Secure Coding via Gaussian Random Fields

TL;DR

This paper investigates phase transitions in Bayesian inference for nonlinear Gaussian random fields and applies the findings to develop a secure coding scheme that achieves the secrecy capacity of the Gaussian wiretap channel.

Contribution

It reveals the critical rate as the channel capacity in nonlinear Gaussian fields and proposes a secure encoding method using random binning that attains secrecy capacity.

Findings

Identifies a phase transition at the channel capacity in Bayesian inference.

Demonstrates secure coding scheme asymptotically achieves secrecy capacity.

Uses replica method to analyze nonlinear Gaussian random fields.

Abstract

Inverse probability problems whose generative models are given by strictly nonlinear Gaussian random fields show the all-or-nothing behavior: There exists a critical rate at which Bayesian inference exhibits a phase transition. Below this rate, the optimal Bayesian estimator recovers the data perfectly, and above it the recovered data becomes uncorrelated. This study uses the replica method from the theory of spin glasses to show that this critical rate is the channel capacity. This interesting finding has a particular application to the problem of secure transmission: A strictly nonlinear Gaussian random field along with random binning can be used to securely encode a confidential message in a wiretap channel. Our large-system characterization demonstrates that this secure coding scheme asymptotically achieves the secrecy capacity of the Gaussian wiretap channel.