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Civil War Book Review

On the Computation of the Rate-Equivocation Region for Wyner's Wire-tap Channels

Meena Gangula

Published Apr 4, 2018• 0 citations
Publisher

Abstract

In a wire tap channel, the transmitter Alice communicates with the legitimate receiver, Bob, in the presence of another receiver, Eve. The goal is to keep the wiretapper, Eve, as ignorant as possible about the information sent to Bob. For communication channels where the adversary, Eve, eavesdrops on the information sent between the legitimate users by wiretapping at the Bob's end, Wyner introduced secure encoding schemes and showed that non zero reliable transmission rate is possible in perfect secrecy. Wyner also obtained the information theoretic achievable rate equivocation region (Rb, DeltaK) for discrete memoryless wire tap channels. The rate equivocation region (Rb, DeltaK) is a trade off between the rate Rb at which information is sent by Alice to Bob, and the level of equivocation (confusion) DeltaK at the wiretapper Eve about this information. In this dissertation, we propose an iterative algorithm to compute the rate equivocation region (Rb, DeltaK) for any finite input finite output discrete memoryless wiretap channels in Wyners model. First, we investigate the achievable region of Wyner's wire tap channel in both Wyners and Csiszar Korner form. We analyze the general form of the achievable region in both forms and show that both forms are equivalent. We also show that calculating the rate equivocation region can be reduced to the computing of a function Gamma Rb defined by Wyner. Computing the function Gamma Rb involves convex programming and there exists no analytic closed form solution. We transform the function Gamma Rb into a minimax problem and obtain the function Gamma Rb in its parametric form. Calculating the parametric expressions in terms of the Lagrange multiplier yields a point on the Gamma Rb curve. Next, we propose an iterative algorithm for computing the parametric expressions. The iterative procedure can be viewed as a generalization of the Arimoto Blahut method. We prove the convergence of the algorithm and derive upper and lower bounds on the optimum value as a stopping criteria of the iterative algorithm. We provide simulation results about the computation of the parametric expressions from the proposed algorithm and of the achievable rate equivocation region. We also analyze the speed of the convergence of the proposed algorithm. We further propose an accelerated iterative algorithm to compute the parametric expressions with improved performance. The iterative procedure can be regarded as a generalization of the accelerated Arimoto Blahut algorithm. We obtain a sufficient condition of the weighted parameter

DOI
10.31390/gradschool_dissertations.4558

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In this paper, we propose an iterative algorithm for calculating Wyner's wire-tap equivocation region. We show that the calculation of the equivocation region can be reduced to the computation of the function Γ(Rb) of the wire-tap channel defined by Wyner. We propose an iterative algorithm for computing Γ(Rb) for a given bit rate Rb, which can be viewed as a generalization of the Arimoto-Blahut method. The convergence of this algorithm is proved and the upper and lower bounds on Γ(Rb) are derived. Simulation results for the calculation of the function Γ(Rb) and the equivocation region are provided.

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Mathematical Problems in Engineering•Jan 1, 2015•0 citations
In a wire tap channel, the transmitter Alice communicates with the legitimate receiver, Bob, in the presence of another receiver, Eve. The goal is to keep the wiretapper, Eve, as ignorant as possible about the information sent to Bob. For communication channels where the adversary, Eve, eavesdrops on the information sent between the legitimate users by wiretapping at the Bob's end, Wyner introduced secure encoding schemes and showed that non zero reliable transmission rate is possible in perfect secrecy. Wyner also obtained the information theoretic achievable rate equivocation region (Rb, DeltaK) for discrete memoryless wire tap channels. The rate equivocation region (Rb, DeltaK) is a trade off between the rate Rb at which information is sent by Alice to Bob, and the level of equivocation (confusion) DeltaK at the wiretapper Eve about this information. In this dissertation, we propose an iterative algorithm to compute the rate equivocation region (Rb, DeltaK) for any finite input finite output discrete memoryless wiretap channels in Wyners model. First, we investigate the achievable region of Wyner's wire tap channel in both Wyners and Csiszar Korner form. We analyze the general form of the achievable region in both forms and show that both forms are equivalent. We also show that calculating the rate equivocation region can be reduced to the computing of a function Gamma Rb defined by Wyner. Computing the function Gamma Rb involves convex programming and there exists no analytic closed form solution. We transform the function Gamma Rb into a minimax problem and obtain the function Gamma Rb in its parametric form. Calculating the parametric expressions in terms of the Lagrange multiplier yields a point on the Gamma Rb curve. Next, we propose an iterative algorithm for computing the parametric expressions. The iterative procedure can be viewed as a generalization of the Arimoto Blahut method. We prove the convergence of the algorithm and derive upper and lower bounds on the optimum value as a stopping criteria of the iterative algorithm. We provide simulation results about the computation of the parametric expressions from the proposed algorithm and of the achievable rate equivocation region. We also analyze the speed of the convergence of the proposed algorithm. We further propose an accelerated iterative algorithm to compute the parametric expressions with improved performance. The iterative procedure can be regarded as a generalization of the accelerated Arimoto Blahut algorithm. We obtain a sufficient condition of the weighted parameter
In this paper, we propose an iterative algorithm for calculating Wyner's wire-tap equivocation region. We show that the calculation of the equivocation region can be reduced to the computation of the function Γ(Rb) of the wire-tap channel defined by Wyner. We propose an iterative algorithm for computing Γ(Rb) for a given bit rate Rb, which can be viewed as a generalization of the Arimoto-Blahut method. The convergence of this algorithm is proved and the upper and lower bounds on Γ(Rb) are derived. Simulation results for the calculation of the function Γ(Rb) and the equivocation region are provided.
Wireless systems are susceptible to eavesdropping as an unwelcome result of the broadcast nature of the wireless medium. Fortunately, the wireless medium also provides endowments including the temporal dispersion, the capability of signal superposition and other properties, to facilitate defending against eavesdropping. The focus of the thesis is to provide schemes for utilizing these properties at the physical layer and their performance analysis from an information-theoretic secrecy perspective. The work is based on Wyner's results on the wire-tap channel, in which the transmission between two legitimate users (Alice and Bob) is eavesdropped upon by Eve. The secrecy level is measured by the equivocation rate at Eve. The system performance is measured by the secrecy capacity. This is the maximum rate at which a message can be sent from Alice to Bob, while constraining the equivocation rate at Eve equal to the communication rate, and thus ensuring that Eve cannot deduce any useful information asymptotically. First, the scenario where a secret message needs to be delivered to the intended receiver within a given delay constraint and without the knowledge of the channel state information (CSI) at the transmitter is considered. In particular, hybrid automatic retransmission request (HARQ) protocols are revisited for a block-fading wire-tap channel via a joint consideration of channel coding, secrecy coding, and retransmission protocols. Two secure HARQ protocols: a repetition time diversity (RTD) scheme with maximal-ratio combining, and an incremental redundancy (INR) scheme based on ratecompatible Wyner secrecy codes are analyzed via an outage-based formulation. The throughput of RTD and INR protocols under probabilistic outage requirements is derived. Asymptotic analysis and numerical computations are given to demonstrate both the benefits of HARQ protocols and the tradeoff between reliability and confidentiality in delay-constrained communications. Second, the effects of interference on information-theoretic secret communication is considered. More specifically, the wiretap channel with a helping interferer (WT-HI) is proposed and studied. Here, an interferer, which does not know the confidential message, helps in ensuring the secrecy of the message by sending independent signals. Achievable secrecy rates and upper bounds on the secrecy capacity are given for both discrete memoryless and Gaussian channels. Interference has conventionally been regarded as a harmful phenomenon that leads to decreasing the rate and the reliability performance of wireless systems. In this work, it is shown that interference can be exploited to assist wireless secrecy. Lastly, the problem of secret key generation over fading channels, in which there exists an additional authenticated, public and error-free feedback channel from Bob to Alice, is considered. A key generation scheme consisting of a communication phase and a key-generation phase is presented. The communication phase is based on layered broadcast coding, which adapts the decoded rate at Bob to the actual channel state without requiring CSI available at Alice. The key-generation phase is based on the messages decoded by Bob after the communication phase and the public feedback. The achievable secrecy key rate and the optimal power distribution over coded layers for maximizing the secrecy key rate are characterized. Theoretical and numerical results show that the broadcast approach outperforms the single-level-coding based approach significantly, and hence demonstrate the role of introducing self-interference in facilitating secret-key generation over slowly fading channels when transmit CSI is not available.
One of the problems of information - theoretic security concerns secure communication over a wiretap channel. The aim in the general wiretap channel model is to maximize the rate of the reliable communication from the source to the legitimate receiver, while keeping the confidential information as secret as possible from the wiretapper (eavesdropper). We introduce and investigate the E - capacity - equivocation region and the E - secrecy capacity function for the wiretap channel, which are, correspondingly, the generalizations of the capacity - equivocation region and secrecy - capacity studied by Csiszár and Körner (1978). The E - capacity equivocation region is the closure of the set of all achievable rate - reliability and equivocation pairs, where the rate - reliability function represents the optimal dependence of rate on the error probability exponent (reliability). By analogy with the notion of E - capacity, we consider the E - secrecy capacity function that for the given E is the maximum rate at which the message can be transmitted being kept perfectly secret from the wiretapper.
The concept of the wiretap channel was first proposed by Wyner. He considered the case in which data is to be transmitted reliably over a discrete memoryless main channel to a legitimate receiver. The wiretapper views the output of the main channel through another discrete memoryless channel. It is assumed that the wiretapper knows the code being used and his only handicap is the additional noise in his signal. The problem is to maximize the transmission rate R to the legitimate receiver and the equivocation d of the wiretapper. In this dissertation, the additive white Gaussian noise wiretap channel is introduced and the set of all achievable (R,d) pairs is determined explicitly through the use of certain special properties of the Gaussian channel. Some useful characterizations of a special class of wiretap channels are also explored. A model of the wiretap channel with feedback is proposed. It turns out that with the introduction of feedback, even when the main channel is inferior to the wiretapper's channel, it is still possible to reliably communicate with the legitimate receiver in complete secrecy. The binary erasure wiretap channel with feedback is examined and inner and outer bounds on the achievable (R,d) region are given. Finally, a scheme for enlarging the capacity region of multiple-access channels using feedback is analyzed. It is shown that conditions under which an enlargement is possible are fairly weak, indicating that feedback can almost increase the capacity region. (Author)
We revisit a channel coding problem where the channel state information (CSI) is rate-limited (or coded) and available to the channel encoder. A wiretapper is added into this model, and the confidential message is intended only for the legal receiver and should be kept from being eavesdropped by the wiretapper. Equivocation analysis is provided to evaluate the level of information leakage to the wiretapper. We characterize an achievable rate-equivocation region as well as an outer bound for this security model. To achieve the rate-equivocation triples, we propose an efficient coding scheme, in which the coded CSI serves as the CSI for the channel encoder, based on Gel’fand and Pinsker’s coding and Wyner’s random coding . Furthermore, an example of Gaussian wiretap channel with rate-limited CSI is presented, of which a lower bound on the secrecy capacity is obtained. By simulation, we find there exists an optimal rate of the coded CSI at which the biggest secrecy transmission rate of the Gaussian case is achieved.