Arbitrarily Varying Channel Research Articles

The Arbitrarily Varying Channel With Colored Gaussian Noise

We address the arbitrarily varying channel (AVC) with colored Gaussian noise. The work consists of three parts. First, we study the general discrete AVC with fixed parameters, a model that combines the AVC and the time-varying channel. We determine both the deterministic code capacity and the random code capacity. Super-additivity is demonstrated, showing that the deterministic code capacity can be strictly larger than the weighted sum of the parametric capacities. In the second part, we consider the arbitrarily varying Gaussian product channel (AVGPC). Hughes and Narayan characterized the random code capacity through min-max optimization leading to a “double” water filling solution. As in the case of the standard Gaussian AVC, the deterministic code capacity is discontinuous in the input constraint, and depends on which of the input or state constraint is higher. As opposed to Shannon's classic water filling solution, it is observed that deterministic coding using independent scalar codes is suboptimal for the AVGPC. Finally, we establish the capacity of the AVC with colored Gaussian noise, where double water filling is performed in the frequency domain. The analysis relies on our preceding results, on the AVC with fixed parameters and the AVGPC.

Secure OFDM System Design and Capacity Analysis Under Disguised Jamming

In this paper, we propose a securely precoded OFDM (SP-OFDM) system for efficient and reliable transmission under disguised jamming, where the jammer intentionally misleads the receiver by mimicking the characteristics of the authorized signal, and causes complete communication failure. More specifically, we bring off a dynamic constellation by introducing secure shared randomness between the legitimate transmitter and receiver, and hence break the symmetricity between the authorized signal and the disguised jamming. We analyze the channel capacities of both the traditional OFDM and SP-OFDM under hostile jamming using the arbitrarily varying channel (AVC) model. It is shown that the deterministic coding capacity of the traditional OFDM is zero under the worst disguised jamming. On the other hand, due to the secure randomness shared between the authorized transmitter and receiver, SP-OFDM can achieve a positive capacity under disguised jamming since the AVC channel corresponding to SP-OFDM is not symmetrizable. A remarkable feature of the proposed SP-OFDM scheme is that while achieving strong jamming resistance, it has roughly the same high spectral efficiency as the traditional OFDM system. The robustness of the proposed SP-OFDM scheme under disguised jamming is demonstrated through both theoretic and numerical analyses.

List-Decoding Capacity of the Gaussian Arbitrarily-Varying Channel.

In this paper, we determine the capacity of the Gaussian arbitrarily-varying channel with a (possibly stochastic) encoder and a deterministic list-decoder under the average probability of error criterion. We assume that both the legitimate and the adversarial signals are restricted by their power constraints. We also assume that there is no path between the adversary and the legitimate user but the adversary knows the legitimate user’s code. We show that for any list size L, the capacity is equivalent to the capacity of a point-to-point Gaussian channel with noise variance increased by the adversary power, if the adversary has less power than L times the transmitter power; otherwise, the capacity is zero. In the converse proof, we show that if the adversary has enough power, then the decoder can be confounded by the adversarial superposition of several codewords while satisfying its power constraint with positive probability. The achievability proof benefits from a novel variant of the Csiszár-Narayan method for the arbitrarily-varying channel.

Reliable Communication under the Influence of a State-Constrained Jammer: An Information-Theoretic Perspective on Receive Diversity

The impact of diversity on reliable communication over arbitrarily varying channels (AVC) is investigated as follows. First, the concept of an identical state-constrained jammer is motivated. Second, it is proved that symmetrizability of binary symmetric AVCs (AVBSC) caused by identical state-constrained jamming is circumvented when communication takes place over at least three orthogonal channels. Third, it is proved that the deterministic capacity of the identical state-constrained AVBSC is continuous and shows super-activation. This effect was hitherto demonstrated only for quantum communication and for classical communication under secrecy constraints.

The Arbitrarily Varying Channel Under Constraints With Side Information at the Encoder

We study the arbitrarily varying channel (AVC) with input and state constraints, when the encoder has state information in a causal or noncausal manner. For the causal state information setting, we develop lower and upper bounds on the random code capacity. A lower bound on the deterministic code capacity is established in the case of a message-averaged input constraint. In the setting where a state constraint is imposed on the jammer, while the user is under no constraints, the random code bounds coincide, and the random code capacity is determined. Furthermore, for this scenario, a generalized non-symmetrizability condition is stated, under which the deterministic code capacity coincides with the random code capacity. For the noncausal state information setting, we determine the random code capacity of the AVC under input and state constraints. In addition, a condition on the channel is stated, under which the deterministic code capacity coincides with the random code capacity.

Analytical Properties of Shannon’s Capacity of Arbitrarily Varying Channels under List Decoding: Super-Additivity and Discontinuity Behavior

The common wisdom is that the capacity of parallel channels is usually additive. This was also conjectured by Shannon for the zero-error capacity function, which was later disproved by constructing explicit counterexamples demonstrating the zero-error capacity to be super-additive. Despite these explicit examples for the zero-error capacity, there is surprisingly little known for nontrivial channels. This paper addresses this question for the arbitrarily varying channel (AVC) under list decoding by developing a complete theory. The list capacity function is studied and shown to be discontinuous, and the corresponding discontinuity points are characterized for all possible list sizes. For parallel AVCs it is then shown that the list capacity is super-additive, implying that joint encoding and decoding for two parallel AVCs can yield a larger list capacity than independent processing of both channels. This discrepancy is shown to be arbitrarily large. Furthermore, the developed theory is applied to the arbitrarily varying wiretap channel to address the scenario of secure communication over AVCs.

Communication in the Presence of a State-Aware Adversary

We study communication systems over the state-dependent channels in the presence of a malicious state-aware jamming adversary. The channel has a memoryless state with an underlying distribution. The adversary introduces a jamming signal into the channel. The message and the entire state sequence are known non-causally to both the encoder and the adversary. This state-aware adversary may choose an arbitrary jamming vector depending on the message and the state vector. Taking an arbitrarily varying channel (AVC) approach, we consider two setups, namely, the discrete memoryless Gel’fand–Pinsker AVC and the additive white Gaussian dirty paper (DP) AVC. We determine the randomized coding capacity of both the AVCs under a maximum probability of error criterion. Similar to other randomized coding setups, we show that the capacity is the same even under the average probability of error criterion. Though the adversary can choose an arbitrary vector jamming strategy, we prove that the adversary cannot affect the rate any worse than when it employs a memoryless strategy, which depends only on the instantaneous state. Thus, the AVC capacity characterization is given in terms of the capacity of the worst memoryless channels with state, induced by the adversary employing such memoryless jamming strategies. For the DP-AVC, it is further shown that among memoryless jamming strategies, none impact the communication more than a memoryless Gaussian jamming strategy which completely disregards the knowledge of the state. Thus, the capacity of the DP-AVC equals that of a standard additive white Gaussian noise (AWGN) channel with two independent sources of AWGN, i.e., the channel noise and the jamming noise.

Equivalence for Networks With Adversarial State

We address the problem of finding the capacity of noisy networks with either independent point-to-point compound channels (CC) or arbitrarily varying channels (AVC). These channels model the presence of a Byzantine adversary, which controls a subset of links or nodes in the network. We derive equivalence results showing that these point-to-point channels with state can be replaced by noiseless bit-pipes without changing the network capacity region. Exact equivalence results are found for the CC model, and for some instances of the AVC, including all nonsymmetrizable AVCs. These results show that a feedback path between the output and input of a CC can increase the equivalent capacity, and that if common randomness can be established between the terminals of an AVC (either by a feedback, a forward path, or via a third-party node), then again the equivalent capacity can increase. This leads to an observation that deleting an edge of arbitrarily small capacity can cause a significant change in network capacity. We also analyze an example involving an AVC for which no fixed-capacity bit-pipe is equivalent.

Super-Activation as a Unique Feature of Secure Communication in Malicious Environments

The wiretap channel models secure communication between two users in the presence of an eavesdropper who must be kept ignorant of transmitted messages. This communication scenario is studied for arbitrarily varying channels (AVCs), in which the legitimate users know only that the true channel realization comes from a pre-specified uncertainty set and that it varies from channel use to channel use in an arbitrary and unknown manner. This concept not only captures the case of channel uncertainty, but also models scenarios in which malevolent adversaries influence or jam the transmission of the legitimate users. For secure communication over orthogonal arbitrarily varying wiretap channels (AVWCs) it has been shown that the phenomenon of super-activation occurs; that is, there are orthogonal AVWCs, each having zero secrecy capacity, which allow for transmission with positive rate if they are used together. It is shown that for such orthogonal AVWCs super-activation is generic in the sense that whenever super-activation is possible, it is possible for all AVWCs in a certain neighborhood as well. As a consequence, a super-activated AVWC is robust and continuous in the uncertainty set, although a single AVWC might not be. Moreover, it is shown that the question of super-activation and the continuity of the secrecy capacity solely depends on the legitimate link. Accordingly, the single-user AVC is subsequently studied and it is shown that in this case, super-activation for non-secure message transmission is not possible making it a unique feature of secure communication over AVWCs. However, the capacity for message transmission of the single-user AVC is shown to be super-additive including a complete characterization. Such knowledge is important for medium access control and in particular resource allocation as it determines the overall performance of a system.

Arbitrarily varying multiple access channels with conferencing encoders: List decoding and finite coordination resources

Communication over channels that may vary in an arbitrary and unknown manner from channel use to channel use is studied. Such channels fall in the framework of arbitrarily varying channels (AVCs), for which it has been shown that the classical deterministic approaches with pre-specified encoder and decoder fail if the AVC is symmetrizable. However, more sophisticated strategies such as common randomness (CR) assisted codes or list decoding are capable to resolve the ambiguity induced by symmetrizable AVCs. AVCs further serve as the indispensable basis for modeling adversarial attacks such as jamming in information theoretic security related communication problems. In this paper, we study the arbitrarily varying multiple access channel (AVMAC) with conferencing encoders, which models the communication scenario with two cooperating transmitters and one receiver. This can be motivated for example by cooperating base stations or access points in future systems. The capacity region of the AVMAC with conferencing encoders is established and it is shown that list decoding allows for reliable communication also for symmetrizable AVMACs. The list capacity region equals the CR-assisted capacity region for large enough list size. Finally, for fixed probability of decoding error the amount of resources, i.e., CR or list size, is quantified and shown to be finite.

Capacity results and super-activation for wiretap channels with active wiretappers

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.

Universal Communication Over Arbitrarily Varying Channels

Consider the problem of universally communicating over an arbitrarily varying channel, i.e., a channel comprised of an unknown, arbitrary sequence of memoryless channels. It is shown that there is a communication system using feedback and common randomness that asymptotically attains, with high probability, the capacity of the time-averaged channel, universally for every sequence of channels. This attainable rate is optimal under certain conditions. While no prior knowledge of the channel sequence is assumed, the capacity of the time-averaged channel meets or exceeds the traditional arbitrarily varying channel (AVC) capacity for every memoryless AVC defined over the same alphabets, and therefore, the system universally attains the random code AVC capacity, without knowledge of the AVC parameters. The presented system combines rateless coding with a universal prediction scheme for the input “prior” distribution, from which the codebook is randomly drawn. Because at each point in time, the future of the channel sequence is unknown to the communicators, the adaptation of the input behavior, by universally predicting the prior, plays a major role in the result.

Anti-Jamming Message-Driven Frequency Hopping—Part II: Capacity Analysis Under Disguised Jamming

This is part II of a two-part paper that explores efficient anti-jamming system design based on message-driven frequency hopping (MDFH). In Part I, we point out that under disguised jamming, where the jammer mimics the authorized signal, MDFH experiences considerable performance losses like other wireless systems. To overcome this limitation, we propose an anti-jamming MDFH scheme (AJ-MDFH), which enhances the jamming resistance of MDFH by enabling shared randomness between the transmitter and the receiver using an AES generated ID sequence transmitted along the information stream. In part II, using the arbitrarily varying channel (AVC) model, we analyze the capacity of MDFH and AJ-MDFH under disguised jamming. We show that under the worst case disguised jamming, as long as the secure ID sequence is unavailable to the jammer (which is ensured by AES), the AVC corresponding to AJ-MDFH is nonsymmetrizable. This implies that the deterministic capacity of AJ-MDFH with respect to the average probability of error is positive. On the other hand, due to lack of shared randomness, the AVC corresponding to MDFH is symmetric, resulting in zero deterministic capacity. We further calculate the capacity of AJ-MDFH and show that it converges as the ID constellation size goes to infinity.

List-Decoding for the Arbitrarily Varying Channel Under State Constraints

List-decoding for arbitrarily varying channels (AVCs) under state constraints is investigated. It is shown that rates within ϵ of the randomized coding capacity of AVCs with input-dependent state can be achieved under maximal error with list-decoding using lists of size O(1/ϵ). Under the average error criterion, an achievable rate and converse bound are given for lists of size L . These bounds are based on two different notions of symmetrizability and do not coincide in general. An example is given which shows that for list size L , the capacity may be positive but strictly smaller than the randomized coding capacity, in contrast to the situation without constraints.

Rateless Codes for AVC Models

The arbitrarily varying channel (AVC) is a channel model whose state is selected maliciously by an adversary. Fixed-blocklength coding assumes a worst-case bound on the adversary's capabilities, which leads to pessimistic results. This paper defines a variable-length perspective on this problem, for which achievable rates are shown that depend on the realized actions of the adversary. Specifically, rateless codes are constructed which require a limited amount of common randomness. These codes are constructed for two kinds of AVC models. In the first the channel state cannot depend on the channel input, and in the second it can. As a by-product, the randomized coding capacity of the AVC with state depending on the transmitted codeword is found and shown to be achievable with a small amount of common randomness. The results for this model are proved using a randomized strategy based on list decoding.

On the Deterministic-Code Capacity of the Two-User Discrete Memoryless Arbitrarily Varying General Broadcast Channel With Degraded Message Sets

An inner bound on the deterministic-code capacity region of the two-user discrete memoryless arbitrarily varying general broadcast channel (AVGBC) was characterized by Jahn, assuming that the common message capacity is nonzero; however, he did not indicate how one could decide whether the latter capacity is positive. Csiszaacuter and Narayan's result for the single-user arbitrarily varying channel (AVC) establishes the missing part in Jahn's characterization. Nevertheless, being based on Ahlswede's elimination technique, Jahn's characterization is not applicable for symmetrizable channels under state constraint. Here, the various notions of symmetrizability for the two-user broadcast AVC are defined. Sufficient non-symmetrizability condition that renders the common message capacity of the AVGBC positive is identified using an approach different from Jahn's. The decoding rules we use establish an achievable region under state and input constraints for the family of degraded message sets codes over the AVGBC

Capacity and decoding rules for the poisson arbitrarily varying channel

The single-user and two-user (multiple-access) Poisson arbitrarily varying channel (AVC) with input and state (peak and average power) constraints, but unlimited in bandwidth, are considered. For both cases, the deterministic and random code capacity with the average probability of error criterion is obtained. In the single-user case, A.D. Wyner's decoder (see ibid., vol.34, p.1449-61, 1988) attains the deterministic-code capacity whereas for the two-user case, a nearest neighbor decoder that belongs to the class of /spl beta/-decoders introduced by I. Csiszar and J. Korner (see Proc. 3rd Czechoslovak-Soviet-Hungarian Seminar on Information Theory, p.25-44, 1980) is shown to attain the deterministic-code capacity region as claimed.

Seminoisy deterministic multiple-access channels: coding theorems for list codes and codes with feedback

Whereas the average error capacity region R/sub a/ for the multiple-access channel (MAC) W: X /spl times/ Y /spl rarr/ Z has been known for a long time, very little is known about the capacity region R/sub m/ for the maximal error concept (as predicted by Ahlswede in 1971). In spite of great efforts during the past three decades even for some special examples of deterministic MAC, for which the maximal error concept coincides with the concept of unique decodability, the progress has been slow. It is known that the permission of list codes can be of great help, even if list sizes are of negligible rates (cf. the arbitrarily varying channel (AVC) and, especially, Shannon's (1948) zero-error capacity problem for the one-way channels). Therefore, it is theoretically appealing to look at their regions R/sub m,l/ for the MAC. For a nice class of deterministic MAC, which we call seminoisy, we completely characterized R/sub m,l/. For these channels, the Y-input is determined uniquely by the output. Dueck's (1978) example with R/sub a/ /spl ne/ R/sub m/ and Vanroose's (1988) noiseless binary switching MAC with R/sub a/ = R/sub m/ fall into this class. Finally, for this class, the capacity region R/sub m,f/, which concerns complete feedback, equals R/sub m,l/.

Identification without randomization

In the theory of identification via noisy channels randomization in the encoding has a dramatic effect on the optimal code size, namely, it grows double-exponentially in the blocklength, whereas in the theory of transmission it has the familiar exponential growth. We consider now instead of the discrete memoryless channel (DMC) more robust channels such as the familiar compound (CC) and arbitrarily varying channels (AVC). They can be viewed as models for jamming situations. We make the pessimistic assumption that the jammer knows the input sequence before he acts. This forces communicators to use the maximal error concept and also makes randomization in the encoding superfluous. Now, for a DMC W by a simple observation, made by Ahlswede and Dueck (1989), in the absence of randomization the identification capacity, say C/sub NRI/(W), equals the logarithm of the number of different row-vectors in W. We generalize this to compound channels. A formidable problem arises if the DMC W is replaced by the AVC W. In fact, for 0-1-matrices only in W we are-exactly as for transmission-led to the equivalent zero-error-capacity of Shannon. But for general W the identification capacity C/sub NRI/(W) is quite different from the transmission capacity C(W). An observation is that the separation codes of Ahlswede (1989) are also relevant here. We present a lower bound on C/sub NRI/(W). It implies for instance for W={(/sub 0 1//sup 1 0/), (/sub /spl delta/ (1-/spl delta/)(1 0)/)}, /spl delta//spl isin/(0, 1/2 ) that C/sub NRI/(W)=1, which is obviously tight. It exceeds C(W), which is known to exceed 1-h(/spl delta/), where h is the binary entropy function. We observe that a separation code, with worst case average list size L~ (which we call an NRA code) can be partitioned into L~2/sup ne/ transmission codes. This gives a nonsingle-letter characterization of the capacity of AVC with maximal probability of error in terms of the capacity of codes with list decoding. We also prove that randomization in the decoding does not increase C/sub I/(W) and C/sub NRI/(W). Finally, we draw attention to related work on source coding.

Correlated sources help transmission over an arbitrarily varying channel

It is well known that the deterministic code capacity (for the average error probability criterion) of an arbitrarily varying channel (AVC) either equals its random code capacity or zero. Here it is shown that if two components of a correlated source are additionally available to the sender and receiver, respectively, the capacity always equals its random code capacity.