Semantic-security capacity for wiretap channels of type II

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 doublyexponentially 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.Index Terms-Erasure wiretap channel, information theoretic security, semantic-security, soft-covering lemma, wiretap channel of type II.

Secret key generation over noisy channels with common randomness

“…We will use the same method in our proofs although the strong secrecy criterion has become more frequent nowadays. Recent works on the wiretap channel employ this approach, e.g., [25], [26], where in particular [26] does not assume the messages have a uniform distribution. December …”

On the capacity of the binary adversarial wiretap channel

“…Semantic secrecy for the Wiretap Channel II with noisy main channel is established in [GCP16]. Although their reliability proof does not apply to our setting with an adversarial main channel, the adversarial nature of the Wiretap Channel II means that the secrecy analysis applies to the AWTC.…”

“…Although their reliability proof does not apply to our setting with an adversarial main channel, the adversarial nature of the Wiretap Channel II means that the secrecy analysis applies to the AWTC. We outline the proof below; for more details, see the original derivation in [GCP16].…”

Secrecy Capacity of the Extended Wiretap Channel II with Noise

“…An extension of wiretap channel II was considered recently in [21,22]. The source message was encoded into the digital sequence X N and transmitted to the legitimate receiver via a DMC.…”

“…A pair of inner-outer bounds was given in [21], while the secrecy capacity with respect to the semantic secrecy criterion was established in [22]. The coding scheme in [21] was based on that developed by Ozarow and Wyner in [14], while the scheme in [22] was by Wyner's soft covering lemma. He et al considered another extension of wiretap channel II in [23].…”