Gaussian Wiretap Channel Research Articles

Impact of Action-Dependent State and Channel Feedback on Gaussian Wiretap Channels

We investigate the state-dependent Gaussian wiretap channel with noncausal channel state information at the transmitter (GWTC-N-CSIT), and explore whether three strategies (i.e., taking action on the state, legitimate receiver’s channel output feedback, and combining the former two strategies together) help to enhance the secrecy capacity of the GWTC-N-CSIT. To be specific, we first determine the secrecy capacity of the GWTC-N-CSIT with noiseless feedback. Next, we derive lower and upper bounds on the secrecy capacity of the GWTC-N-CSIT with action-dependent state. Finally, we derive lower and upper bounds on the secrecy capacity of the GWTC-N-CSIT with both action-dependent state and noiseless feedback, and show that these bounds meet for a special case. Numerical results of this paper indicate that all three strategies enhance the secrecy capacity of the GWTC-N-CSIT. The study of this paper offers new options for enhancing the secrecy rates of the state-dependent wiretap channel models.

Impact of Action-Dependent State and Channel Feedback on Gaussian Wiretap Channels

Impact of Action-Dependent State and Channel Feedback on Gaussian Wiretap Channels

Double-Layered Random Coding for Secret Key Generation in Gaussian Wiretap Channels

Most of the existing information-theoretic secure key generation schemes have characterized maximum achievable secrecy rate or secrecy capacity over noiseless public broadcast channels. However, in many practical systems even legitimate nodes have to communicate over noisy channels. In this letter, we propose a secret key generation scheme based on secure network coding in Gaussian wiretap channels. We prove that even when the channel from the sender to the receiver is weaker than that to the eavesdropper, secret communication at a positive rate is still possible if the condition $\sigma _{1} is satisfied. Moreover, we establish an upper bound on the secrecy leakage rate after privacy amplification using double-layered random coding and joint typicality encoding and decoding over noisy public channels. The proof of achievability and the converse is given to demonstrate that the proposed scheme can achieve information-theoretic security.

Semantically Secure Lattice Codes for Compound MIMO Channels

We consider compound multi-input multi-output (MIMO) wiretap channels where minimal channel state information at the transmitter (CSIT) is assumed. Code construction is given for the special case of isotropic mutual information, which serves as a conservative strategy for general cases. Using the flatness factor for MIMO channels, we propose lattice codes universally achieving the secrecy capacity of compound MIMO wiretap channels up to a constant gap (measured in nats) that is equal to the number of transmit antennas. The proposed approach improves upon existing works on secrecy coding for MIMO wiretap channels from an error probability perspective, and establishes information theoretic security (in fact semantic security). We also give an algebraic construction to reduce the code design complexity, as well as the decoding complexity of the legitimate receiver. Thanks to the algebraic structures of number fields and division algebras, our code construction for compound MIMO wiretap channels can be reduced to that for Gaussian wiretap channels, up to some additional gap to secrecy capacity.

A Partially Coupled LDPC Coded Scheme for the Gaussian Wiretap Channel

In this letter, we propose an effective partially coupled low-density parity-check coded scheme to reduce the security gap for the Gaussian wiretap channel. In our scheme, codewords are partially coupled by exclusive OR adding the message bits from a codeword to the parity check bits from its consecutive codeword. An iterative decoding algorithm is also proposed to utilize the correlations between codewords. Based on this design, error propagation effect between codewords is achieved to worsen the performance at the eavesdropper, which receives messages via a poor conditioned channel. Numerical results demonstrate that the proposed scheme outperforms the punctured scheme in terms of the equivocation rate and the security gap.

Wiretap Code Design by Neural Network Autoencoders

In industrial machine type communications, an increasing number of wireless devices communicate under reliability, latency, and confidentiality constraints, simultaneously. From information theory, it is known that wiretap codes can asymptotically achieve reliability (vanishing block error rate (BLER) at the legitimate receiver Bob) while also achieving secrecy (vanishing information leakage (IL) to an eavesdropper Eve). However, under finite block length, there exists a tradeoff between the BLER at Bob and the IL at Eve. In this work, we propose a flexible wiretap code design for degraded Gaussian wiretap channels under finite block length, which can change the operating point on the Pareto boundary of the tradeoff between BLER and IL given specific code parameters. To attain this goal, we formulate a multi-objective programming problem, which takes the BLER at Bob and the IL at Eve into account. During training, we approximate the BLER by the mean square error and the IL by schemes based on Jensen's inequality and the Taylor expansion and then solve the optimization problem by neural network autoencoders. Simulation results show that the proposed scheme can find codes outperforming polar wiretap codes (PWC) with respect to both BLER and IL simultaneously. We show that the codes found by the autoencoders could be implemented with real modulation schemes with only small losses in performance.

Secrecy Capacity of Colored Gaussian Noise Channels With Feedback

In this paper, the finite-order autoregressive moving average (ARMA) Gaussian wiretap channel with noiseless causal feedback is considered, in which an eavesdropper receives noisy observations of signals in both forward and feedback channels. It is shown that the generalized Schalkwijk–Kailath scheme, a capacity-achieving coding scheme for the Gaussian feedback channel, achieves the same maximum rate for the same channel even with the presence of an eavesdropper. Therefore, the secrecy capacity is equal to the feedback capacity without the presence of an eavesdropper for the Gaussian feedback channel. Furthermore, the results are extended to the additive white Gaussian noise (AWGN) channel with quantized feedback. It is shown that the proposed coding scheme achieves a positive secrecy rate. Our result implies that as the amplitude of the quantization noise decreases to zero, the secrecy rate converges to the capacity of the AWGN channel.

Generating a Binary Symmetric Channel for Wiretap Codes

In this paper, we fill a void between information theoretic security and practical coding over the Gaussian wiretap channel using a three-stage encoder/decoder technique. Security is measured using Kullback–Leibler divergence and resolvability techniques along with a limited number of practical assumptions regarding the eavesdropper’s decoder. The results specify a general coding recipe for obtaining both secure and reliable communications over the Gaussian wiretap channel, and one specific set of concatenated codes is presented as a test case for the sake of providing simulation-based evaluation of security and reliability over the network. It is shown that there exists a threshold in signal-to-noise ratio (SNR) over a Gaussian channel, such that receivers experiencing SNR below the threshold have no practical hope of receiving information about the message when the three-stage coding technique is applied. Results further indicate that the two innermost encoding stages successfully approximate a binary symmetric channel, allowing the outermost encoding stage (e.g., a wiretap code) to focus solely on secrecy coding over this approximated channel.

Implications of Coding Layers on Physical-Layer Security: A Secrecy Benefit Approach.

In this work, we consider the pros and cons of using various layers of keyless coding to achieve secure and reliable communication over the Gaussian wiretap channel. We define a new approach to information theoretic security, called practical secrecy and the secrecy benefit, to be used over real-world channels and finite blocklength instantiations of coding layers, and use this new approach to show the fundamental reliability and security implications of several coding mechanisms that have traditionally been used for physical-layer security. We perform a systematic/structured analysis of the effect of error-control coding, scrambling, interleaving, and coset coding, as coding layers of a secrecy system. Using this new approach, scrambling and interleaving are shown to be of no effect in increasing information theoretic security, even when measuring the effect at the output of the eavesdropper’s decoder. Error control coding is shown to present a trade-off between secrecy and reliability that is dictated by the chosen code and the signal-to-noise ratios at the legitimate and eavesdropping receivers. Finally, the benefits of secrecy coding are highlighted, and it is shown how one can shape the secrecy benefit according to system specifications using combinations of different layers of coding to achieve both reliable and secure throughput.

Wiretap Channels: Nonasymptotic Fundamental Limits

This paper investigates the maximal secret communication rate over a wiretap channel subject to reliability and secrecy constraints at a given blocklength. New achievability and converse bounds are derived, which are uniformly tighter than existing bounds, and lead to the tightest bounds on the second-order coding rate for discrete memoryless and Gaussian wiretap channels. The exact second-order coding rate is established for semi-deterministic wiretap channels, which characterizes the optimal tradeoff between reliability and secrecy in the finite-blocklength regime. Underlying our achievability bounds are two new privacy amplification results, which not only refine the existing results, but also achieve stronger notions of secrecy.

On the Gaussian Multiple Access Wiretap Channel and the Gaussian Wiretap Channel With a Helper: Achievable Schemes and Upper Bounds

We study deterministic approximations of the Gaussian two-user multiple access wiretap channel (G-MAC-WT) and the Gaussian wiretap channel with a helper (G-WT-H). These approximations enable results beyond the recently shown 2/3 and 1/2 secure degrees of freedom (s.d.o.f.) for the G-MAC-WT and the G-WT-H, respectively. While the s.d.o.f. were obtained by real interference alignment, our approach uses signal-scale alignment. We show achievable schemes which are independent of the rationality of the channel gains. Moreover, our results can differentiate between channel strengths, in particular, between both users, and will establish secrecy rates dependent on this difference. We can show that the resulting achievable secrecy rates tend to the s.d.o.f. for vanishing channel gain differences. Moreover, we extend previous and develop new techniques to prove generalized s.d.o.f. bounds for varying channel strengths and show that our achievable schemes reach the bounds for certain channel gain parameters. We believe that our analysis is the next step toward a constant-gap analysis of the G-MAC-WT and the G-WT-H.

Codebook-Based Hybrid Beamforming Design for MISOME Wiretap Channel

For practical implementation of multi-input multi-output systems, hybrid analog-digital beamforming may address the challenges of hardware limitation in fully digital beamforming (FDB). In this letter, in order to tackle the potential security issues in large-scale multi-antenna systems without exorbitant hardware cost, we focus on the codebook-based hybrid beamforming design in multi-input single-output multi-eavesdropper Gaussian wiretap channel to maximize the secrecy rate. However, the secrecy rate maximization (SRM) problem is non-convex by nature due to the codebook constraint imposed by analog beamforming. To handle it, we first reformulate the SRM problem into a combinatorial inner product maximization problem. Then, a combinatorial matching pursuit algorithm is proposed to solve this problem, in which the analog beamformer is designed independently of the digital beamformer. With the obtained analog beamformer, we show that the optimal digital beamformer admits a closed-form expression. Simulation results reveal that the proposed algorithm, though suffering from the inherent precision loss of the codebook, can achieve the near-optimal performance of the optimal FDB.

Ratio of Two Envelopes Taken From $\alpha$ -$\mu$ , $\eta$ -$\mu$ , and $\kappa$ -$\mu$ Variates and Some Practical Applications

In this paper, new, exact expressions for the probability density functions and the cumulative distribution functions of the ratio of random envelopes involving the $\alpha $ - $\mu $ , $\eta $ - $\mu $ , and $\kappa $ - $\mu $ fading distributions are derived. The expressions are obtained in terms of easily computable infinite series and also in terms of the multivariable Fox H-function. Some special cases of these ratios, namely, Hoyt/Hoyt, $\eta $ - $\mu $ /Nakagami- ${m}$ , $\kappa $ - $\mu $ /Nakagami- ${m}$ , $\kappa $ - $\mu / \eta $ - $\mu $ with an integer $\mu $ for the $\eta $ - $\mu $ variate, $\eta $ - $\mu / \eta $ - $\mu $ with an integer $\mu $ for only one of the $\eta $ - $\mu $ , and their reciprocals are found in novel exact closed-form expressions . In addition, simple closed-form expressions for the asymptotes of the probability density functions and cumulative distribution functions of all ratios, both for the lower and upper tails of the distributions are derived. In the same way, asymptotes for the bit error rate on a binary signaling channel are obtained in closed-form expressions. To demonstrate the practical utility of these new formulations, an application example is provided. In particular, the secrecy capacity of a Gaussian wire-tap channel used for device-to-device and vehicle-to-vehicle communications is characterized using data obtained from field measurements conducted at 5.8 GHz.

Almost Universal Codes for MIMO Wiretap Channels

Despite several works on secrecy coding for fading and MIMO wiretap channels from an error probability perspective, the construction of information-theoretically secure codes over such channels remains an open problem. In this paper, we consider a fading wiretap channel model where the transmitter has only partial statistical channel state information. Our channel model includes static channels, i.i.d. block fading channels, and ergodic stationary fading with fast decay of large deviations for the eavesdropper's channel. We extend the flatness factor criterion from the Gaussian wiretap channel to fading and MIMO wiretap channels, and establish a simple design criterion where the normalized product distance / minimum determinant of the lattice and its dual should be maximized simultaneously. Moreover, we propose concrete lattice codes satisfying this design criterion, which are built from algebraic number fields with constant root discriminant in the single-antenna case, and from division algebras centered at such number fields in the multiple-antenna case. The proposed lattice codes achieve strong secrecy and semantic security for all rates $R<C_b-C_e-\kappa$, where $C_b$ and $C_e$ are Bob and Eve's channel capacities respectively, and $\kappa$ is an explicit constant gap. Furthermore, these codes are almost universal in the sense that a fixed code is good for secrecy for a wide range of fading models. Finally, we consider a compound wiretap model with a more restricted uncertainty set, and show that rates $R<\bar{C}_b-\bar{C}_e-\kappa$ are achievable, where $\bar{C}_b$ is a lower bound for Bob's capacity and $\bar{C}_e$ is an upper bound for Eve's capacity for all the channels in the set.

Optical Wiretap Channel With Input-Dependent Gaussian Noise Under Peak- and Average-Intensity Constraints

This paper studies the optical wiretap channel with input-dependent Gaussian noise, in which the main distortion is caused by an additive Gaussian noise whose variance depends on the current signal strength. Subject to nonnegativity and peak-intensity constraints on the channel input, we first present a practical optical wireless communication scenario for which the considered wiretap channel is stochastically degraded. We then study the secrecy-capacity-achieving input distribution of this wiretap channel and prove it to be discrete with a finite number of mass points, one of them located at the origin. Moreover, we show that the entire rate-equivocation region of this wiretap channel is also obtained by discrete input distributions with a finite support. Similar to the case of the Gaussian wiretap channel under a peak-power constraint, here too, we observe that under nonnegativity and peak-intensity constraints, there is a tradeoff between the secrecy capacity and the capacity in the sense that both may not be achieved simultaneously. Furthermore, we prove the optimality of discrete input distributions in the presence of an additional average intensity constraint. Finally, we shed light on the asymptotic behavior of the secrecy capacity in the low- and high-intensity regimes. In the low-intensity regime, the secrecy capacity scales quadratically with the peak-intensity constraint. On the other hand, in the high-intensity regime, the secrecy capacity does not scale with the constraint.

Randomized Serially Concatenated LDGM Codes for the Gaussian Wiretap Channel

We study the application of a special class of low-density parity-check codes to the wiretap channel. We construct a randomized coding scheme based on serially concatenated low-density generator matrix codes and their duals extending the approach used for convolutional and turbo codes. Furthermore, we propose an efficient iterative decoder for this scheme utilizing a joint iterative message passing algorithm. We demonstrate via numerical examples that this approach outperforms other available practical coding alternatives for the Gaussian wiretap channel in terms of the resulting security gap.

Achieving Secrecy Capacity of the Gaussian Wiretap Channel With Polar Lattices

In this paper, an explicit scheme of wiretap coding based on polar lattices is proposed to achieve the secrecy capacity of the additive white Gaussian noise (AWGN) wiretap channel. First, polar lattices are used to construct secrecy-good lattices for the mod- $\Lambda _{s}$ Gaussian wiretap channel (GWC). Then, we propose an explicit shaping scheme to remove this mod- $\Lambda _{s}$ front end and extend polar lattices to the genuine GWC. The shaping technique is based on the lattice Gaussian distribution, which leads to a binary asymmetric channel at each level for the multilevel lattice codes. By employing the asymmetric polar coding technique, we construct an AWGN-good lattice and a secrecy-good lattice with optimal shaping simultaneously. As a result, the encoding complexity for the sender and the decoding complexity for the legitimate receiver are both $O(N\log N\log (\log N))$ . The proposed scheme is proven to be semantically secure.

Linear Precoder Design for an MIMO Gaussian Wiretap Channel With Full-Duplex Source and Destination Nodes

This paper investigates and quantifies the advantages of a Full-Duplex (FD) transmitter/receiver pair in improving the secrecy rate of the system. We consider a linear precoder design for a multiple-input multiple-output Gaussian wiretap channel, which comprises two legitimate nodes, i.e., Alice and Bob, operating in FD mode and exchanging confidential messages in the presence of a passive eavesdropper. Using the sum secrecy degrees of freedoms (sum SDoFs) as metric, we formulate an optimization problem with respect to Alice's and Bob's precoding matrices. In order to solve this problem, we first propose a cooperative secrecy transmission scheme, whose feasible set is sufficient to achieve the maximum sum SDoF. Based on that feasible set, we then determine in closed form the maximum achievable sum SDoF and also provide a method for constructing the precoding matrix pair, which achieves the maximum sum SDoF. The latter pair would be near-optimal in terms of the achievable secrecy sum rate in the high signal-to-noise ratio (SNR) regime. By providing the maximum achievable sum SDoF as a function of the number of antennas, one could select the optimal system parameters to further maximize the achievable sum SDoF. We use simulations to evaluate the performance of the proposed precoding matrices in realistic channel scenarios and at various levels of the SNR.

Secrecy Capacity of Colored Gaussian Noise Channels with Feedback

In this paper, the k-th order autoregressive moving average (ARMA(k)) Gaussian wiretap channel with noiseless causal feedback is considered, in which an eavesdropper receives noisy observations of the signals in both forward and feedback channels. It is shown that a variant of the generalized Schalkwijk-Kailath scheme, a capacity-achieving coding scheme for the feedback Gaussian channel, achieves the same maximum rate for the same channel with the presence of an eavesdropper. Therefore, the secrecy capacity is equal to the feedback capacity without the presence of an eavesdropper for the feedback channel. Furthermore, the results are extended to the additive white Gaussian noise (AWGN) channel with quantized feedback. It is shown that the proposed coding scheme achieves a positive secrecy rate. As the amplitude of the quantization noise decreases to zero, the secrecy rate converges to the capacity of the AWGN channel.

Efficient Secure Channel Coding based on QPP-Block-LDPC Codes

An encrypting block low-density parity-check code based on quadratic permutation polynomials (QPPs) (QPP-Block-LDPC) is proposed in this paper. The parity-check matrix of the encrypting QPP-Block-LDPC code is composed of a group of permutations, which can be mapped to QPPs. The coefficients of the QPPs, which are served as secret keys, are utilized to randomize the permutation submatrices. A secure channel coding scheme is presented using the QPP-Block-LDPC code to produce the error-correcting ciphertexts. The coding can achieve more efficient implementation and provide an acceptable security level for the communication over the Gaussian wiretap channel. Simulation results demonstrate that this coding offers good trade-offs between the error performance and the security level.