# One-Shot Perfect Secret Key Agreement for Finite Linear Sources

## TL;DR

This paper characterizes the minimum public discussion needed for all terminals to agree on a maximum-length secret key in a finite linear source model, using linear functions and non-interactive protocols.

## Contribution

It provides a non-asymptotic, one-shot characterization of communication complexity for secret key agreement in finite linear sources, introducing a linear, non-interactive protocol approach.

## Key findings

- Minimum discussion is achieved by a linear, non-interactive protocol.
- The protocol involves linear processing of private observations before discussion.
- Secret key is a linear function of all observations.

## Abstract

We consider a non-asymptotic (one-shot) version of the multiterminal secret key agreement problem on a finite linear source model. In this model, the observation of each terminal is a linear function of an underlying random vector composed of finitely many i.i.d. uniform random variables. Restricting the public discussion to be a linear function of the terminals' observations, we obtain a characterization of the communication complexity (minimum number of symbols of public discussion) of generating a secret key of maximum length. The minimum discussion is achieved by a non-interactive protocol in which each terminal first does a linear processing of its own private observations, following which the terminals all execute a discussion-optimal communication-for-omniscience protocol. The secret key is finally obtained as a linear function of the vector of all observations.