Block Fading Wiretap Channels Research Articles

Secure Degrees of Freedom of MIMO Two-Way Wiretap Channel With no CSI Anywhere

This article considers a two-way multiple-input multiple-output (MIMO) Rayleigh block fading wiretap channel with two full-duplex nodes (Alice and Bob) exchanging messages simultaneously and wiretapped by a $ {N}_{ {e}}$ -antennas eavesdropper (Eve). The channel state remains unchanged over a coherence interval T and is unknown to all terminals, including Alice, Bob, and Eve, at the beginning of the coherence interval. We first show the expression of the optimal signal waveform for Alice and Bob, which maximizes the sum secrecy rate, should be the products of independent random diagonal matrices and isotropically distributed unitary matrices. Then, conditioned on large T and small $ {N}_{ {e}}$ , the upper-bound for the secure degree of freedom (s.d.o.f.) of the two-way wiretap channel is derived, and a constant-norm-signaling (CNS) scheme is presented to achieve this theoretical bound. Finally, we formulate an optimization problem of the s.d.o.f. achieved by the CNS scheme for general cases with arbitrary T and $ {N}_{ {e}}$ , and solve this problem by exploring the monotonicity of the achievable s.d.o.f.. Our result shows that the optimal s.d.o.f. can be independent of coherence time T and becomes the product of the numbers of legitimate nodes’s transmitter antennas (NLNTA) for large $ {N}_{ {e}}$ and small NLNTA.

Securing Multi-User Broadcast Wiretap Channels With Finite CSI Feedback

In this work, we investigate the problem of secure broadcasting over block-fading wiretap channels with limited channel knowledge at the transmitter. More particularly, we analyze the effect of having a finite rate feedback on the throughput of multi-user broadcast wiretap channels. We consider that the transmitter is only provided by a $b$-bits feedback of the main channel state information (CSI) sent by each legitimate receiver, at the beginning of each fading block, over error-free public links with limited capacity. Also, we assume that the transmitter is aware of the statistics of the eavesdropper's CSI but not of its channel's realizations. Under these assumptions of CSI uncertainty, we characterize the ergodic secrecy capacity of the system when a common message is broadcasted to all legitimate receivers, the ergodic secrecy sum-capacity when multiple independent messages are transmitted, and the ergodic secrecy capacity region for the broadcast channel with confidential messages (BCCM). In all three scenarios, we show that as long as the transmitter has some knowledge of the main CSI, obtained even through a 1-bit CSI feedback, a non-zero secrecy rate can still be achieved. The impact of having the feedback sent over a binary erasure channel (BEC) is also investigated for the BCCM case. Here again, and even with the possibility of having the feedback bits erased, a positive secrecy rate can still be achieved as long as the erasure event is not a probability-one event. An asymptotic analysis of the obtained results is provided for the high SNR regime, and the scaling law of the system, when the number of legitimate receivers is large, is also presented.

Secrecy Transmission Scheme Based on 2-D Polar Coding Over Block Fading Wiretap Channels

This letter discusses a secrecy transmission scheme based on 2-D polar coding over block fading wiretap channels. Unlike the previous approaches using 1-D polar coding, we propose to encode the secret bits in two dimensions to adapt to variation of the instantaneous secrecy capacity. In the proposed scheme, intra-block coding in one dimension is used to guarantee reliability, and cross-block coding in the other dimension is used to combat eavesdropping. Theoretical analysis demonstrates that our scheme is able to achieve the maximum perfect secrecy rate asymptotically. Simulation results show that the equivocation rate of eavesdropper is close to the secrecy capacity of the system with finite codelength.

Secure Multiple-Antenna Block-Fading Wiretap Channels With Limited CSI Feedback

In this paper, we investigate the ergodic secrecy capacity of a block-fading wiretap channel with limited channel knowledge at the transmitter. We consider that the legitimate receiver, the eavesdropper, and the transmitter are equipped with multiple antennas and that the receiving nodes are aware of their respective channel matrices. On the other hand, the transmitter is only provided by a B-bit feedback of the main channel state information. The feedback bits are sent by the legitimate receiver, at the beginning of each fading block, over an error-free public link with limited capacity. The statistics of the main and the eavesdropper channel state information are known at all nodes. Assuming an average transmit power constraint, we establish upper and lower bounds on the ergodic secrecy capacity. Then, we present a framework to design the optimal codebooks for feedback and transmission. In addition, we show that the proposed lower and upper bounds coincide asymptotically, as the capacity of the feedback link becomes large, i.e., B→∞, hence fully characterizing the ergodic secrecy capacity in this case. Besides, we analyze the asymptotic behavior of the presented secrecy rates, at high signal-to-noise ratio, and evaluate the gap between the bounds.

Rate Adaptation for Incremental Redundancy Secure HARQ

This paper studies secure communication based on incremental redundancy (INR) secure hybrid automatic retransmission request (HARQ) protocol over block-fading wiretap channels. The transmitter has no instantaneous channel state information (CSI) available from either main channel or the eavesdropper channel, hence the coding rates cannot be adapted to instantaneous channel conditions. We investigate the outage performance for two schemes of INR secure HARQ protocols: case 1) when there exists two reliable multibit feedback channels from both legitimate receiver and eavesdropper to the transmitter carrying a function of outdated CSI, and case 2) when there is a multibit feedback channel only from legitimate receiver. In both cases, we demonstrate that using the information carried via multibit feedback channels, the transmitter can adapt the coding rates in order to achieve a better secrecy throughput using a smaller number of transmissions comparing to the ACK/NACK feedback channel model. For some parameters, our rate adaptation protocol achieves a strictly positive secrecy throughput whereas it is equal to zero for the protocol with ACK/NACK feedback. We show that for some set of parameters, the loss of secrecy throughput between case 1 and case 2 is very small compared to the gain provided by both protocols.

Secure Degrees of Freedom of MIMO Rayleigh Block Fading Wiretap Channels With No CSI Anywhere

We consider the block Rayleigh fading multiple-input multiple-output (MIMO) wiretap channel with no prior channel state information (CSI) available at any of the terminals. The channel gains remain constant within a coherence interval of $T$ symbols, and then change to another independent realization in the next coherence interval. The transmitter, the legitimate receiver, and the eavesdropper have $n_{t}$ , $n_{r}$ , and $n_{e} $ antennas, respectively. We determine the exact secure degrees of freedom (s.d.o.f.) of this system when $T\geq 2\min(n_{t},n_{r} )$ . We show that, in this case, the s.d.o.f. is exactly equal to $(\min(n_{t},n_{r})-n_{e})^{+}(T-\min(n_{t},n_{r} ))/T$ . The first term in this expression can be interpreted as the eavesdropper with $n_{e}$ antennas taking away $n_{e} $ antennas from both the transmitter and the legitimate receiver. The second term can be interpreted as a fraction of the s.d.o.f. being lost due to the lack of CSI at the legitimate receiver. In particular, the fraction loss, $\min(n_{t},n_{r} ) /T$ , can be interpreted as the fraction of channel uses dedicated to training the legitimate receiver for it to learn its own CSI. We prove that this s.d.o.f. can be achieved by employing a constant norm channel input, which can be viewed as a generalization of discrete signalling to multiple dimensions.

On the Secrecy Capacity of Block Fading Channels With a Hybrid Adversary

We consider a block fading wiretap channel, where a transmitter attempts to send messages securely to a receiver in the presence of a hybrid half-duplex adversary, which arbitrarily decides to either jam or eavesdrop the transmitter-to-receiver channel. We provide bounds to the secrecy capacity for various possibilities on receiver feedback and show special cases where the bounds are tight. We show that, without any feedback from the receiver, the secrecy capacity is zero if the transmitter-to-adversary channel stochastically dominates the effective transmitter-to-receiver channel. However, the secrecy capacity is nonzero even when the receiver is allowed to feed back only one bit at the end of each block. Our novel achievable strategy improves the rates proposed in the literature for the nonhybrid adversarial model. We also analyze the effect of multiple adversaries and delay constraints on the secrecy capacity. We show that our novel time sharing approach leads to positive secrecy rates even under strict delay constraints.

Construction A of Lattices Over Number Fields and Block Fading (Wiretap) Coding

We propose a lattice construction from totally real and complex multiplication fields, which naturally generalizes Construction A of lattices from p-ary codes obtained from the cyclotomic field Q(ζ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> ), p a prime, which in turn contains the so-called Construction A of lattices from binary codes as a particular case. We focus on the maximal totally real subfield Q(ζ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sup> + ζ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-r</sup> ) of the cyclotomic field Q(ζ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sup> ), r ≥ 1. Our construction has applications to coset encoding of algebraic lattice codes for block fading channels, and in particular for block fading wiretap channels.

Ergodic Secret Message Capacity of the Wiretap Channel with Finite-Rate Feedback

We study the secret message capacity of an ergodic block fading wiretap channel with partial channel state information at the transmitter and perfect channel state information at the receivers, under both a short term power constraint (STPC) and a long term power constraint (LTPC). We consider that in addition to the statistics of the main and the eavesdropper channel state information (CSI), the sender is provided by the legitimate receiver with a q-bit feedback, at the beginning of each coherence block, through an error-free public channel, with capacity q bits. We establish upper and lower bounds on the secrecy capacity. We show that the lower and the upper bounds coincide asymptotically as q → ∞. When applied to Rayleigh fading channels, we show that, a 4-bit feedback achieves about 90% of the secrecy capacity when perfect main CSI is available at the transmitter. Finally, asymptotic analysis at high and low Signal-to-Noise Ratio (SNR) is presented. It is found that the capacity is bounded at high-SNR, whereas at asymptotically low-SNR, the lower bounds and the upper bound scale linearly with SNR under STPC. Furthermore, subject to LTPC, the capacity at low-SNR is equal to the capacity of the main channel without secrecy constraint and with perfect CSI at both the transmitter and the receiver, under a mild condition on the fading statistics. We also show that a positive secrecy rate is achievable even when the feedback is at the end of each coherence block and q=1.

A Broadcast Approach for Fading Wiretap Channels

A (layered) broadcast approach is studied for the fading wiretap channel without the channel state information (CSI) at the transmitter. Two broadcast schemes, based on superposition coding and embedded coding, respectively, are developed to encode information into a number of layers and use stochastic encoding to keep the corresponding information secret from an eavesdropper. The layers that can be successfully and securely transmitted are determined by the channel states to the legitimate receiver and the eavesdropper. The advantage of these broadcast approaches is that the transmitter does not need to know the CSI to the legitimate receiver and the eavesdropper, but the scheme still adapts to the channel states of the legitimate receiver and the eavesdropper. Three scenarios of block fading wiretap channels with stringent delay constraints are studied, in which either the legitimate receiver's channel, the eavesdropper's channel, or both channels are fading. For each scenario, the secrecy rate that can be achieved via the broadcast approach developed in this paper is derived, and the optimal power allocation over the layers (or the conditions on the optimal power allocation) is also characterized. A notion of probabilistic secrecy, which characterizes the probability that a certain secrecy rate of decoded messages is achieved during one block, is also introduced and studied for scenarios when the eavesdropper's channel is fading. Numerical examples are provided to demonstrate the impact of the CSI at the transmitter and the channel fluctuations of the eavesdropper on the average secrecy rate. These examples also demonstrate the advantage of the proposed broadcast approach over the compound channel approach.

On the Secrecy Degrees of Freedom of the Multiantenna Block Fading Wiretap Channels

We consider a practical scenario of the Gaussian multiantenna wiretap channel where a transmitter with no channel state information wishes to send a confidential message to its legitimate receiver in the presence of an eavesdropper. It has been known that the secrecy capacity of such a channel does not scale with signal-to-noise ratio under general conditions. Taking into account the different temporal fading structures at the legitimate receiver and the eavesdropper, we characterize lower and upper bounds on the secrecy degrees of freedom (s.d.o.f.) of the channel at hand. Our results show that a positive s.d.o.f. can be ensured whenever two receivers experience the asynchronous variation. Remarkably, simple linear precoding schemes provide the optimal s.d.o.f. in most cases of interest by aligning either the confidential signal at the eavesdropper or the artificial noise at the legitimate receiver.

On the Throughput of Secure Hybrid-ARQ Protocols for Gaussian Block-Fading Channels

The focus of this paper is an information-theoretic study of retransmission protocols for reliable packet communication under a secrecy constraint. The hybrid automatic retransmission request (HARQ) protocol is revisited for a block-fading wire-tap channel, in which two legitimate users communicate over a block-fading channel in the presence of a passive eavesdropper who intercepts the transmissions through an independent block-fading channel. In this model, the transmitter obtains a 1-bit ACK/NACK feedback from the legitimate receiver via an error-free public channel. Both reliability and confidentiality of secure HARQ protocols are studied by the joint consideration of channel coding, secrecy coding, and retransmission protocols. In particular, the error and secrecy performance of repetition time diversity (RTD) and incremental redundancy (INR) protocols are investigated based on good Wyner code sequences, which ensure that the confidential message is decoded successfully by the legitimate receiver and is kept in total ignorance by the eavesdropper for a given set of channel realizations. This paper first illustrates that there exists a good rate-compatible Wyner code family which ensures a secure INR protocol. Next, two types of outage probabilities, connection outage and secrecy outage probabilities are defined in order to characterize the tradeoff between the reliability of the legitimate communication link and the confidentiality with respect to the eavesdropper's link. For a given connection/secrecy outage probability pair, an achievable throughput of secure HARQ protocols is derived for block-fading channels. Finally, both asymptotic analysis and numerical computations demonstrate the benefits of HARQ protocols to throughput and secrecy.