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IEEE International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob)

An Iterative Algorithm for Wyner’s Wire-Tap Equivocation Region

Meena Gangula

Published Oct 1, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.1109/wimob.2019.8923306

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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.
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 derive an exponentially decaying upper-bound on the unnormalized amount of information leaked to the wire-tapper in Wyner's wire-tap channel setting. We characterize the exponent of the bound as a function of the randomness used by the encoder. This exponent matches that of the recent work of Hayashi [12] which is, to the best of our knowledge, the best exponent that exists in the literature. Our proof (like those of [17], [18]) is exclusively based on an i.i.d. random coding construction while that of [12], in addition, requires the use of random universal hash functions.
Based on Arimoto’s work in 1972, we propose an iterative algorithm for computing the capacity of a discrete memoryless classical-quantum channel with a finite input alphabet and a finite dimensional output, which we call the Blahut–Arimoto algorithm for classical-quantum channel, and an input cost constraint is considered. We show that, to reach ε accuracy, the iteration complexity of the algorithm is upper bounded by log n log ε ε where n is the size of the input alphabet. In particular, when the output state { ρ x } x ∈ X is linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an ε accurate solution with a complexity of O ( m 3 log n log ε ε ) , and O ( m 3 log ε log ( 1 − δ ) ε D ( p * | | p N 0 ) ) in the special case, where m is the output dimension, D ( p * | | p N 0 ) is the relative entropy of two distributions, and δ is a positive number. Numerical experiments were performed and an approximate solution for the binary two-dimensional case was analysed.
Based on Arimoto's work in 1978 [1], we propose an iterative algorithm for computing the capacity of a discrete memoryless classical-quantum channel with a finite input alphabet and a finite dimensional output, which we call the Blahut-Arimoto algorithm for classical-quantum channel, and an input cost constraint is considered. We show that to reach ε accuracy, the iteration complexity of the algorithm is up bounded by log n log ε ε where n is the size of the input alphabet. In particular, when the output state {ρx}x∈Xis linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an ε accurate solution with a complexity of O(m3log n log ε/ε), and O(m3log ε log(1-δ)D(ε/p*||pN(0))) in the special case, where m is the output dimension and D(p*||pN(0)) is the relative entropy of two distributions and δ is a positive number.