Channel Resolvability Research Articles

Quantum soft-covering lemma with applications to rate-distortion coding, resolvability and identification via quantum channels

We propose a quantum soft-covering problem for a given general quantum channel and one of its output states, which consists in finding the minimum rank of an input state needed to approximate the given channel output. We then prove a one-shot quantum covering lemma in terms of smooth min-entropies by leveraging decoupling techniques from quantum Shannon theory. This covering result is shown to be equivalent to a coding theorem for rate distortion under a posterior (reverse) channel distortion criterion by two of the present authors. Both one-shot results directly yield corollaries about the i.i.d. asymptotics, in terms of the coherent information of the channel. The power of our quantum covering lemma is demonstrated by two additional applications: first, we formulate a quantum channel resolvability problem, and provide one-shot as well as asymptotic upper and lower bounds. Second, we provide new upper bounds on the unrestricted and simultaneous identification capacities of quantum channels, in particular separating for the first time the simultaneous identification capacity from the unrestricted one, proving a long-standing conjecture of the last author.

Smoothing of Binary Codes, Uniform Distributions, and Applications.

The action of a noise operator on a code transforms it into a distribution on the respective space. Some common examples from information theory include Bernoulli noise acting on a code in the Hamming space and Gaussian noise acting on a lattice in the Euclidean space. We aim to characterize the cases when the output distribution is close to the uniform distribution on the space, as measured by the Rényi divergence of order α∈(1,∞]. A version of this question is known as the channel resolvability problem in information theory, and it has implications for security guarantees in wiretap channels, error correction, discrepancy, worst-to-average case complexity reductions, and many other problems. Our work quantifies the requirements for asymptotic uniformity (perfect smoothing) and identifies explicit code families that achieve it under the action of the Bernoulli and ball noise operators on the code. We derive expressions for the minimum rate of codes required to attain asymptotically perfect smoothing. In proving our results, we leverage recent results from harmonic analysis of functions on the Hamming space. Another result pertains to the use of code families in Wyner's transmission scheme on the binary wiretap channel. We identify explicit families that guarantee strong secrecy when applied in this scheme, showing that nested Reed-Muller codes can transmit messages reliably and securely over a binary symmetric wiretap channel with a positive rate. Finally, we establish a connection between smoothing and error correction in the binary symmetric channel.

Variable-Length Resolvability for General Sources and Channels.

We introduce the problem of variable-length (VL) source resolvability, in which a given target probability distribution is approximated by encoding a VL uniform random number, and the asymptotically minimum average length rate of the uniform random number, called the VL resolvability, is investigated. We first analyze the VL resolvability with the variational distance as an approximation measure. Next, we investigate the case under the divergence as an approximation measure. When the asymptotically exact approximation is required, it is shown that the resolvability under two kinds of approximation measures coincides. We then extend the analysis to the case of channel resolvability, where the target distribution is the output distribution via a general channel due to a fixed general source as an input. The obtained characterization of channel resolvability is fully general in the sense that, when the channel is just an identity mapping, it reduces to general formulas for source resolvability. We also analyze the second-order VL resolvability.

Multiple Access Channel Resolvability Codes From Source Resolvability Codes

We show that the problem of code construction for multiple access channel (MAC) resolvability can be reduced to the simpler problem of code construction for source resolvability. Specifically, we propose a MAC resolvability code construction that relies on a combination of multiple source resolvability codes, used in a black-box manner, and leverages randomness recycling implemented via distributed hashing and block-Markov coding. Since explicit source resolvability codes are known, our results also yield the first explicit coding schemes that achieve the entire MAC resolvability region for any discrete memoryless multiple-access channel with binary input alphabets.

Physical-layer Security from Channel Resolvability

This paper discusses the performance of achieving physical layer security from channel resolvability in terms of secrecy and offers a kind of classification for different detection methods, including conventional hypothesis testing and active hypothesis testing. The difference between them is whether or not the detection time is a random variable. Moreover, this paper has a discussion on the applications of the practical case studies based on polar code from the channel resolvability, such as wireless channels with or without state information, point to point channel with an eavesdropper, multiple access channel with an eavesdropper, and communication networks at the physical layer. The channel resolvability is shown to be a promising method to achieve secrecy and covertness in the physical-layer.

Minimax Converse for Identification via Channels

A minimax converse for the identification via channels is derived. By this converse, a general formula for the identification capacity, which coincides with the transmission capacity, is proved without the assumption of the strong converse property. Furthermore, the optimal second-order coding rate of the identification via channels is characterized when the type I error probability is non-vanishing and the type II error probability is vanishing. Our converse is built upon the so-called partial channel resolvability approach; however, the minimax argument enables us to circumvent a flaw reported in the literature.

Covert Identification Over Binary-Input Discrete Memoryless Channels

This paper considers the covert identification problem in which a sender aims to reliably convey an identification (ID) message to a set of receivers via a binary-input discrete memoryless channel (BDMC), and simultaneously to guarantee that the communication is covert with respect to a warden who monitors the communication via another independent BDMC. We prove a square-root law for the covert identification problem. This states that an ID message of size exp(exp(Θ(√{ n})) can be transmitted over n channel uses. We then characterize the exact pre-constant in the Θ(·) notation. This constant is referred to as the covert identification capacity. We show that it equals the recently developed covert capacity in the standard covert communication problem, and somewhat surprisingly, the covert identification capacity can be achieved without any shared key between the sender and receivers. The achievability proof relies on a random coding argument with pulse-position modulation (PPM), coupled with a second stage which performs code refinements. The converse proof relies on an expurgation argument as well as results for channel resolvability with stringent input constraints.

A Unified Framework for One-Shot Achievability via the Poisson Matching Lemma

We introduce a fundamental lemma called the Poisson matching lemma, and apply it to prove one-shot achievability results for various settings, namely channels with state information at the encoder, lossy source coding with side information at the decoder, joint source-channel coding, broadcast channels, distributed lossy source coding, multiple access channels, channel resolvability and wiretap channels. Our one-shot bounds improve upon the best known one-shot bounds in most of the aforementioned settings (except multiple access channels, channel resolvability and wiretap channels, where we recover bounds comparable to the best known bounds), with shorter proofs in some settings even when compared to the conventional asymptotic approach using typicality. The Poisson matching lemma replaces both the packing and covering lemmas, greatly simplifying the error analysis. This paper extends the work of Li and El Gamal on Poisson functional representation, which mainly considered variable-length source coding settings, whereas this paper studies fixed-length settings, and is not limited to source coding, showing that the Poisson functional representation is a viable alternative to typicality for most problems in network information theory.

Cooperative Resolvability and Secrecy in the Cribbing Multiple-Access Channel

We study channel resolvability for the discrete memoryless multiple-access channel with cribbing, i.e., the characterization of the amount of randomness required at the inputs to approximately produce a chosen i.i.d. output distribution according to Kullback-Leibler divergence. We analyze resolvability rates when one encoder cribs (i) the input of the other encoder; or the output of the other encoder, (ii) non-causally, (iii) causally, or (iv) strictly-causally. For scenarios (i)-(iii), we exactly characterize the channel resolvability region. For (iv), we provide inner and outer bounds for the channel resolvability region; the crux of our achievability result is to handle the strict causality constraint with a block-Markov coding scheme in which dependencies across blocks are suitably hidden. Finally, we leverage the channel resolvability results to derive achievable secrecy rate regions for each of the cribbing scenarios under strong secrecy constraints.

Secure Communication Over Fully Quantum Gel’fand-Pinsker Wiretap Channel

In this work we study the problem of secure communication over a fully quantum Gel'fand-Pinsker channel. The best known achievability rate for this channel model in the classical case was proven by Goldfeld, Cuff and Permuter in [Goldfeld, Cuff, Permuter, 2016]. We generalize the result of [Goldfeld, Cuff, Permuter, 2016]. One key feature of the results obtained in this work is that all the bounds obtained are in terms of error exponent. We obtain our achievability result via the technique of simultaneous pinching. This in turn allows us to show the existence of a simultaneous decoder. Further, to obtain our encoding technique and to prove the security feature of our coding scheme we prove a bivariate classical-quantum channel resolvability lemma and a conditional classical-quantum channel resolvability lemma. As a by product of the achievability result obtained in this work, we also obtain an achievable rate for a fully quantum Gel'fand-Pinsker channel in the absence of Eve. The form of this achievable rate matches with its classical counterpart. The Gel'fand-Pinsker channel model had earlier only been studied for the classical-quantum case and in the case where Alice (the sender) and Bob (the receiver) have shared entanglement between them.

Rényi Resolvability and Its Applications to the Wiretap Channel

The conventional channel resolvability problem refers to the determination of the minimum rate required for an input process so that the output distribution approximates a target distribution in either the total variation distance or the relative entropy. In contrast to previous works, in this paper, we use the (normalized or unnormalized) Rényi divergence (with the Rényi parameter in[0, 2] U {∞}) to measure the level of approximation. We also provide asymptotic expressions for normalized Rényi divergence when the Rényi parameter is larger than or equal to 1 as well as (lower and upper) bounds for the case when the same parameter is smaller than 1. We characterize the Rényi resolvability, which is defined as the minimum rate required to ensure that the Rényi divergence vanishes asymptotically. The Rényi resolvabilities are the same for both the normalized and unnormalized divergence cases. In addition, when the Rényi parameter smaller than 1, consistent with the traditional case where the Rényi parameter is equal to 1, the Rényi resolvability equals the minimum mutual information over all input distributions that induce the target output distribution. When the Rényi parameter is larger than 1 the Rényi resolvability is, in general, larger than the mutual information. The optimal Rényi divergence is proven to vanish at least exponentially fast for both of these two cases, as long as the code rate is larger than the Rényi resolvability. The optimal exponential rate of decay for i.i.d. random codes is also characterized exactly. We apply these results to the wiretap channel, and completely characterize the optimal tradeoff between the rates of the secret and non-secret messages when the leakage measure is given by the (unnormalized) Rényi divergence. This tradeoff differs from the conventional setting when the leakage is measured by the traditional mutual information.

Empirical and Strong Coordination via Soft Covering With Polar Codes

We design polar codes for empirical coordination and strong coordination in two-node networks. Our constructions hinge on the fact that polar codes enable explicit low-complexity schemes for soft covering. We leverage this property to propose explicit and low-complexity coding schemes that achieve the capacity regions of both empirical coordination and strong coordination for sequences of actions taking value in an alphabet of prime cardinality. Our results improve previously known polar coding schemes, which (i) were restricted to uniform distributions and to actions obtained via binary symmetric channels for strong coordination, (ii) required a non-negligible amount of common randomness for empirical coordination, and (iii) assumed that the simulation of discrete memoryless channels could be perfectly implemented. As a by-product of our results, we obtain a polar coding scheme that achieves channel resolvability for an arbitrary discrete memoryless channel whose input alphabet has prime cardinality.

Strong Coordination Over Multi-Hop Line Networks Using Channel Resolvability Codebooks

We analyze the problem of strong coordination over a multi-hop line network in which the node initiating the coordination is a terminal network node. We assume that each node has access to a certain amount of randomness that is local to the node, and that the nodes also have shared common randomness, which are used together with explicit hop-by-hop communication to achieve information-theoretic strong coordination. We derive the trade-offs among the required rates of communication on the network links, the rates of local randomness available at network nodes, and the rate of common randomness to realize strong coordination. We present an achievable coding scheme built using multiple layers of channel resolvability codes, and establish several settings in which this scheme offers the best possible trade-offs among network resources.

Strong Secrecy for Interference Channels Based on Channel Resolvability

Interference channels with confidential messages are studied under strong secrecy constraints, based on the framework of channel resolvability theory. It is shown that if the random binning rate for securing a confidential message is above the resolution of its corresponding wiretapped channel, strong secrecy can be guaranteed. The information-spectrum method introduced by Han and Verdu is generalized to an arbitrary interference channel to obtain a direct channel resolvability result as a first step. For stationary and memoryless channels with discrete output alphabets, the results show that the achievable rates under weak and strong secrecy constraints are the same. This result is then generalized to channels with continuous output alphabets by deriving a reverse direction of Pinsker’s inequality to bound the secrecy measure from above by a function of the variational distance of relevant distributions. As an application, Gaussian interference channels are studied in which the agreement between the best known weak and strong secrecy rate regions also appear. Following the footsteps of Csiszar, Hayashi and of Bloch and Laneman, these results provide further evidence that channel resolvability is a powerful and general framework for strong secrecy analysis in multiuser networks.

Achieving Full Secure Degrees-of-Freedom for the MISO Wiretap Channel With an Unknown Eavesdropper

In this paper, we study the achievable secure degrees-of-freedom (sdof) for the multiple-input single-output (MISO) wiretap channel with an unknown eavesdropper. It is assumed that the eavesdropper’s (Eve’s) channel state information (CSI) is unknown to the transmitter (Alice) and legitimate receiver (Bob). Recent studies have shown that the achievable sdof in the sense of strong secrecy is zero when Eve’s number of antennas is equal to or more than Bob’s number of antennas, which is the scenario considered in this paper. To this end, we propose a novel precoding technique and a coding strategy that together achieve full sdof in the sense of strong secrecy without knowing Eve’s CSI and without using artificial noise. The proposed precoding method uses the CSI of the Alice-Bob channel in a nonlinear fashion, which makes the transmitted symbols undecodable at Eve. The proposed coding scheme is based on the channel resolvability concept and ensures strong secrecy. Achieving full sdof with an unknown Eve’s CSI is significant, because it is contrary to what is believed about the achievable sdof for the MISO wiretap channel in the sense of strong secrecy. We also show that the proposed scheme achieves near Alice-Bob’s channel capacity in the sense of strong secrecy with a probability approaching one at finite signal-to-noise ratio.

Secure Multiplex Coding With Dependent and Non-Uniform Multiple Messages

The secure multiplex coding (SMC) is a technique to remove rate loss in the coding for wire-tap channels and broadcast channels with confidential messages caused by the inclusion of random bits into transmitted signals. SMC replaces the random bits by other meaningful secret messages, and a collection of secret messages serves as the random bits to hide the rest of messages. In the previous researches, multiple secret messages were assumed to have independent and uniform distributions, which is difficult to be ensured in practice. We remove this restrictive assumption by a generalization of the channel resolvability technique. We also give practical construction techniques for SMC by using an arbitrary given error-correcting code as an ingredient, and channel-universal coding of SMC. By using the same principle as the channel-universal SMC, we give coding for the broadcast channel with confidential messages universal to both channel and source distributions.

Covert Communication Over Noisy Channels: A Resolvability Perspective

We consider the situation in which a transmitter attempts to communicate reliably over a discrete memoryless channel while simultaneously ensuring covertness (low probability of detection) with respect to a warden, who observes the signals through another discrete memoryless channel. We develop a coding scheme based on the principle of channel resolvability, which generalizes and extends prior work in several directions. First, it shows that, irrespective of the quality of the channels, it is possible to communicate on the order of $\sqrt{n}$ reliable and covert bits over $n$ channel uses if the transmitter and the receiver share on the order of $\sqrt{n}$ key bits; this improves upon earlier results requiring on the order of $\sqrt{n}\log n$ key bits. Second, it proves that, if the receiver's channel is "better" than the warden's channel in a sense that we make precise, it is possible to communicate on the order of $\sqrt{n}$ reliable and covert bits over $n$ channel uses without a secret key; this generalizes earlier results established for binary symmetric channels. We also identify the fundamental limits of covert and secret communications in terms of the optimal asymptotic scaling of the message size and key size, and we extend the analysis to Gaussian channels. The main technical problem that we address is how to develop concentration inequalities for "low-weight" sequences; the crux of our approach is to define suitably modified typical sets that are amenable to concentration inequalities.

$E_{\gamma}$-Resolvability

The conventional channel resolvability refers to the minimum rate needed for an input process to approximate the channel output distribution in total variation distance. In this paper, we study $E_{\gamma }$ -resolvability, in which total variation is replaced by the more general $E_{\gamma }$ distance. A general one-shot achievability bound for the precision of such an approximation is developed. Let $Q_{\sf X|U}$ be a random transformation, $n$ be an integer, and $E\in (0,+\infty )$ . We show that in the asymptotic setting where $\gamma =\exp (nE)$ , a (nonnegative) randomness rate above $\inf _{Q_{\sf U}: D(Q_{\sf X}\|{\pi }_{\sf X})\le E} \{D(Q_{\sf X}\|{\pi }_{\sf X})+I(Q_{\sf U},Q_{\sf X|U})-E\}$ is sufficient to approximate the output distribution ${\pi }_{\sf X}^{\otimes n}$ using the channel $Q_{\sf X|U}^{\otimes n}$ , where $Q_{\sf U}\to Q_{\sf X|U}\to Q_{\sf X}$ , and is also necessary in the case of finite $\mathcal {U}$ and $\mathcal {X}$ . In particular, a randomness rate of $\inf _{Q_{\sf U}}I(Q_{\sf U},Q_{\sf X|U})-E$ is always sufficient. We also study the convergence of the approximation error under the high-probability criteria in the case of random codebooks. Moreover, by developing simple bounds relating $E_{\gamma }$ and other distance measures, we are able to determine the exact linear growth rate of the approximation errors measured in relative entropy and smooth Renyi divergences for a fixed-input randomness rate. The new resolvability result is then used to derive: 1) a one-shot upper bound on the probability of excess distortion in lossy compression, which is exponentially tight in the i.i.d. setting; 2) a one-shot version of the mutual covering lemma; and 3) a lower bound on the size of the eavesdropper list to include the actual message and a lower bound on the eavesdropper false-alarm probability in the wiretap channel problem, which is (asymptotically) ensemble-tight.

Error-Control Coding for Physical-Layer Secrecy

The renewed interest for physical-layer security techniques has put forward a new role for error-control codes. In addition to ensuring reliability, carefully designed codes have been shown to provide a level of information-theoretic secrecy, by which the amount of information leaked to an adversary may be controlled. The ability to achieve information-theoretic secrecy relies on the study of alternative coding mechanisms, such as channel resolvability and privacy amplification, in which error-control codes are exploited as a means to shape the distribution of stochastic processes. This use of error-control codes, which goes much beyond that of correcting errors, creates numerous new design challenges. The objective of this paper is threefold. First, the paper aims at providing system engineers with explicit tools to build simple secrecy codes in order to stimulate interest and foster their integration in communication system prototypes. Second, it aims at providing coding and information theorists with a synthetic overview of the theoretical concepts and techniques for secrecy. Finally, it aims at highlighting the open challenges and opportunities faced for the integration of these codes in practical systems.

Nonasymptotic and Second-Order Achievability Bounds for Coding With Side-Information

We present a novel nonasymptotic or finite blocklength achievability bounds for three side-information problems in network information theory. These include: 1) the Wyner-Ahlswede-Korner (WAK) problem of almost-lossless source coding with rate-limited side-information; 2) the Wyner–Ziv (WZ) problem of lossy source coding with side-information at the decoder; and 3) the Gel’fand–Pinsker (GP) problem of channel coding with noncausal state information available at the encoder. The bounds are proved using ideas from channel simulation and channel resolvability. Our bounds for all three problems improve on all previous nonasymptotic bounds on the error probability of the WAK, WZ, and GP problems—in particular those derived by Verdu. Using our novel nonasymptotic bounds, we recover the general formulas for the optimal rates of these side-information problems. Finally, we also present achievable second-order coding rates by applying the multidimensional Berry–Esseen theorem to our new nonasymptotic bounds. Numerical results show that the second-order coding rates obtained using our nonasymptotic achievability bounds are superior to those obtained using existing finite blocklength bounds.