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IEEE International Symposium on Information Theory (ISIT)

On arbitrarily varying wiretap channels for different classes of secrecy measures

Holger Boche, Rafael F. Schaefer, H. Vincent Poor

Published Jun 1, 2014• 12 citations
Publisher

Abstract

The wiretap channel models secure communication in the presence of an eavesdropper who must be kept ignorant of transmitted messages. In this paper, the arbitrarily varying wiretap channel (AVWC), in which the channel may vary in an unknown and arbitrary manner from channel use to channel use, is considered. For arbitrarily varying channels (AVCs) the capacity might differ depending on whether deterministic or common randomness (CR) assisted codes are used. The AVWC has been studied for both coding strategies and the relation between the corresponding secrecy capacities has been established. However, a characterization of the CR-assisted secrecy capacity itself or even a general CR-assisted achievable secrecy rate remain open in general for weak and strong secrecy. Here, the secrecy measure of high decoding error at the eavesdropper is considered, where the eavesdropper is further assumed to know channel states and to adapt its decoding strategy accordingly. For this secrecy measure a general CR-assisted achievable secrecy rate is established. The relation between secrecy capacities for different secrecy measures is discussed: The weak and strong secrecy capacities are smaller than or equal to the one for high decoding error. It is conjectured that this relation can be strict for certain channels.

DOI
10.1109/isit.2014.6875259

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The wiretap channel models secure communication in the presence of an eavesdropper who must be kept ignorant of transmitted messages. In this paper, the arbitrarily varying wiretap channel (AVWC), in which the channel may vary in an unknown and arbitrary manner from channel use to channel use, is considered. For arbitrarily varying channels (AVCs) the capacity might differ depending on whether deterministic or common randomness (CR) assisted codes are used. The AVWC has been studied for both coding strategies and the relation between the corresponding secrecy capacities has been established. However, a characterization of the CR-assisted secrecy capacity itself or even a general CR-assisted achievable secrecy rate remain open in general for weak and strong secrecy. Here, the secrecy measure of high decoding error at the eavesdropper is considered, where the eavesdropper is further assumed to know channel states and to adapt its decoding strategy accordingly. For this secrecy measure a general CR-assisted achievable secrecy rate is established. The relation between secrecy capacities for different secrecy measures is discussed: The weak and strong secrecy capacities are smaller than or equal to the one for high decoding error. It is conjectured that this relation can be strict for certain channels.
The wiretap channel models secure communication in the presence of a non-legitimate eavesdropper who has to be kept ignorant. In this paper, the arbitrarily varying wiretap channel (AVWC) is studied, in which the channel to both legitimate receiver and eavesdropper may vary in an unknown and arbitrary manner from channel use to channel use. It has been shown that for AVCs coordination between the transmitter and legitimate receiver based on common randomness (CR) is indispensable for reliable communication. Approaches taken so far yield CR-assisted strategies where the needed amount of CR increases unbounded with the block length. In this paper, it is shown that if we allow for a small but non-vanishing average probability of error and information leakage (in terms of weak secrecy), the amount of CR is always finite and independent of the block length. The corresponding secrecy capacity equals the one with asymptotically vanishing performance requirements. Furthermore, it is shown that the average decoding error at the eavesdropper can be made arbitrarily close to 1 regardless of the applied decoding strategy.
The classical wiretap channel models secure communication in the presence of a nonlegitimate wiretapper who has to be kept ignorant. Traditionally, the wiretapper is passive in the sense that he only tries to eavesdrop the communication using his received channel output. In this paper, more powerful active wiretappers are studied. In addition to eavesdropping, these wiretappers are able to influence the communication conditions of all users by controlling the corresponding channel states. Since legitimate transmitters and receivers do not know the actual channel realization or the wiretapper's strategy of influencing the channel states, they are confronted with arbitrarily varying channel (AVC) conditions. The corresponding secure communication scenario is, therefore, given by the arbitrarily varying wiretap channel (AVWC). In the context of AVCs, common randomness (CR) has been shown to be an important resource for establishing reliable communication, in particular, if the AVC is symmetrizable. But availability of CR also affects the strategy space of an active wiretapper as he may or may not exploit the common randomness for selecting the channel states. Several secrecy capacity results are derived for the AVWC. In particular, the CR-assisted secrecy capacity of the AVWC with an active wiretapper exploiting CR is established and analyzed in detail. Finally, it is demonstrated for active wiretappers how two orthogonal AVWCs, each useless for transmission of secure messages, can be super-activated to a useful channel allowing for secure communication at nonzero secrecy rates. To the best of our knowledge, this is not possible for passive wiretappers and, further, provides the first example of such super-activation, which has been expected to appear only in the area of quantum communication. Such knowledge is particularly important as it provides valuable insights for the design and the medium access control of future wireless communication systems.
We define the common randomness-assisted capacity of an arbitrarily varying wiretap channel (AVWC) when the eavesdropper is kept ignorant about the common randomness. We prove a multi-letter capacity formula for this model. We prove that, if enough common randomness is used, the capacity formula can be given a single-shot form again. We then consider the opposite extremal case, where no common randomness is available, and derive the capacity. It is known that the capacity of the system can be discontinuous under these circumstances. We prove here that it is still stable in the sense that it is continuous around its positivity points. We further prove that discontinuities can only arise if the legal link is symmetrizable and characterize the points where it is positive. These results shed new light on the design principles of communication systems with embedded security features. At last, we investigate the effect of super-activation of the message transmission capacity of AVWCs under the average error criterion. We give a complete characterization of those AVWCs that may be super-activated. The effect is thereby also related to the (conjectured) super-activation of the common randomness assisted capacity of AVWCs with an eavesdropper that gets to know the common randomness. Super-activation is based on the idea of wasting a few bits of non-secret messages in order to enable provably secret transmission of a large bulk of data, a concept that may prove to be of further importance in the design of communication systems. In this paper, we provide further insight into this phenomenon by providing a class of codes that is capacity achieving and does not convey any information to the eavesdropper.
For communication over arbitrarily varying channels (AVC), common randomness is an important resource to establish reliable communication, especially if the AVC is symmetrizable. In this paper the arbitrarily varying wiretap channel (AVWC) with active wiretapper is studied. Here the wiretapper is active in the sense that it can exploit the knowledge about the common randomness to control the channel conditions of the legitimate users. The common randomness assisted secrecy capacity of the AVWC with active wiretapper is analyzed and is related to the corresponding secrecy capacity of the AVWC with passive wiretapper. If the active secrecy capacity is positive, it equals the corresponding passive secrecy capacity. The case of zero active capacity is also studied.
We give a complete characterization of the secrecy capacity of arbitrarily varying wiretap channels (AVWCs) with correlated random coding under a strong secrecy criterion where the eavesdropper may also know the correlated randomness. We obtain that the correlated random coding secrecy capacity is continuous as a function of the AVWC. We show that the deterministic coding secrecy capacity of the AVWC either equals 0 or the correlated random coding secrecy capacity. For the case that only a weak secrecy criterion is applied, a complete characterization of the corresponding secrecy capacity for deterministic codes is possible. In the proof of the secrecy capacity formula for correlated random codes, we apply an auxiliary channel which is compound from the sender to the intended receiver and varies arbitrarily from the sender to the eavesdropper. We discuss the relation between the usual mutual information secrecy criterion and a criterion formulated in terms of total variation distance, and investigate the robustness of the AVWC model.
The secrecy capacity problems over the general arbitrarily varying wiretap channel (AVWC), with respect to the maximal decoding error probability and strong secrecy criterion, are considered, where the channel state sequence may be known or unknown at the receiver. In the mean time, it is always assumed that the channel state sequence is known at the eavesdropper and unknown at the transmitter. Capacity results of both stochastic code (with random encoder and deterministic decoder) and random code (with random encoder and decoder) are discussed. This model includes the previous models of classic AVWC as special cases. Single-letter lower bounds on the secrecy capacities are given, which are proved to be the secrecy capacities when the main channel is less noisy than the wiretap channel. The coding scheme is based on Csiszar's almost independent coloring scheme and Ahlswede's elimination technique. Moreover, a new kind of typical sequence with respect to states is defined for this coding scheme. It is concluded that the secrecy capacity of stochastic code is identical to that of random code when the receiver knows the state sequence. Meanwhile, random code may achieve larger secrecy capacity when the state sequence is unknown by the receiver.