Coded Cooperative Data Exchange for a Secret Key

TL;DR

This paper studies the minimum number of public transmissions needed for clients to generate a secret key securely, proving NP-hardness of linear schemes and characterizing optimal performance and limitations.

Contribution

It establishes the NP-hardness of secret key generation with linear codes and characterizes the optimal and suboptimal performance of linear coding schemes.

Findings

Linear coding schemes are proven to be NP-hard to optimize.

Complete characterization of the best possible linear coding performance.

Linear codes can be strictly suboptimal compared to non-linear schemes.

Abstract

We consider a coded cooperative data exchange problem with the goal of generating a secret key. Specifically, we investigate the number of public transmissions required for a set of clients to agree on a secret key with probability one, subject to the constraint that it remains private from an eavesdropper. Although the problems are closely related, we prove that secret key generation with fewest number of linear transmissions is NP-hard, while it is known that the analogous problem in traditional cooperative data exchange can be solved in polynomial time. In doing this, we completely characterize the best possible performance of linear coding schemes, and also prove that linear codes can be strictly suboptimal. Finally, we extend the single-key results to characterize the minimum number of public transmissions required to generate a desired integer number of statistically independent secret keys.