Network Coding and Wireless Physical-layer ... - Jacobs University

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Chapter 6: Wireless Physical-layer Secret-key Generation (WPSG) in Relay Networks: Information Theoretic Limits, Key Extension, and Security Protocol 81 when there is only one or two paths, the correlation is significantly high. In [34], two rigorous metrics have been developed for quantifying the information theoretic limits of key generation in the scenario depicted in Fig. 6.3. Given the users have noisy channel estimates, which are denoted by ĥa, ĥa ′, ĥb, and ĥc, we refer to the maximum number of independent key bits that can be generated per channel observation as I K = I(h a ; h a ′), where I(x; y) denotes the mutual information between x and y. Likewise, the maximum number of independent key bits that can be generated and are secure from an eavesdropper is given by I SK = I(ĥa, ĥa ′|ĥb, ĥc) [76]. In [34], closed-form expressions for I K and I SK are derived for correlated complex Gaussian vector channels. The generated key bits that are not secure are called vulnerable key bits. The number of vulnerable key bits is therefore I V K = I K − I SK . To illustrate the key generation in Fig. 6.3, consider a simple scenario with scalar channels. Alice sends a pilot signal x to Bob, who derives the channel estimate ĥa of the channel h a from the received signal y b = h a x + n b , where n b denotes complex Gaussian noise on Bob’s side. After that, Bob sends x to Alice, who derives the estimate ĥa ′ of h a ′ in a similar manner. The number of available key bits I K per channel observation that can be generated by Alice and Bob is given by I K = I(ĥa; ĥa ′). If the time between Alice’s transmission and Bob’s is short and the same frequency band is used, we can assume reciprocity or h a = h a ′. Assuming further that h a is Rayleigh-distributed with a standard deviation of 0.5, we obtain a simulation result in Fig. 6.4 showing the relationship between I K and the signal-to-noise ratio, which is the ratio of the power of x to the Gaussian noise power at Bob and Alice. We also compute the result when the key is derived from the envelopes of channel parameters only, such that I K = I(|ĥa|; |ĥa ′|) [76]. 6.3 Possible Key Extensions In a general wireless network, the length of the key can be extended by having it generated from several transmission routes instead of only one. Although it is obvious that the total key length is the summation of the lengths from all paths, the key length from each path
Chapter 6: Wireless Physical-layer Secret-key Generation (WPSG) in Relay Networks: 82 Information Theoretic Limits, Key Extension, and Security Protocol 14 12 between channel estimates between channel envelope estimates Mutual Information 10 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 20 SNR(dB) Figure 6.4: A simulation result showing the mutual information between the channel estimates of Alice and Bob, and between their channel envelope estimates is not easily derived when there are relays in between. For instance, if there is one relay in the path, as illustrated in Fig. 6.5, the channel parameter h a between the transmitter and the receiver becomes h aT = h a1 · h a2 , which is the product of two complex Gaussian random variables. If we let the real parts of h aT , h a1 , and h a2 be h aT r , h a1r , and h a2r , respectively, and the imaginary parts be h aT i , h a1i , and h a2i , respectively, we have h aT i = h a1i h a2r + h a1r h a2i . (6.1) If we denote the random variables representing the random processes that generate h aT i , h a1i h a2r , and h a1r h a2i by Z, X, and Y respectively, the probability density function f Z (z) will be the convolution of f X (x) and f Y (y). f Z (z) = ∫ ∞ −∞ f X (z − y)f Y (y)dy . (6.2) In our case, X and Y have the same probability density function f X (x) = f Y (x) = 4 K 0(4x) π , (6.3)
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Network Coding and Wireless Physica
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Abstract The general abstraction of
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Contents Abstract i Abstract ii Dec
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vi CONTENTS 5.4 Degree Distribution
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List of Figures 2.1 A basic digital
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x LIST OF FIGURES 5.11 Comparison o
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Part I Motivation, Overview, and In
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Chapter 1: Motivation and Overview
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Chapter 2 Introduction to Digital C
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Chapter 2: Introduction to Digital
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Page 46 and 47: 33 The emergence of scalable video
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Page 132 and 133: Bibliography [1] 3GPP, “3GPP; Tec
Page 134 and 135: BIBLIOGRAPHY 121 [21] F.A. Hayek,
Page 136 and 137: BIBLIOGRAPHY 123 [43] M. Friedman a
Page 138 and 139: BIBLIOGRAPHY 125 [63] S. Jaggi, P.
Page 140: BIBLIOGRAPHY 127 [81] X. Sun, X. Wu
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