A Framework for Partial Secrecy

TL;DR

This paper explores the theoretical limits of partial secrecy in communication systems where secret keys are limited, analyzing how to maximize distortion for eavesdroppers and providing a linear programming characterization.

Contribution

It introduces an information-theoretic region characterizing the tradeoff between secret key rate and eavesdropper distortion, with a simplified linear program formulation.

Findings

Derived a region for secret key rate versus eavesdropper distortion

Provided a linear program for the tradeoff characterization

Illustrated the approach with a Bernoulli-p bits example

Abstract

We consider theoretical limits of partial secrecy in a setting where an eavesdropper attempts to causally reconstruct an information sequence with low distortion based on an intercepted transmission and the past of the sequence. The transmitter and receiver have limited secret key at their disposal but not enough to establish perfect secrecy with a one-time pad. From another viewpoint, the eavesdropper is acting as an adversary, competing in a zero-sum repeated game against the sender and receiver of the secrecy system. In this case, the information sequence represents a sequence of actions, and the distortion function captures the payoff of the game. We give an information theoretic region expressing the tradeoff between secret key rate and max-min distortion for the eavesdropper. We also simplify this characterization to a linear program. As an example, we discuss how to optimally use secret key to hide Bernoulli-p bits from an eavesdropper so that they incur maximal Hamming distortion.