Semantic-Security Capacity for Wiretap Channels of Type II

TL;DR

This paper derives the semantic-security capacity for the wiretap channel of type II with a noisy main channel, introducing a novel soft covering lemma that ensures high-probability codebook performance against an eavesdropper with subset access.

Contribution

It establishes the semantic-security capacity for WTC II and introduces a stronger soft covering lemma with doubly-exponentially small failure probability.

Findings

Secrecy capacity equals semantic-security capacity for WTC II.

Achievable rates match the weak-secrecy capacity of the classic WTC with an erasure channel.

The stronger soft-covering lemma is a key tool for security proofs.

Abstract

The secrecy capacity of the type II wiretap channel (WTC II) with a noisy main channel is currently an open problem. Herein its secrecy-capacity is derived and shown to be equal to its semantic-security (SS) capacity. In this setting, the legitimate users communicate via a discrete-memoryless (DM) channel in the presence of an eavesdropper that has perfect access to a subset of its choosing of the transmitted symbols, constrained to a fixed fraction of the blocklength. The secrecy criterion is achieved simultaneously for all possible eavesdropper subset choices. The SS criterion demands negligible mutual information between the message and the eavesdropper's observations even when maximized over all message distributions. A key tool for the achievability proof is a novel and stronger version of Wyner's soft covering lemma. Specifically, a random codebook is shown to achieve the soft-covering phenomenon with high probability. The probability of failure is doubly-exponentially small in the blocklength. Since the combined number of messages and subsets grows only exponentially with the blocklength, SS for the WTC II is established by using the union bound and invoking the stronger soft-covering lemma. The direct proof shows that rates up to the weak-secrecy capacity of the classic WTC with a DM erasure channel (EC) to the eavesdropper are achievable. The converse follows by establishing the capacity of this DM wiretap EC as an upper bound for the WTC II. From a broader perspective, the stronger soft-covering lemma constitutes a tool for showing the existence of codebooks that satisfy exponentially many constraints, a beneficial ability for many other applications in information theoretic security.