Gaussian Wiretap Channel Research Articles

Formally Unimodular Packings for the Gaussian Wiretap Channel

Formally Unimodular Packings for the Gaussian Wiretap Channel

Physical Layer Security: Channel Sounding Results for the Multi-Antenna Wiretap Channel.

Many physical-layer security works in the literature rely on purely theoretical work or simulated results to establish the value of physical-layer security in securing communications. We consider the secrecy capacity of a wireless Gaussian wiretap channel using channel sounding measurements to analyze the potential for secure communication in a real-world scenario. A multi-input, multi-output, multi-eavesdropper (MIMOME) system is deployed using orthogonal frequency division multiplexing (OFDM) over an 802.11n wireless network. Channel state information (CSI) measurements were taken in an indoor environment to analyze time-varying scenarios and spatial variations. It is shown that secrecy capacity is highly affected by environmental changes, such as foot traffic, network congestion, and propagation characteristics of the physical environment. We also present a numerical method for calculating MIMOME secrecy capacity in general and comment on the use of OFDM with regard to calculating secrecy capacity.

Amplitude Constrained Vector Gaussian Wiretap Channel: Properties of the Secrecy-Capacity-Achieving Input Distribution.

This paper studies the secrecy capacity of an n-dimensional Gaussian wiretap channel under a peak power constraint. This work determines the largest peak power constraint R¯n, such that an input distribution uniformly distributed on a single sphere is optimal; this regime is termed the low-amplitude regime. The asymptotic value of R¯n as n goes to infinity is completely characterized as a function of noise variance at both receivers. Moreover, the secrecy capacity is also characterized in a form amenable to computation. Several numerical examples are provided, such as the example of the secrecy-capacity-achieving distribution beyond the low-amplitude regime. Furthermore, for the scalar case (n=1), we show that the secrecy-capacity-achieving input distribution is discrete with finitely many points at most at the order of R2σ12, where σ12 is the variance of the Gaussian noise over the legitimate channel.

Short Blocklength Wiretap Channel Codes via Deep Learning: Design and Performance Evaluation

We design short blocklength codes for the Gaussian wiretap channel under information-theoretic security guarantees. Our approach consists in decoupling the reliability and secrecy constraints in our code design. Specifically, we handle the reliability constraint via an autoencoder, and handle the secrecy constraint with hash functions. For blocklengths smaller than or equal to 128, we evaluate through simulations the probability of error at the legitimate receiver and the leakage at the eavesdropper for our code construction. This leakage is defined as the mutual information between the confidential message and the eavesdropper’s channel observations, and is empirically measured via a neural network-based mutual information estimator. Our simulation results provide examples of codes with positive secrecy rates that outperform the best known achievable secrecy rates obtained non-constructively for the Gaussian wiretap channel. Additionally, we show that our code design is suitable for the compound and arbitrarily varying Gaussian wiretap channels, for which the channel statistics are not perfectly known but only known to belong to a pre-specified uncertainty set. These models not only capture uncertainty related to channel statistics estimation, but also scenarios where the eavesdropper jams the legitimate transmission or influences its own channel statistics by changing its location.

Design of Practical Scrambling Schemes for Physical-Layer Security

In physical-layer security, coding and modulation are employed to prevent successful eavesdropping without the need for secret keys. Most promising coding approaches apply channel coding in combination with information-bit scrambling, which leads to error multiplication at the descrambling process. In this letter, the strategies of shift-register based scrambling and block-wise scrambling (using matrix multiplications) are assessed for a Gaussian wiretap channel scenario w.r.t. the security gap metric. We show that the former leads to an unequal protection of communicated message bits. This becomes apparent only upon the evaluation of individual bit positions’ error rates and security gaps. In contrast, the more general block scrambling enables schemes that provide equal protection. An optimized scrambling matrix design is presented, which yields the best error-rate performance. In addition to the theoretical examination of error propagation in the descrambling process, numerical results that combine low-density parity-check codes and scrambling support those findings.

Two-Way Full-Duplex Gaussian Channels With or Without Eavesdropper: Revisit

In this paper, the role of feedback in two-way full-duplex Gaussian channel (TW-FD-GC) is revisited. First, we propose a practical coding scheme for the TW-FD-GC, which is based on the noisy type of the well-known Schalkwijk-Kailath (SK) feedback scheme. We show that though this scheme cannot achieve the capacity of the TW-FD-GC, the codeword length that achieves desired decoding error probability is extremely short. Then, we further prove that the proposed scheme is secure by itself, i.e., the achievable rate region of the proposed scheme is also an achievable <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">secrecy</i> rate region of the two-way full-duplex Gaussian wiretap channel (TW-FD-GWTC). In addition, for the TW-FD-GWTC, numerical result shows that the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">secrecy</i> sum rate of our proposed scheme can be as large as those achieved by existing schemes in the literature for some cases. Next, we extend the proposed scheme to the TW-FD-GC with Gaussian state interference non-causally known by the corresponding transmitter. We show that the interference can be perfectly canceled by this extended scheme. Finally, we prove that this extended scheme is also secure by itself, and numerical result shows that this extended scheme may perform better than the existing ones in the literature.

Secrecy Capacity of a Gaussian Wiretap Channel With ADCs is Always Positive

We consider a complex Gaussian wiretap channel with finite-resolution analog-to-digital converters (ADCs) at both the legitimate receiver and the eavesdropper. For this channel, we show that a positive secrecy rate is always achievable as long as the channel gains at the legitimate receiver and at the eavesdropper are different, regardless of the quantization levels of the ADCs. For the achievability, we first consider the case of the one-bit ADCs at the legitimate receiver and apply a binary input distribution where the two input points have the same phase when the channel gain at the legitimate receiver is less than that at the eavesdropper, and otherwise the opposite phase. Then the result is generalized for the case of arbitrary finite-resolution ADCs at the legitimate receiver by translating the input distribution appropriately. We also provide numerical lower bounds on the achievable secrecy rates, and analyze the tendency of the secrecy rates according to the channel difference, the power constraint, and the quantization levels for some cases. For the special case of the real Gaussian wiretap channel with one-bit ADCs at both the legitimate receiver and the eavesdropper, we show that our choice of phase does not lose any optimality for the Wyner code.

On the Optimality of the Stationary Solution of Secrecy Rate Maximization for MIMO Wiretap Channel

To achieve perfect secrecy in a multiple-input multiple-output (MIMO) Gaussian wiretap channel (WTC), we need to find its secrecy capacity and optimal signaling, which involves solving a difference of convex functions program known to be non-convex for the non-degraded case. To deal with this, a class of existing solutions have been developed but only local optimality is guaranteed by standard convergence analysis. Interestingly, our extensive numerical experiments have shown that these local optimization methods <i>indeed achieve global optimality</i>. In this letter, we provide an analytical proof for this observation. To achieve this, we show that the Karush-Kuhn-Tucker (KKT) conditions of the secrecy rate maximization problem admit a unique solution for <i>both degraded and non-degraded cases</i>. Motivated by this, we also propose a low-complexity algorithm to find a stationary point. Numerical results are presented to verify the theoretical analysis.

On the Zero-Outage Secrecy-Capacity of Dependent Fading Wiretap Channels

It is known that for a slow fading Gaussian wiretap channel without channel state information at the transmitter and with statistically independent fading channels, the outage probability of any given target secrecy rate is non-zero, in general. This implies that the so-called zero-outage secrecy capacity (ZOSC) is zero and we cannot transmit at any positive data rate reliably and confidentially. When the fading legitimate and eavesdropper channels are statistically dependent, this conclusion changes significantly. Our work shows that there exist dependency structures for which positive zero-outage secrecy rates (ZOSR) are achievable. In this paper, we are interested in the characterization of these dependency structures and we study the system parameters in terms of the number of observations at legitimate receiver and eavesdropper as well as average channel gains for which positive ZOSR are achieved. First, we consider the setting that there are two paths from the transmitter to the legitimate receiver and one path to the eavesdropper. We show that by introducing a proper dependence structure among the fading gains of the three paths, we can achieve a zero secrecy outage probability (SOP) for some positive secrecy rate. In this way, we can achieve a non-zero ZOSR. We conjecture that the proposed dependency structure achieves maximum ZOSR. To better understand the underlying dependence structure, we further consider the case where the channel gains are from finite alphabets and systematically and globally solve the ZOSC. In addition, we apply the rearrangement algorithm to solve the ZOSR for continuous channel gains. The results indicate that the legitimate link must have an advantage in terms of the number of antennas and average channel gains to obtain positive ZOSR. The results motivate further studies into the optimal dependency structures.

Self-Secure Capacity-Achieving Feedback Schemes of Gaussian Multiple-Access Wiretap Channels With Degraded Message Sets

It has been shown that the SK scheme, which was proposed by Schalkwijk and Kailath, is a self-secure capacity-achieving (SSCA) feedback scheme for the Gaussian wiretap channel, i.e., the SK scheme not only achieves the feedback capacity of the Gaussian channel, but also is secure by itself and achieves the feedback secrecy capacity of the Gaussian wiretap channel. For the multi-user wiretap channels, very recently, it has been shown that Ozarow’s capacity-achieving feedback scheme for the two-user Gaussian multiple-access channel (GMAC) is the SSCA feedback scheme for the two-user Gaussian multiple-access wiretap channel (GMAC-WT). In this paper, first, we propose a SSCA feedback scheme for the two-user GMAC-WT with degraded message sets (GMAC-WT-DMS). Next, we extend the above scheme to the two-user GMAC-WT-DMS with noncausal channel state information at the transmitters (NCSIT), and show that the extended scheme is also a SSCA feedback scheme. Finally, we derive outer bounds on the secrecy capacity regions of the two-user GMAC-WT-DMS with or without NCSIT, and numerical results show the rate gains by the feedback.

Security Gap Improvement of BICM Systems Through Bit-Labeling Optimization for the Gaussian Wiretap Channel

Security Gap Improvement of BICM Systems Through Bit-Labeling Optimization for the Gaussian Wiretap Channel

New Result on the Feedback Capacity of the Action-Dependent Dirty Paper Wiretap Channel.

The Gaussian wiretap channel with noncausal state interference available at the transmitter, which is also called the dirty paper wiretap channel (DP-WTC), has been extensively studied in the literature. Recently, it has been shown that taking actions on the corrupted state interference of the DP-WTC (also called the action-dependent DP-WTC) helps to increase the secrecy capacity of the DP-WTC. Subsequently, it has been shown that channel feedback further increases the secrecy capacity of the action-dependent DP-WTC (AD-DP-WTC), and a sub-optimal feedback scheme is proposed for this feedback model. In this paper, a two-step hybrid scheme and a corresponding new lower bound on the secrecy capacity of the AD-DP-WTC with noiseless feedback are proposed. The proposed new lower bound is shown to be optimal (achieving the secrecy capacity) and tighter than the existing one in the literature for some cases, and the results of this paper are further explained via numerical examples.

Learning end-to-end codes for the BPSK-constrained Gaussian wiretap channel

Finite-length codes are learned for the Gaussian wiretap channel in an end-to-end manner assuming that the communication parties are equipped with deep neural networks (DNNs), and communicate through binary phase-shift keying (BPSK) modulation scheme. The goal is to find codes via DNNs which allow a pair of transmitter and receiver to communicate reliably and securely in the presence of an adversary aiming at decoding the secret messages. Following the information-theoretic secrecy principles, the security is evaluated in terms of mutual information utilizing a deep learning tool called MINE (mutual information neural estimation). System performance is evaluated for different DNN architectures, designed based on the existing secure coding schemes, at the transmitter. Numerical results demonstrate that the legitimate parties can indeed establish a secure transmission in this setting as the learned codes achieve points on almost the boundary of the equivocation region.

Improved physical layer security of visible light communications with focused light emitters

A conventional secure visible light communication (VLC) system that is composed of one transmitter, one friendly jammer with multiple light-emitting-diodes, one legitimate receiver and one eavesdropper is considered. We propose the use of an optical lens at the legitimate transmitter as an additional effective and practical security measure. We investigate the secrecy rate of a Gaussian wiretap channel VLC scenario inside a room. First, we consider friendly jamming through optimal power allocation between the information and the jamming signal. Then, we investigate the improved security through the use of the optical lens into the system. The use of this collimating thin lens spatially limits the transmitted information to a confined and adjustable area that is beneath the transmitter. Through simulation results we demonstrate that the use of the optical lens can significantly improves the secrecy rate by focusing the information transmission. We also design and implement an experimental test-bed and demonstrate the applicability of the proposed approach. We experimentally validate that the system is practically realizable, and the numerical simulations are consistent with our measurement results. Although the proposed scenario necessitates the receiving unit to be located in a stationary position, we showcase that the hardware implementation is simple, practical, and significantly lower in cost as opposed to a beam-forming type VLC system that offers directional beam forming to send the information towards a dynamic receiver. Furthermore, through a careful adjustment of the distance between the transmitter and the lens, it is possible to control the size of the region where a receiver can access the data. Both experimental and simulation results show that, with the utilization of a lens at the transmitter, VLC is a promising candidate for applications where the secrecy is a critical concern.

Optimal Secure GDoF of Symmetric Gaussian Wiretap Channel With a Helper

We study a symmetric Gaussian wiretap channel with a helper, where a confidential message is sent from a transmitter to a legitimate receiver, in the presence of a helper and an eavesdropper, under a weak notion of secrecy constraint. For this setting, we characterize the optimal secure generalized degrees-of-freedom (GDoF). The result reveals that, adding a helper can significantly increase the secure GDoF of the wiretap channel. The result is supported by a new converse and a new scheme. In the proposed scheme, the helper sends a cooperative jamming signal at a specific power level and direction. In this way, it minimizes the penalty in GDoF incurred by the secrecy constraint. In the secure rate analysis, the techniques of noise removal and signal separation are used.

Adding Common Randomness Can Remove the Secrecy Penalty in GDoF

In communication networks secrecy constraints usually incur an extra limit in capacity or generalized degrees-of-freedom (GDoF), in the sense that a penalty in capacity or GDoF is incurred due to the secrecy constraints. Over the past decades a significant amount of effort has been made by the researchers to understand the limits of secrecy constraints in communication networks. In this work, we focus on how to remove the secrecy penalty in communication networks, i.e., how to remove the GDoF penalty due to secrecy constraints. We begin with three basic settings: a two-user symmetric Gaussian interference channel with confidential messages, a symmetric Gaussian wiretap channel with a helper, and a two-user symmetric Gaussian multiple access wiretap channel. Interestingly, in this work we show that adding common randomness at the transmitters can totally remove the penalty in GDoF or GDoF region of the three settings considered here. The results reveal that adding common randomness at the transmitters is a powerful way to remove the secrecy penalty in communication networks in terms of GDoF performance. Common randomness can be generated offline before the real-time message communication. The role of the common randomness is to jam the information signal at the eavesdroppers, without causing too much interference at the legitimate receivers. To accomplish this role, a new method of Markov chain-based interference neutralization is proposed in the achievability schemes utilizing common randomness. From the practical point of view, we need to minimize the amount of common randomness used for removing the secrecy penalty in terms of GDoF performance. With this motivation, for most of the cases we characterize the minimal GDoF of common randomness to remove secrecy penalty, based on our derived converses and achievability.

New Capacity Results for Fading Gaussian Multiuser Channels With Statistical CSIT

In this paper, fast fading Gaussian multiuser channels are considered. If the channel state information (CSI) is perfectly known to the transmitter, capacities have been derived for many cases in which the channels satisfy certain information-theoretic orders such as degradedness, or strong/very strong interference. We study the case when only the statistics of the CSI are known at the transmitter, which is an open problem in general. The main contributions of this paper are the following: First, we derive a sufficient condition to construct equivalent (having the same capacity region as the original channel) degraded Gaussian broadcast channels, which is based on the marginal distributions of the channel gains. To achieve this goal, we leverage three schemes: coupling, maximal coupling, and copulas, in addition to the same marginal property. The underlying idea of all schemes is to obtain an equivalent channel by changing the joint distribution in such a way that it satisfies a certain information-theoretic order while ensuring that the marginal distributions of the channels to different users are not changed. The construction of this equivalent multiuser channel allows us to directly apply existing capacity results. We then further derive the capacity regions of Gaussian interference channels with strong and very strong interferences, and also the secrecy capacity of Gaussian wiretap channels, while in all cases, the transmitters know only the statistics of the channels. Several practical examples such as Rayleigh fading and Nakagami- ${m}$ fading illustrate the applicability of the derived results.

An Encryption-Aware PHY Security Framework for 4-Node Gaussian Wiretap Channels With Joint Power Constraint

In traditional physical layer security paradigm, no leakage of confidential information to the eavesdropper is tolerated regardless of whether the message is encrypted or not. This will result in an achievable secure transmission rate that is significantly smaller than the channel capacity. This article presents a novel approach that allows treatment of physical layer security in conjunction with encryption to achieve a flexible trade-off of system resources. We propose a novel framework to model the interplay between secured transmission rate and error probability in error prone ciphertexts. To this end, we use the concept of rate-equivocation region to establish such a connection. To clearly describe the application of our framework, we consider the case of 4-node Gaussian wiretap channel. For such a channel, we characterize the rate-equivocation region in different scenarios, and then use it to study the achievable rate of encryption-aware physical layer security. The obtained results show that, for a fixed transmission power, the prior knowledge of encryption can significantly increase the secured transmission rate. In addition, encryption-aware physical layer security can achieve a target transmission rate at a reduced transmission power compared to the conventional encryption-agnostic physical layer security.

Semantic Security via Seeded Modular Coding Schemes and Ramanujan Graphs

A novel type of functions called biregular irreducible functions is introduced and applied as security components (instead of, e.g., universal hash functions) in seeded modular wiretap coding schemes, whose second component is an error-correcting code. These schemes are called modular BRI schemes. An upper bound on the semantic security information leakage of modular BRI schemes in a one-shot setting is derived which separates the effects of the biregular irreducible function on the one hand and the error-correcting code plus the channel on the other hand. The effect of the biregular irreducible function is described by the second-largest eigenvalue of an associated stochastic matrix. A characterization of biregular irreducible functions is given in terms of connected edge-disjoint biregular graphs. It allows for the construction of new biregular irreducible functions from families of edge-disjoint Ramanujan graphs, which are shown to exist. A frequently used arithmetic universal hash function can be converted into a biregular irreducible function for certain parameters. Sequences of Ramanujan biregular irreducible functions are constructed which exhibit an optimal trade-off between the size of the regularity set and the rate of decrease of the associated second-largest eigenvalue. Together with the one-shot bound on the information leakage, the existence of these sequences implies an asymptotic coding result for modular BRI schemes applied to discrete and Gaussian wiretap channels. It shows that the separation of error correction and security as done in a modular BRI scheme is secrecy capacity-achieving for discrete and Gaussian wiretap channels. The same holds for a derived construction where the seed is generated locally by the sender and reused several times. Finally, optimal sequences of biregular irreducible functions used in the above constructions must be nearly Ramanujan.

On Secure Communications Over Gaussian Wiretap Channels via Finite-Length Codes

Practical codes for the Gaussian wiretap channel are designed aiming at satisfying information-theoretic metrics to ensure security against a passive eavesdropper (Eve). Specifically, a design criterion is introduced for the coset coding scheme in order to satisfy a strong secrecy condition described with the mutual information between the secret message and Eve's observation. In addition, mutual information neural estimation (MINE) powered from deep learning tools is applied in order to directly compute the information-theoretic security constraint, and verify the proposed solutions. It is shown that finite-length coset codes can indeed ensure secure transmission from an information-theoretic perspective.