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IEEE Transactions on Information Theory

Strong Secrecy of Arbitrarily Varying Wiretap Channel With Constraints

Yiqi Chen, Dan Nong He, Chenhao Ying, Yuan Luo

Published Mar 23, 2022• 6 citations
Publisher

Abstract

The strong secrecy transmission problem of the arbitrarily varying wiretap channel (AVWC) with input and state constraints is investigated in this paper. First, a stochastic-encoder code lower bound of the strong secrecy capacity is established by applying the type argument and Csiszár’s almost independent coloring lemma. Then, a superposition stochastic-encoder code lower bound of the secrecy capacity is provided. The superposition stochastic-encoder code lower bound can be larger than the ordinary stochastic-encoder code lower bound. Random code lower and upper bounds of the secrecy capacity of the AVWC with constraints are further provided. Based on these results, we further consider a special case of the model, namely severely less noisy AVWC, and give the stochastic-encoder code and random code capacities. It is proved that the stochastic-encoder code capacity of the AVWC with constraints is either equal to or strictly smaller than the corresponding random code capacity, which is consistent with the property of the ordinary AVC. Finally, some numerical examples are presented to better illustrate our capacity results. Compared to the soft covering lemma that requires the codewords to be generated i.i.d., our method has more relaxed requirements regarding codebooks. It is proved that the good codebooks for secure transmission can be generated by choosing codewords randomly from a given type set, which is critical when considering the AVWC with constraints.

DOI
10.1109/tit.2022.3161808

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arXiv:1601.03660•IEEE Globecom Workshops (GC Wkshps)•Oct 20, 2016•43 citations
The strong secrecy transmission problem of the arbitrarily varying wiretap channel (AVWC) with input and state constraints is investigated in this paper. First, a stochastic-encoder code lower bound of the strong secrecy capacity is established by applying the type argument and Csiszár’s almost independent coloring lemma. Then, a superposition stochastic-encoder code lower bound of the secrecy capacity is provided. The superposition stochastic-encoder code lower bound can be larger than the ordinary stochastic-encoder code lower bound. Random code lower and upper bounds of the secrecy capacity of the AVWC with constraints are further provided. Based on these results, we further consider a special case of the model, namely severely less noisy AVWC, and give the stochastic-encoder code and random code capacities. It is proved that the stochastic-encoder code capacity of the AVWC with constraints is either equal to or strictly smaller than the corresponding random code capacity, which is consistent with the property of the ordinary AVC. Finally, some numerical examples are presented to better illustrate our capacity results. Compared to the soft covering lemma that requires the codewords to be generated i.i.d., our method has more relaxed requirements regarding codebooks. It is proved that the good codebooks for secure transmission can be generated by choosing codewords randomly from a given type set, which is critical when considering the AVWC with constraints.
The strong secrecy transmission problem of arbitrarily varying wiretap channel (AVWC) with input and state constraints is investigated in this paper. A stochastic code lower bound of the secrecy capacity is established by applying the type argument and Csiszár's almost independent coloring lemma, which includes the result of the ordinary AVWC as special case. Our codebook generation without the i.i.d. assumption of the soft covering lemma is to ensure the reliable communication over the AVWC with constraints. It is proved that the 0good codebooks for the strong secrecy transmission can be generated by choosing codewords randomly from a given type set, which is critical in this model.
In this work the arbitrarily varying wiretap channel AVWC under the average error criterion and the strong secrecy criterion is studied. We show that in the case of a non-symmetrisable channel to the legitimate receiver the deterministic code secrecy capacity equals the random code secrecy capacity and thus we establish a result for the AVWC similar to that of Ahlswede's dichotomy for ordinary AVCs. We derive a lower bound on the random code secrecy capacity in the case of a best channel to the eavesdropper. We further prove upper bounds on the deterministic code secrecy capacity, which in special cases results in explicit expressions of the secrecy capacity.
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.
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.
We give a complete characterization of the secrecy capacity of arbitrarily varying wiretap channels (AVWCs) with correlated random coding. We apply two alternative strong secrecy criteria, which both lead to the same formula and which coincide in the case of deterministic codes. On the basis of the derived formula, we show 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, we derive a complete characterization of the corresponding secrecy capacity for deterministic codes. 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.
Determining a single-letter secrecy-capacity formula for the arbitrarily varying wiretap channel (AVWTC) is an open problem largely because of two main challenges. Not only does it capture the difficulty of the compound wiretap channel (another open problem), it also requires that secrecy is ensured with respect to exponentially many possible channel state sequences. By extending the strong soft-covering lemma, recently derived by the authors, to the heterogeneous scenario, this paper accounts for the exponential number of secrecy constraints while only imposing single-letter constraints on the communication rate. Through this approach, we derive a single-letter characterization of the correlated-random (CR)-assisted semantic-security (SS) capacity of an AVWTC with a type constraint on the allowed state sequences. The allowed state sequences are the ones in a typical set around a single constraining type. The stringent SS requirement is established by showing that the mutual information between the message and the eavesdropper's observations is negligible even when maximized over all message distributions, choices of state sequences, and realizations of the CR-code. Both the achievability and the converse proofs of the type-constrained coding theorem rely on stronger claims than actually required. The direct part establishes a novel single-letter lower bound on the CR-assisted SS-capacity of an AVWTC with state sequences constrained by any convex and closed set of state probability mass functions. This bound achieves the best known single-letter secrecy rates for a corresponding compound wiretap channel over the same constraint set. In contrast to other single-letter results in the AVWTC literature, this paper does not assume the existence of a best channel to the eavesdropper. Instead, SS follows by leveraging the heterogeneous version of the strong soft-covering lemma and a CR-code reduction argument. Optimality is a consequence of a max-inf upper bound on the CR-assisted SS-capacity of an AVWTC with state sequences constrained to any collection of type-classes. When adjusted to the aforementioned compound WTC, the upper bound simplifies to a max-min structure, thus strengthening the previously best known single-letter upper bound by Liang et al. that has a min-max form. The proof of the upper bound uses a novel distribution coupling argument. The capacity formula shows that the legitimate users effectively see an averaged main channel, while security must be ensured versus an eavesdropper with perfect channel state information. An example visualizes our single-letter results, and their relation to the past multi-letter secrecy-capacity characterization of the AVWTC is highlighted.