the discussion rate optimality of the tree-packing protocol.

Original language | English |
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Article number | 8633870 |

Pages (from-to) | 5080-5093 |

Number of pages | 14 |

Journal | IEEE Transactions on Information Theory |

Volume | 65 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2019 |

Externally published | Yes |

### Bibliographical note

Funding Information:Manuscript received March 13, 2018; revised January 19, 2019; accepted January 25, 2019. Date of publication February 4, 2019; date of current version July 12, 2019. C. Chan was supported by a grant from the University Grants Committee of the Hong Kong Special Administrative Region, China, under Project 21203318. M. Mukherjee and N. Kashyap were supported in part by the Department of Science and Technology, Government of India, through the Swarnajayanti Fellowship. This paper was presented in part at the 2017 IEEE International Symposium on Information Theory [1].

Publisher Copyright:

© 2019 IEEE.

## Keywords

- Secret key agreement
- hypergraphical sources
- multiterminal source model
- secrecy capacity