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Software - Science, Technology and Engineering

Semantic-Security Capacity for the Physical Layer via Information Theory

Ziv Goldfeld, Paul Cuff, Haim Henri Permuter

Published Jun 1, 2016• 5 citations
Publisher

Abstract

Physical layer security can ensure secure communication over noisy channels in the presence of an eavesdropper with unlimited computational power. We adopt an information theoretic variant of semantic-security (SS) (a cryptographic gold standard), as our secrecy metric and study the open problem of the type II wiretap channel (WTC II) with a noisy main channel is, whose secrecy-capacity is unknown even under looser metrics than SS. Herein the secrecy-capacity is derived and shown to be equal to its SS capacity. In this setting, the legitimate users communicate via a discrete-memory less (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 block length. The secrecy criterion is achieved simultaneously for all possible eavesdropper subset choices. On top of that, SS requires 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, the lemma shows that a random codebook achieves the soft-covering phenomenon with high probability. The probability of failure is doubly-exponentially small in the block length. Since the combined number of messages and subsets grows only exponentially with the block length, 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.

DOI
10.1109/swste.2016.12

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Physical layer security can ensure secure communication over noisy channels in the presence of an eavesdropper with unlimited computational power. We adopt an information theoretic variant of semantic-security (SS) (a cryptographic gold standard), as our secrecy metric and study the open problem of the type II wiretap channel (WTC II) with a noisy main channel is, whose secrecy-capacity is unknown even under looser metrics than SS. Herein the secrecy-capacity is derived and shown to be equal to its SS capacity. In this setting, the legitimate users communicate via a discrete-memory less (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 block length. The secrecy criterion is achieved simultaneously for all possible eavesdropper subset choices. On top of that, SS requires 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, the lemma shows that a random codebook achieves the soft-covering phenomenon with high probability. The probability of failure is doubly-exponentially small in the block length. Since the combined number of messages and subsets grows only exponentially with the block length, 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.
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.
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.
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. On top of that, SS requires 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, the lemma shows that a random codebook achieves 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.
We propose a new proof method for direct coding theorems for wiretap channels where the eavesdropper has access to a quantum version of the transmitted signal on an infinite-dimensional Hilbert space and the legitimate parties communicate through a classical channel or a classical input, quantum output (cq) channel. The transmitter input can be subject to an additive cost constraint, which specializes to the case of an average energy constraint. This method yields errors that decay exponentially with increasing block lengths. Moreover, it provides a guarantee of a quantum version of semantic security, which is an established concept in classical cryptography and physical layer security. Therefore, it complements existing works which either do not prove the exponential error decay or use weaker notions of security. The main part of this proof method is a direct coding result on channel resolvability which states that there is only a doubly exponentially small probability that a standard random codebook does not solve the channel resolvability problem for the cq channel. Semantic security has strong operational implications meaning essentially that the eavesdropper cannot use its quantum observation to gather any meaningful information about the transmitted signal. We also discuss the connections between semantic security and various other established notions of secrecy.
We investigate an approach to physical-layer security based on the premise that the coding mechanism for secrecy over noisy channels is fundamentally tied to the notion of resolvability. Instead of considering capacity-based constructions, which associate to each message a sub-code whose rate approaches the capacity of the eavesdropper’s channel, we consider resolvability-based constructions, which associate to each message a sub-code whose rate is beyond the resolvability of the eavesdropper’s channel. We provide evidence that resolvability is a more powerful and perhaps more fundamental coding mechanism for secrecy by developing results that hold for strong secrecy metrics and arbitrary channels. Specifically, we show that, at least for binary symmetric wiretap channels, random capacity-based constructions fail to approach the strong secrecy capacity while resolvability-based constructions achieve it. We then obtain the secrecy-capacity region of arbitrary broadcast channels with confidential messages and a cost constraint for strong secrecy metrics, which generalizes existing results. Finally, we specialize our results to study the secrecy capacity of wireless channels with perfect channel state information, compound and mixed channels, as well as the secret-key capacity of source models for secret-key agreement. By tying secrecy to resolvability, we obtain achievable rates for stronger secrecy metrics and with simpler proofs than previously derived. Index Terms information-theoretic security, wiretap channel, secret-key agreement, information-spectrum, resolv-ability, wireless channels.
We propose a new proof method for direct coding theorems for wiretap channels where the eavesdropper has access to a quantum version of the transmitted signal on an infinite dimensional Hilbert space. This method yields errors that decay exponentially with increasing block lengths. Moreover, it provides a guarantee of a quantum version of semantic security, which is an established concept in classical cryptography and physical layer security. Semantic security has strong operational implications meaning essentially that the eavesdropper cannot use its quantum observation to gather any meaningful information about the transmitted signal. Therefore, it complements existing works which either do not prove the exponential error decay or use weaker notions of security. The main part of this proof method is a direct coding result on channel resolvability which states that there is only a doubly exponentially small probability that a standard random codebook does not solve the channel resolvability problem for the classical-quantum channel.