An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> B. Secrecy Capacity of Single-Antenna Wiretap Channels <s> We consider the situation in which digital data is to be reliably transmitted over a discrete, memoryless channel (dmc) that is subjected to a wire-tap at the receiver. We assume that the wire-tapper views the channel output via a second dmc). Encoding by the transmitter and decoding by the receiver are permitted. However, the code books used in these operations are assumed to be known by the wire-tapper. The designer attempts to build the encoder-decoder in such a way as to maximize the transmission rate R, and the equivocation d of the data as seen by the wire-tapper. In this paper, we find the trade-off curve between R and d, assuming essentially perfect (“error-free”) transmission. In particular, if d is equal to Hs, the entropy of the data source, then we consider that the transmission is accomplished in perfect secrecy. Our results imply that there exists a C s > 0, such that reliable transmission at rates up to C s is possible in approximately perfect secrecy. <s> BIB001 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> B. Secrecy Capacity of Single-Antenna Wiretap Channels <s> Wyner's results for discrete memoryless wire-tap channels are extended to the Gaussian wire-tap channel. It is shown that the secrecy capacity Cs is the difference between the capacities of the main and wire.tap channels. It is further shown that Rd= Cs is the upper boundary of the achievable rate-equivocation region. <s> BIB002 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> B. Secrecy Capacity of Single-Antenna Wiretap Channels <s> Given two discrete memoryless channels (DMC's) with a common input, it is desired to transmit private messages to receiver 1 rate R_{1} and common messages to both receivers at rate R_{o} , while keeping receiver 2 as ignorant of the private messages as possible. Measuring ignorance by equivocation, a single-letter characterization is given of the achievable triples (R_{1},R_{e},R_{o}) where R_{e} is the equivocation rate. Based on this channel coding result, the related source-channel matching problem is also settled. These results generalize those of Wyner on the wiretap channel and of Korner-Marton on the broadcast Channel. <s> BIB003 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> B. Secrecy Capacity of Single-Antenna Wiretap Channels <s> We consider the transmission of confidential data over wireless channels with multiple communicating parties. Based on an information-theoretic problem formulation in which two legitimate partners communicate over a quasi-static fading channel and an eavesdropper observes their transmissions through another independent quasi-static fading channel, we define the secrecy capacity in terms of outage probability and provide a complete characterization of the maximum transmission rate at which the eavesdropper is unable to decode any information. In sharp contrast with known results for Gaussian wiretap channels (without feedback), our contribution shows that in the presence of fading information-theoretic security is achievable even when the eavesdropper has a better average signal-to-noise ratio (SNR) than the legitimate receiver ? fading thus turns out to be a friend and not a foe. <s> BIB004 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> B. Secrecy Capacity of Single-Antenna Wiretap Channels <s> We consider the secure transmission of information over an ergodic fading channel in the presence of an eavesdropper. Our eavesdropper can be viewed as the wireless counterpart of Wyner's wiretapper. The secrecy capacity of such a system is characterized under the assumption of asymptotically long coherence intervals. We analyze the full channel state information (CSI) case, where the transmitter has access to the channel gains of the legitimate receiver and eavesdropper, and the main channel CSI scenario, where only the legitimate receiver channel gain is known at the transmitter. In each scenario, the secrecy capacity is obtained along with the optimal power and rate allocation strategies. We then propose a low-complexity on/off power allocation strategy that achieves near-optimal performance with only the main channel CSI. More specifically, this scheme is shown to be asymptotically optimal as the average SNR goes to infinity, and interestingly, is shown to attain the secrecy capacity under the full CSI assumption. Remarkably, our results reveal the positive impact of fading on the secrecy capacity and establish the critical role of rate adaptation, based on the main channel CSI, in facilitating secure communications over slow fading channels. <s> BIB005 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> B. Secrecy Capacity of Single-Antenna Wiretap Channels <s> The fading broadcast channel with confidential messages (BCC) is investigated, where a source node has common information for two receivers (receivers 1 and 2), and has confidential information intended only for receiver 1. The confidential information needs to be kept as secret as possible from receiver 2. The broadcast channel from the source node to receivers 1 and 2 is corrupted by multiplicative fading gain coefficients in addition to additive Gaussian noise terms. The channel state information (CSI) is assumed to be known at both the transmitter and the receivers. The parallel BCC with independent subchannels is first studied, which serves as an information-theoretic model for the fading BCC. The secrecy capacity region of the parallel BCC is established, which gives the secrecy capacity region of the parallel BCC with degraded subchannels. The secrecy capacity region is then established for the parallel Gaussian BCC, and the optimal source power allocations that achieve the boundary of the secrecy capacity region are derived. In particular, the secrecy capacity region is established for the basic Gaussian BCC. The secrecy capacity results are then applied to study the fading BCC. The ergodic performance is first studied. The ergodic secrecy capacity region and the optimal power allocations that achieve the boundary of this region are derived. The outage performance is then studied, where a long-term power constraint is assumed. The power allocation is derived that minimizes the outage probability where either the target rate of the common message or the target rate of the confidential message is not achieved. The power allocation is also derived that minimizes the outage probability where the target rate of the confidential message is not achieved subject to the constraint that the target rate of the common message must be achieved for all channel states. <s> BIB006 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> B. Secrecy Capacity of Single-Antenna Wiretap Channels <s> In this paper, we investigate the ergodic secrecy capacity of the fast fading Gaussian wiretap channel when only the statistics of the channel state information are known at the transmitter. We derive conditions for the existence of degradedness and a positive ergodic secrecy capacity under the usual stochastic order, the convex order, and the increasing convex order between the legitimate and eavesdropper channels. For more general orders, we prove the secrecy capacity of layered erasure wiretap channels and propose a layered signaling for the achievable scheme, and we derive an upper bound on the capacity for fast fading Gaussian wiretap channels. Finally, the numerical results show that under Nakagami-m fast fading channels, the proposed layered signaling outperforms the Gaussian codebook in several cases. In particular, in certain cases, the Gaussian codebook can achieve only a zero secrecy rate, whereas the proposed scheme achieves positive secrecy rates. Therefore, the connectivity of wireless networks can be significantly improved by the proposed scheme. <s> BIB007
The notion of secrecy capacity was originally introduced by Wyner BIB001 for degraded wiretap channels. In the scenario studied in BIB001 , the main channel and the eavesdropper's channel are assumed to be discrete memoryless channels (DMCs) where Eve's observation of the transmitted signal is a degraded version of the signal received by Bob. Wyner showed that the secrecy capacity for this scenario can be expressed as the difference of the mutual information between Alice's and Bob's signals with that of Alice and Eve, maximized over all input distributions. An extension of the notion of secrecy capacity to the case of additive white Gaussian noise (AWGN) channels is studied in BIB002 . It is shown that, for the case of a Gaussian wiretap channel, a nonzero secrecy capacity can be attained if Bob receives the signal through a less noisy channel. While the scenarios considered in BIB001 and BIB002 put the eavesdropper at a disadvantage, Csiszar and Korner have studied the more general problem of secrecy for non-degraded channels in BIB003 . In this channel model, referred to as broadcast channel with confidential messages, the message transmitted by Alice contains a common message intended to both Bob and Eve as well as a secret message, which is intended for Bob only, and is needed to be kept secret from Eve. It is shown that the secrecy capacity over a discrete memoryless wiretap channel (DMWC) is given by where U is an auxiliary random variable. With a given P YZ |X (y, z |x ), the secrecy capacity is achieved by maximizing the difference of the mutual information terms corresponding to the main and the eavesdropper's channels over all joint In order to study the physical layer security in wireless communication scenarios, we need to extend the wiretap channel to a model that takes into account the fading phenomenon. Assuming a narrowband transmission, fading is modeled as multiplicative gains over the main and eavesdropper's channels, denoted by H B and H E , respectively. The first characterization of the role of fading in providing physical layer security is given in BIB004 where quasi-static fading channels are considered towards the legitimate receiver and the eavesdropper. It is shown via analyzing the secrecy outage probability that, in presence of fading, secure transmission is possible even in the scenarios where the eavesdropper has a better signal-to-noise ratio (SNR). Liang et al. BIB006 obtain the ergodic secrecy capacity region for the fading broadcast channel with confidential messages (BCC) by deriving the boundary achieving optimal power allocation strategies. The secrecy capacity over block fading channels is derived in BIB005 under the assumption that each coherence interval is long enough to allow for invoking proper random coding arguments. Given that the transmitter knows either the main CSI (MCSI) or both the main and the eavesdropper CSI (ECSI) perfectly, an achievable secrecy rate can be formulated as BIB005 : where [a] + = max{a, 0}. It is shown in BIB005 that the secrecy capacity that is achieved under the full CSI assumption serves as an upper bound on the secrecy capacity when only MCSI is known at the transmitter. This is due to the fact that with the knowledge of the instantaneous ECSI, the transmitter is capable of realizing a more efficient transmission, which provides additional gains in terms of secrecy rates. For instance, when perfect knowledge of both channels are available, the transmitter can optimize the transmit power according to the instantaneous values of the channel gains H B and H E , providing a higher secrecy rate with respect to the case where the optimization of the transmit power is carried out according to the main channel only. Under the assumption that the transmitter does not know the realizations of the channels, and it only has their statistics, assuming that each receiver knows its own channel, the secrecy capacity can be obtained with the aid of the results for the case of DMWC BIB003 as where U satisfies the Markov chain relationship U → X → (Y , H B ), (Z , H E ). This is obtained by treating H B and H E as channel outputs at the legitimate receiver and at the eavesdropper, respectively BIB007 . Using the chain rule of mutual information and noting the fact that H B and H E are independent of X and U, the expression in (5) can be modified as Determining the optimal joint distribution P UX (u, x ) and the resulting exact secrecy capacity is an open problem, however, by ignoring the channel prefixing, the performance can be quantified using an achievable ergodic secrecy rate as given by