In this paper, we study cyclic shift symmetric wiretap channels in which the channels between Alice and Bob and Alice and Eve are both cyclic shift symmetric. We characterize the rate-equivocation region by determining the optimal selection of rate splitting U and channel prefixing V for these channels. We show that optimal U and V are determined via cyclic shifts of the solution of an auxiliary optimization problem that involves only one auxiliary random variable. We find the cardinality bound on the necessary auxiliary variable and formulate the problem as a constrained optimization problem. We determine the optimality conditions for the binary-input cyclic shift symmetric wiretap channels. We find the optimum by inspecting each point of the I(X;Y) - I(X; Z) function and ruling out the sub-optimal candidates that satisfy the optimality conditions. In particular, we address BSC-BEC and BEC-BSC wiretap channels. By using the optimality conditions, we determine the optimal selections of U and V for the rate-equivocation regions of these channels.
In this paper, we study cyclic shift symmetric wiretap channels in which the channels between Alice and Bob and Alice and Eve are both cyclic shift symmetric. We characterize the rate-equivocation region by determining the optimal selection of rate splitting U and channel prefixing V for these channels. We show that optimal U and V are determined via cyclic shifts of the solution of an auxiliary optimization problem that involves only one auxiliary random variable. We find the cardinality bound on the necessary auxiliary variable and formulate the problem as a constrained optimization problem. We determine the optimality conditions for the binary-input cyclic shift symmetric wiretap channels. We find the optimum by inspecting each point of the I(X;Y) - I(X; Z) function and ruling out the sub-optimal candidates that satisfy the optimality conditions. In particular, we address BSC-BEC and BEC-BSC wiretap channels. By using the optimality conditions, we determine the optimal selections of U and V for the rate-equivocation regions of these channels.
Characterization of the rate-equivocation region of a general wiretap channel involves two auxiliary random variables:U, for rate splitting andV, for channel prefixing. In this paper, we explore specific classes of wiretap channels for which the evaluation of the rate-equivocation region is simpler. We show that if the wiretap channel is more capable,V=Xis optimal and the boundary of the rate-equivocation region is achieved by varying rate splittingUalone. Conversely, we show under a mild condition that if the wiretap channel is not more capable, thenV=Xis strictly suboptimal. Next, we focus on the class of cyclic shift symmetric wiretap channels. We show that optimal rate splittingUthat achieves the boundary of the rate-equivocation region is uniform with cardinality |X| and the prefix channel between optimalUandVis expressed as cyclic shifts of the solution of an auxiliary optimization problem over a single variable. We provide a special class of cyclic shift symmetric wiretap channels for whichU=φ is optimal. We apply our results to the binary-input cyclic shift symmetric wiretap channels and thoroughly characterize the rate-equivocation regions of the BSC-BEC and BEC-BSC wiretap channels.
Csiszár and Körner's characterization of the rate-equivocation region of a general wiretap channel involves two auxiliary random variables: U, which represents rate splitting and V, which represents channel prefixing. For some channels, one or both of these auxiliary random variables are unnecessary, simplifying the expression and evaluation of the rate-equivocation region. In this paper, we provide new conditions under which channel prefixing or rate splitting does not improve the rate-equivocation region. In particular, we show that when the main channel is more capable than the eavesdropping channel, channel prefixing is unnecessary; the entire rate-equivocation region can be achieved by rate splitting alone. Conversely, we show under a mild assumption that if the main receiver is not more capable, then channel prefixing is strictly necessary. Moreover, we show that if the main channel is more capable but not less noisy, then rate splitting is strictly necessary. Next, we focus on the set of cyclic shift symmetric channels. We prove that for these channels, if in addition I(X; Y) - I(X;Z) is maximized at the uniform distribution, then rate splitting is unnecessary. Our results apply to BSC-BEC and BEC-BSC wiretap channels. We identify the conditions on the parameters of the BSC and BEC under which channel prefixing and/or rate splitting are unnecessary.
This work develops new achievable rate regions for the two way wiretap channel. In our setup, Alice and Bob wish to exchange messages securely in the presence of a passive eavesdropper Eve. In the full-duplex scenario, our achievability argument relies on allowing the two users to jointly optimize their channel prefixing distributions, such that the new channel conditions are favorable compared to that of Eve. Random binning and private key sharing over the channel are then used to exploit the secrecy advantage available in the equivalent cascade channel and to distribute the available secrecy rate among users. For the half-duplex case, we introduce the idea of randomized scheduling and establish the significant gain it offers in terms of the achievable secrecy sum-rate. We also quantify the gains that can be leveraged from the proposed schemes in the modulo-2 and Gaussian channels via numerical results in certain selected scenarios.
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
We consider the Gaussian wiretap channel with amplitude and variance constraints on the channel input. We first show that the entire rate-equivocation region of the Gaussian wiretap channel with an amplitude constraint is obtained by discrete input distributions with finite support. We prove this result by considering the existing single-letter description of the rate-equivocation region, and showing that discrete distributions with finite support exhaust this region. Our result highlights an important difference between the peak power (amplitude) constrained and the average power (variance) constrained cases. Although, in the average power constrained case, both the secrecy capacity and the capacity can be achieved simultaneously, our results show that in the peak power constrained case, in general, there is a tradeoff between the secrecy capacity and the capacity, in the sense that, both may not be achieved simultaneously. We also show that under sufficiently small amplitude constraints the possible tradeoff between the secrecy capacity and the capacity does not exist and they are both achieved by the symmetric binary distribution. Finally, we prove the optimality of discrete input distributions in the presence of an additional variance constraint.
The problem of secure lossy source-channel wiretapping with arbitrarily correlated side informations at both receivers is investigated. This scenario consists of an encoder (referred to as Alice) that wishes to compress a source and send it through a noisy channel to a legitimate receiver (referred to as Bob). In this context, Alice must simultaneously satisfy the desired requirements on the distortion level at Bob, and the equivocation rate at the eavesdropper (referred to as Eve). This setting can be seen as a generalization of the conventional problems of secure source coding with side information at the decoders, and the wiretap channel. Inner and outer bounds on the rate-distortion-equivocation region for the case of arbitrary channels and side informations are derived. In some special cases of interest, it is shown that separation holds. By means of an appropriate coding, the presence of any statistical difference among the side informations, the channel noises, and the distortion at Bob can be fully exploited in terms of secrecy.