Alice and Bob want to share a secret key and to communicate an independent message, both of which they desire to be kept secret from an eavesdropper Eve. We study this problem of secret communication and secret key generation when two resources are available -- correlated sources at Alice, Bob, and Eve, and a noisy broadcast channel from Alice to Bob and Eve which is independent of the sources. We are interested in characterizing the fundamental trade-off between the rates of the secret message and secret key. We present an achievable solution and prove its optimality for the parallel channels and sources case when each sub-channel and source component satisfies a degradation order (either in favor of the legitimate receiver or the eavesdropper). This includes the case of jointly Gaussian sources and an additive Gaussian channel, for which the secrecy region is evaluated.
In this paper, the individual secrecy of two-way wiretap channel is investigated, where two legitimate users' messages are separately guaranteed secure against an external eavesdropper. For one thing, in some communication scenarios, the joint secrecy is impossible to achieve both positive secrecy rates of two users. For another, the individual secrecy satisfies the secrecy demand of many practical communication systems. Thus, firstly, an achievable secrecy rate region is derived for the general two-way wiretap channel with individual secrecy. In a deterministic channel, the region with indivi…
We consider secrecy obtained when one transmits on a Gaussian Wiretap channel above the secrecy capacity. Instead of equivocation, we consider probability of error as the criterion of secrecy. The usual channel codes are considered for transmission. The rates obtained can reach the channel capacity. We show that the "confusion" caused to the Eve when the rate of transmission is above capacity of the Eve's channel is similar to the confusion caused by using the wiretap channel codes used below the secrecy capacity.
We analyze physical-layer security based on the premise that the coding mechanism for secrecy over noisy channels is tied to the notion of channel resolvability. Instead of considering capacity-based constructions, which associate to each message a sub-code that operates just below the capacity of the eavesdropper's channel, we consider channel-resolvability-based constructions, which associate to each message a sub-code that operates just above the resolvability of the eavesdropper's channel. Building upon the work of Csiszar and Hayashi, we provide further evidence that channel resolvability…
We find the secrecy capacity of the 2-2-1 Gaussian MIMO wire-tap channel, which consists of a transmitter and a receiver with two antennas each, and an eavesdropper with a single antenna. We determine the secrecy capacity of this channel by proposing an achievable scheme and then developing a tight upper bound that meets the proposed achievable secrecy rate. We show that, for this channel, Gaussian signalling in the form of beam-forming is optimal, and no pre-processing of information is necessary.
In this work, we consider achievable secrecy rates for symmetric $K$-user ($K \ge 3$) interference channels with confidential messages. We find that nested lattice codes and layered coding are useful in providing secrecy for these channels. Achievable secrecy rates are derived for very strong interference. In addition, we derive the secure degrees of freedom for a range of channel parameters. As a by-product of our approach, we also demonstrate that nested lattice codes are useful for K-user symmetric interference channels without secrecy constraints in that they yield higher degrees of freedo…
We consider a discrete memoryless broadcast channel consists of two users and a sender. The sender has two independent confidential messages for each user. We extend the work of Liu et al.\ on broadcast channels with two confidential messages with weak secrecy criterion to strong secrecy. Our results are based on an extension of the techniques developed by Hou and Kramer on bounding Kullback-Leibler divergence in context of \textit{resolvability} and \textit{effective secrecy}.
We investigate the effects of user cooperation on the secrecy of broadcast channels by considering a cooperative relay broadcast channel. We show that user cooperation can increase the achievable secrecy region. We propose an achievable scheme that combines Marton's coding scheme for broadcast channels and Cover and El Gamal's compress-and-forward scheme for relay channels. We derive outer bounds for the rate-equivocation region using auxiliary random variables for single-letterization. Finally, we consider a Gaussian channel and show that both users can have positive secrecy rates, which is n…
In this paper, we consider the achievable sum secrecy rate in full-duplex wiretap channel in the presence of an eavesdropper and imperfect channel state information (CSI). We assume that the users participating in full-duplex communication and the eavesdropper have single antenna each. The users have individual transmit power constraints. They also transmit jamming signals to improve the secrecy rates. We obtain the achievable perfect secrecy rate region by maximizing the sum secrecy rate. We also obtain the corresponding optimum powers of the message signals and the jamming signals. Numerical…
We investigate the secrecy capacity of an ergodic fading wiretap channel in which the main channel is correlated with the eavesdropper channel. In this study, the full Channel State Information (CSI) is assumed, and thus the transmitter knows the channel gains of the legitimate receiver and the eavesdropper. By analyzing the resulting secrecy capacity we quantify the loss of the secrecy capacity due to the correlation. In addition, we study the asymptotic behavior of the secrecy capacity as Signal-to-Noise Ratio (SNR) tends to infinity. The capacity of an ordinary fading channel logarithmicall…
A Gaussian multiple-input multiple-output (MIMO) wiretap channel model is considered. The input is a two-antenna transmitter, while the outputs are the legitimate receiver and an eavesdropper, both equipped with multiple antennas. All channels are assumed to be known. The problem of obtaining the optimal input covariance matrix that achieves secrecy capacity subject to a power constraint is addressed, and a closed-form expression for the secrecy capacity is obtained.
Polar lattices, which are constructed from polar codes, are provably good for the additive white Gaussian noise (AWGN) channel. In this work, we propose a new polar lattice construction that achieves the secrecy capacity under the strong secrecy criterion over the mod-$Λ$ Gaussian wiretap channel. This construction leads to an AWGN-good lattice and a secrecy-good lattice simultaneously. The design methodology is mainly based on the equivalence in terms of polarization between the $Λ/Λ'$ channel in lattice coding and the equivalent channel derived from the chain rule of mutual information in mu…
This paper deals with the secrecy capacity of the radio channel in interference-limited regime. We assume that interferers are uniformly scattered over the network area according to a Point Poisson Process and the channel model consists of path-loss, log-normal shadowing and Nakagami-m fading. Both the probability of non-zero secrecy capacity and the secrecy outage probability are then derived in closed-form expressions using tools of stochastic geometry and higher-order statistics. Our numerical results show how the secrecy metrics are affected by the disposition of the desired receiver, the …
An alternative or supplementary approach named as physical layer security has been proposed to afford an extra security layer on top of the conventional cryptography technique. In this paper, an overview of secrecy performance investigations over the classic Alice-Bob-Eve wiretap fading channels is conducted. On the basis of the classic wiretap channel model, we have comprehensively listed and thereafter compared the existing works on physical layer secrecy analysis considering the small-scale, large-scale, composite, and cascaded fading channel models. Exact secrecy metrics expressions, inclu…
Incorporating the physical layer characteristics to secure communications has received considerable attention in recent years. Moreover, cooperation with some nodes of network can give benefits of multiple-antenna systems, increasing the secrecy capacity of such channels. In this paper, we consider cooperative wiretap channel with the help of an Amplify and Forward (AF) relay to transmit confidential messages from source to legitimate receiver in the presence of an eavesdropper. In this regard, the secrecy capacity of AF relying is derived, assuming the relay is subject to a peak power constra…
This work investigates the effect of finite-alphabet source input on the secrecy rate of a multi-antenna wiretap system. Existing works have characterized maximum achievable secrecy rate or secrecy capacity for single and multiple antenna systems based on Gaussian source signals and secrecy code. Despite the impracticality of Gaussian sources, the compact closed-form expression of mutual information between linear channel Gaussian input and corresponding output has led to broad application of Gaussian input assumption in physical secrecy analysis. For practical considerations, we study the eff…
This paper is concerned with the general multiple access wiretap channel and the existence of codes that accomplish reliability and strong secrecy. Information leakage to the eavesdropper is assessed by the variational distance metric, whereas the average error probability is bounded by modifying Feinsteins Lemma. We derive an achievable strong secrecy rate region by utilizing the resolvability theory for the multiple-access channel in terms of information-spectrum methods.
This paper simplifies an existing coding scheme for the two-receiver discrete memoryless broadcast channel with complementary receiver side information where there is a passive eavesdropper and individual secrecy is required. The existing coding scheme is simplified in two steps by replacing Wyner secrecy coding with Carleial-Hellman secrecy coding. The resulting simplified scheme is free from redundant message splits and random components. Not least, the simplified scheme retains the existing achievable individual secrecy rate region. Finally, its construction simplicity helps us gain additio…
A multiple-input single-output (MISO) wiretap channel model is considered, that includes a multi-antenna transmitter, a single-antenna legitimate receiver and a single-antenna eavesdropper. For the scenario in which spherical uncertainty for both the legitimate and the eavesdropper channels is included, the problem of finding the optimal input covariance that maximizes the worst-case secrecy rate subject to a power constraint, is considered, and an explicit expression for the maximum worst-case secrecy rate is provided.
In this paper, we consider the achievable sum secrecy rate in MISO (multiple-input-single-output) {\em full-duplex} wiretap channel in the presence of a passive eavesdropper and imperfect channel state information (CSI). We assume that the users participating in full-duplex communication have multiple transmit antennas, and that the users and the eavesdropper have single receive antenna each. The users have individual transmit power constraints. They also transmit jamming signals to improve the secrecy rates. We obtain the achievable perfect secrecy rate region by maximizing the worst case sum…
In this paper, we study the secrecy rates over weak Gaussian interference channels for different transmission schemes. We focus on the low-SNR regime and obtain the minimum bit energy E_b/N_0_min values, and the wideband slope regions for both TDMA and multiplexed transmission schemes. We show that secrecy constraints introduce a penalty in both the minimum bit energy and the slope regions. Additionally, we identify under what conditions TDMA or multiplexed transmission is optimal. Finally, we show that TDMA is more likely to be optimal in the presence of secrecy constraints.
The secrecy capacity of a discrete memoryless Gaussian Wire-Tap Channel when the input is from a finite complex constellation is studied. It is shown that the secrecy capacity curves of a finite constellation plotted against the SNR, for a fixed noise variance of the eavesdropper's channel has a global maximum at an internal point. This is in contrast to what is known in the case of Gaussian codebook input where the secrecy capacity curve is a bounded, monotonically increasing function of SNR. Secrecy capacity curves for some well known constellations like BPSK, 4-QAM, 16-QAM and 8-PSK are plo…
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 channel state information (CSI) is assumed to be known at both the transmitter and the receivers. The secrecy capacity region is first established for the parallel Gaussian BCC, and the optimal source power allocations that achieve the boundary of the secrecy capacity region are derived…
Recently, the secrecy capacity of the multi-antenna wiretap channel was characterized by Khisti and Wornell [1] using a Sato-like argument. This note presents an alternative characterization using a channel enhancement argument. This characterization relies on an extremal entropy inequality recently proved in the context of multi-antenna broadcast channels, and is directly built on the physical intuition regarding to the optimal transmission strategy in this communication scenario.
Theoretical studies on physical layer security often adopt the secrecy outage probability as the performance metric for wireless communications over quasi-static fading channels. The secrecy outage probability has two limitations from a practical point of view: a) it does not give any insight into the eavesdropper's decodability of confidential messages; b) it cannot characterize the amount of information leakage to the eavesdropper when an outage occurs. Motivated by the limitations of the secrecy outage probability, we propose three new secrecy metrics for secure transmissions over quasi-sta…
This work studies the secrecy-capacity of a scalar-Gaussian wiretap channel with an amplitude constraint on the input. It is known that for this channel, the secrecy-capacity-achieving distribution is discrete with finitely many points. This work improves such result by showing an upper bound of the order $\frac{\mathsf{A}}{σ_1^2}$ where $\mathsf{A}$ is the amplitude constraint and $σ_1^2$ is the variance of the Gaussian noise over the legitimate channel.
The secrecy performance of a source-channel model is studied in the context of lossy source compression over a noisy broadcast channel. The source is causally revealed to the eavesdropper during decoding. The fidelity of the transmission to the legitimate receiver and the secrecy performance at the eavesdropper are both measured by a distortion metric. Two achievability schemes using the technique of hybrid coding are analyzed and compared with an operationally separate source-channel coding scheme. A numerical example is provided and the comparison results show that the hybrid coding schemes …
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 feedb…
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, t…
We give a complete characterization of the correlated random coding secrecy capacity of arbitrarily varying wiretap channels (AVWCs). We apply two alternative strong secrecy criteria, which both lead to the same multi-letter formula. The difference of these criteria lies in the treatment of correlated randomness, they coincide in the case of uncorrelated codes. On the basis of the derived formula, we show that the correlated random coding secrecy capacity is continuous as a function of the AVWC, in contrast to the discontinuous uncorrelated coding secrecy capacity. In the proof of the secrecy …
Imperfect secrecy in communication systems is investigated. Instead of using equivocation as a measure of secrecy, the distortion that an eavesdropper incurs in producing an estimate of the source sequence is examined. The communication system consists of a source and a broadcast (wiretap) channel, and lossless reproduction of the source sequence at the legitimate receiver is required. A key aspect of this model is that the eavesdropper's actions are allowed to depend on the past behavior of the system. Achievability results are obtained by studying the performance of source and channel coding…
Different polar coding schemes are proposed for the memoryless degraded broadcast channel under different reliability and secrecy requirements: layered decoding and/or layered secrecy. In this setting, the transmitter wishes to send multiple messages to a set of legitimate receivers keeping them masked from a set of eavesdroppers. The layered decoding structure requires receivers with better channel quality to reliably decode more messages, while the layered secrecy structure requires eavesdroppers with worse channel quality to be kept ignorant of more messages. The implementation of the propo…
In this work we study the secrecy capacity of Gaussian multiple-input multiple-output (MIMO) wiretap channels (WTCs) with a finite memory, subject to a per-symbol average power constraint on the MIMO channel input. MIMO channels with finite memory are very common in wireless communications as well as in wireline communications (e.g., in communications over power lines). To derive the secrecy capacity of the Gaussian MIMO WTC with finite memory we first construct an asymptotically-equivalent block-memoryless MIMO WTC, which is then transformed into a set of parallel, independent, memoryless MIM…
We derive a lower bound on the secrecy capacity of the compound wiretap channel with channel state information at the transmitter which matches the general upper bound on the secrecy capacity of general compound wiretap channels given by Liang et al. and thus establishing a full coding theorem in this case. We achieve this with a stronger secrecy criterion and the maximum error probability criterion, and with a decoder that is robust against the effect of randomisation in the encoding. This relieves us from the need of decoding the randomisation parameter which is in general not possible withi…
Strong secrecy capacity of compound wiretap channels is studied. The known lower bounds for the secrecy capacity of compound finite-state memoryless channels under discrete alphabets are extended to arbitrary uncertainty sets and continuous alphabets under the strong secrecy criterion. The conditions under which these bounds are tight are given. Under the saddle-point condition, the compound secrecy capacity is shown to be equal to that of the worst-case channel. Based on this, the compound Gaussian MIMO wiretap channel is studied under the spectral norm constraint and without the degradedness…
In this paper, the one-sided secrecy of two-way wiretap channel with feedback is investigated, where the confidential messages of one user through multiple transmissions is guaranteed secure against an external eavesdropper. For one thing, one-sided secrecy satisfies the secure demand of many practical scenarios. For another, the secrecy is measured over many blocks since the correlation between eavesdropper's observation and the confidential messages in successive blocks, instead of secrecy measurement of one block in previous works. Thus, firstly, an achievable secrecy rate region is derived…
Recent results have shown that structured codes can be used to construct good channel codes, source codes and physical layer network codes for Gaussian channels. For Gaussian channels with secrecy constraints, however, efforts to date rely on random codes. In this work, we advocate that structured codes are useful for providing secrecy, and show how to compute the secrecy rate when structured codes are used. In particular, we solve the problem of bounding equivocation rates with one important class of structured codes, i.e., nested lattice codes. Having established this result, we next demonst…
This paper investigates physical layer security (PLS) in the intelligent reflecting surface (IRS)-assisted multiple-user uplink channel. Since the instantaneous eavesdropper's channel state information (CSI) is unavailable, the secrecy rate can not be measured. In this case, existing investigations usually focus on the maximization of the minimum (max-min) of signal to interference plus noise power ratio (SINRs) among multiple users, and do not consider secrecy outage probability caused by eavesdroppers. In this paper, we first formulate the minimization of the maximum (min-max) secrecy outage…
The secrecy outage performance of wireless communication systems operating over spatially correlated composite fading channels is analyzed in this paper. We adopt a multiplicative composite channel model for both the legitimate communication link and the link between the eavesdropper and the legitimate transmitter, consisting of Nakagami-$m$ distributed small-scale fading and shadowing (large-scale fading) modeled by the Gamma distribution. We consider the realistic case where small-scale fading between the links is independent, but shadowing is arbitrarily correlated, and present novel analyt…
In this paper, the physical layer security aspects of a wireless framework over α-\k{appa}-μ shadowed (AKMS) fading channel are examined by acquiring closed-form novel expressions of average secrecy capacity, secure outage probability (SOP), and strictly positive secrecy capacity. The lower bound of SOP is derived along with the asymptotic expression of SOP at the high signal-to-noise ratio regime in order to achieve secrecy diversity gain. Capitalizing on these expressions, the consequences due to the simultaneous occurrence of fading and shadowing are quantified. Finally, Monte-Carlo simulat…
We study the security of communication between a single transmitter and multiple receivers in a broadcast channel in the presence of an eavesdropper. We consider several special classes of channels. As the first model, we consider the degraded multi-receiver wiretap channel where the legitimate receivers exhibit a degradedness order while the eavesdropper is more noisy with respect to all legitimate receivers. We establish the secrecy capacity region of this channel model. Secondly, we consider the parallel multi-receiver wiretap channel with a less noisiness order in each sub-channel, where t…
This paper studies the problem of secure communication over broadcast channels under the individual secrecy constraints. That is, the transmitter wants to send two independent messages to two legitimate receivers in the presence of an eavesdropper, while keeping the eavesdropper ignorant of each message (i.e., the information leakage from each message to the eavesdropper is made vanishing). Building upon Carleial-Hellman's secrecy coding, Wyner's secrecy coding, the frameworks of superposition coding and Marton's coding together with techniques such as rate splitting and indirect decoding, ach…
We consider lattice coding for the Gaussian wiretap channel, where the challenge is to ensure reliable communication between two authorized parties while preventing an eavesdropper from learning the transmitted messages. Recently, a measure called the secrecy function of a lattice coding scheme was proposed as a design criterion to characterize the eavesdropper's probability of correct decision. In this paper, the family of formally unimodular lattices is presented and shown to possess the same secrecy function behavior as unimodular and isodual lattices. Based on Construction A, we provide a …
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 first consider the full Channel State Information (CSI) case, where the transmitter has access to the channel gains of the legitimate receiver and the eavesdropper. The secrecy capacity under this full CSI assumption serves as an upper bound for the secrecy capacity when only …
The wiretap channel model of Wyner is one of the first communication models with both reliability and security constraints. Capacity-achieving schemes for various models of the wiretap channel have received considerable attention in recent literature. In this paper, we show that capacity of the general (not necessarily degraded or symmetric) wiretap channel under a "strong secrecy constraint" can be achieved using a transmission scheme based on polar codes. We also extend our construction to the case of broadcast channels with confidential messages defined by Csisz{á}r and K{örner}, achieving …
We consider the secure transmission in ergodic fast-Rayleigh fading multiple-input single-output single-antennaeavesdropper (MISOSE) wiretap channels. We assume that the statistics of both the legitimate and eavesdropper channels is the only available channel state information at the transmitter (CSIT). By introducing a new secrecy capacity upper bound, we prove that the secrecy capacity is achieved by Gaussian input without prefixing. To attain this, we form another MISOSE channel for upper-bounding, and tighten the bound by finding the worst correlations between the legitimate and eavesdropp…
In this paper the secrecy problem in the cognitive statedependent interference channel is considered. In this scenario we have a primary and a cognitive transmitter-receiver pairs. The cognitive transmitter has the message of the primary sender as side information. In addition, the state of the channel is known at the cognitive encoder. So, the cognitive encoder uses this side information to cooperate with the primary transmitter and sends its individual message confidentially. An achievable rate region and an outer bound for the rate region in this channel are derived. The results are extende…
This paper presents a detailed secrecy outage performance analysis of correlated multi-antenna wiretap channels with three popular diversity combining schemes, namely maximal ratio combining, selection combining, and equal gain combining at the legitimate receiver. For single input multiple output wiretap channels, both exact and asymptotic expressions are derived for the secrecy outage probability (SOP) of these systems. The findings suggest that, compared with the scenario where all the channels are independent, correlation of the main channels alone increases the SOP by a factor of 1/ det(U…
This paper presents a polar coding scheme to achieve secrecy in block fading binary symmetric wiretap channels without the knowledge of instantaneous channel state information (CSI) at the transmitter. For this model, a coding scheme that hierarchically utilizes polar codes is presented. In particular, on polarization of different binary symmetric channels over different fading blocks, each channel use is modeled as an appropriate binary erasure channel over fading blocks. Polar codes are constructed for both coding over channel uses for each fading block and coding over fading blocks for cert…
This paper studies the capacity of the multi-antenna or multiple-input multiple-output (MIMO) secrecy channels with multiple eavesdroppers having single/multiple antennas. It is known that the MIMO secrecy capacity is achievable with the optimal transmit covariance matrix that maximizes the minimum difference between the channel mutual information of the secrecy user and those of the eavesdroppers. The MIMO secrecy capacity computation can thus be formulated as a non-convex max-min problem, which cannot be solved efficiently by standard convex optimization techniques. To handle this difficulty…
The coding problem for wiretap channels with causal channel state information available at the encoder and/or the decoder is studied under the strong secrecy criterion. This problem consists of two aspects: one is due to naive wiretap channel coding and the other is due to one-time pad cipher based on the secret key agreement between Alice and Bob using the channel state information. These two aspects are closely related to each other and investigated in details from the viewpoint of achievable rates with strong secrecy. The general wiretap channel with various directions of channel state info…
One key metric for physical layer security is the secrecy capacity. This is the maximum rate that a system can transmit with perfect secrecy. For a Multiple Input Multiple Output (MIMO) system (a newer technology for 5G, 6G and beyond) the secrecy capacity is not fully understood. For a Gaussian MIMO channel, the secrecy capacity is a non-convex optimisation problem for which a general solution is not available. Previous work by the authors showed that the secrecy capacity of a MIMO system with a single eavesdrop antenna is concave to a cut off point. In this work, which extends the previous p…
A broadcast channel (BC) where the decoders cooperate via a one-sided link is considered. One common and two private messages are transmitted and the private message to the cooperative user should be kept secret from the cooperation-aided user. The secrecy level is measured in terms of strong secrecy, i.e., a vanishing information leakage. An inner bound on the capacity region is derived by using a channel-resolvability-based code that double-bins the codebook of the secret message, and by using a likelihood encoder to choose the transmitted codeword. The inner bound is shown to be tight for s…
In many information-theoretic channel coding problems, adding an input cost constraint to the operational setup amounts to restricting the optimization domain in the capacity formula. This paper shows that, in contrast to common belief, such a simple modification does not hold for the cost-constrained (CC) wiretap channel (WTC). The secrecy-capacity of the discrete memoryless (DM) WTC without cost constraints is described by a single auxiliary random variable. For the CC DM-WTC, however, we show that two auxiliaries are necessary to achieve capacity. Specifically, we first derive the secrecy-c…
Reliable and secure communication is an important aspect of modern fiber optic communication. In this work we consider a multi-mode fiber (MMF) channel wiretapped by an eavesdropper. We assume the transmitter knows the legitimate channel, but statistical knowledge of the eavesdropper's channel only. We propose a transmission scheme with artificial noise (AN) for such a channel. In particular, we formulate the corresponding optimization problem which aims to maximize the average secrecy rate and develop an algorithm to solve it. We apply this algorithm to actual measured MMF channels. As real f…
