IEEE 802.15.4

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The universal testbed (Section 3.2) is applied at the premises of our research group, which is an office area in a university building. Alice is positioned at a predestined access point position. Bob and Eve are mounted on an automated antenna positioning setup, which is located at several predestined “end-device” positions (cf. Figure 4.8). For this, we choose positions which are representative for security-related IoT devices, such as doorknobs (keyless entry systems), window frames (perimeter fence intrusion sensor), and wall (motion detectors) positions. Due to a lack of space, in this version of the paper we restrict ourselves to a description of one representative realization of all experiments.

Figure 4.9 illustrated the 24 positions of the testbeds in the office environment. Table 4.1 summarizes the results of all 24 experiments. The maximum correlation a potential attacker can achieve is 0.7. Interestingly, there is not pattern visible where the best position might be. The positioning (distance to A) where the highest correlated observation occurs lies in the range of 10 mm to 250 mm. The first zero correlation does also not occur at the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle 0.4\mu} distance (50 mm) all the time and seems to be not a predictable property. In the appendix, we provide detailed illustrations of the results in Figures 12.3-12.72.

We perform mobile, long-time narrow-band channel measurements on 2.4 GHz (wavelength 12.5 cm). The data exchange protocol is implemented on three Raspberry Pi 2 platforms (credit card-sized computer). All devices are equipped with a CC2531 USB enabled IEEE 802.15.4 communication interface[1]. The CC2531 is a true SoC solution for IEEE 802.15.4 applications, that is compatible to network layer standards for resource-constrained devices: ZigBee, WirelessHART, and 6LoWPAN. The platform is equipped with proprietary PCB antennas, i.e., Meandered Inverted-F antenna (MIFA), with the size of 5 × 12 mm. Such antennas provide good performance with a small form factor. The platform and antenna design are widely used in commercial products and suited for systems where ultra-low-power consumption is required.


Figure 4.8: The testbed includes several experimental setups for performance evaluation as well as for security analysis. Alice (X), Bob (Y) and Eve (Z) are mounted on a automated antenna positioning system.

In order to establish common channel probing, Alice periodically sends data frames to Bob and waits for acknowledgments. Eve also receives these request-response pairs. When receiving a probe, all three devices extract Received Signal Strength Indicators (RSSI) values and, thus, can measure a channel-dependent sequence over time. For evaluation of the channel measurements, we store and process the realizations of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle v_k := (x_k, y_k, z_k)^T} , locally on a monitoring laptop.

Table 4.2 lists the relevant parameters of our measurement setup. We obtain a complete realization of vk on every sampling interval Ts = 100 msec. The protocol ensures that Alice, Bob, and Eve can probe the channel within a probing duration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle T_p < 5 msec} . We want to analyze the joint statistical properties of the samples with respect to the position of Eve in the scene. As a consequence, we apply an automated antenna positioning system, which is constructed from a low-reflective material, cf. Figure 4.8. It moves the antenna of Eve on a linear guide towards the fixed antenna of Bob in step size Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \Delta_d = 5 mm } with accuracy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \hat{\Delta_d} = \pm 0.05mm} . The variable distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \Delta_{BE}} ranges from 0 to 30 cm in order to provide 60 different locations. Alice’s antenna is placed orthogonal to the linear guiding at a fixed distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \Delta_{AB} = 5 m} . For each position of Eve’s antenna, we record at least N samples.

Alice and Bob extract the common randomness Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle x_k} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle y_k} from a time-varying channel. Since we aim for meaningful and reproducible results, we have to create an environment which provides the joint stationarity to the random process. Therefore, with a distance of 10 cm to Alice’s antenna, we deploy a curtain of 30×30 cm aluminum strips that continuously rotates at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \approx 0.1} rotations per second, cf. Figure 4.8. However, the rotation itself inserts a deterministic component into the channel. The evolution of the self-dependence of channel gains — we show exemplary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle x_k^{ds}} — is illustrated in Figure 3.21. It shows that the mutual information decays rapidly and vanishes after four samples, corresponding to approximately 400 ms. However, due to the continuously rotating curtain of aluminum strips, we discover strong stochastical dependencies after 96 samples, corresponding to approximately 9.6 s. Therefore, we adapt a random source (Unix file /dev/urandom) to the motor controller and program the instrument to rotate with random speed between 0.240 rad/s and 1 rad/s in random directions and with random interval lengths Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle 0^\circ, 1^\circ, . . . 60^\circ} (uniformly distributed). Figure 3.21 shows that no strong stochastical dependencies are given anymore.

Table 4.1 Evaluation results of 23 different channel measurements. The table summarizes the position of the first zero correlation of E as well as the maximum correlation of E as well as its position. The position is defined as the distance between E’s and A’s position.


The testbed includes several experimental setups for performance evaluation as well as for security analysis. Alice (X), Bob (Y) and Eve (Z) are mounted on a automated antenna positioning system.

Evaluation

We now use the experimental measurements to evaluate and compare the results of the Pearson correlation, mutual information, as well as the achievable bound of the secret-key capacity, as a function of attacker’s distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \Delta_{BE}} to Bob. We interpret the original measurements as realizations of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle v_k} . In addition, we have the decorrelated and downsampled outcomes, denoted by the processes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle v_{k}^{de}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle v_k^{ds}} , respectively. The decorrelated samples are obtained by a linear prediction of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle N_m = 30} . To generate the i.i.d. random vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle v_k^{ds}} we downsample Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle v_k} by the factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle N_m = 30} . We have already outlined the necessity of i.i.d. random vectors to obtain accurate estimations. This is not given for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle v_k} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle v_k^{de}} , however, they provide valid approximations, as the results indicate later on. We present three Figures 4.10, 4.11, 4.12 with three Subfigures a)-c) each, which are arranged in a 3x3 matrix on the next page. The rows denote the Figures as follows.

  1. Fig. 4.10 illustrates the results for the original process Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle v_k} .
  2. Fig. 4.11 shows the results for the downsampled process Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle v_k^{ds}} of (2.15).
  3. Fig. 4.12 depicts the results for the decorrelated process Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle v_k^{de}} of (2.18).

The columns constitute Subfigures as follows. For convenience, we introduce generic labels Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle X\in\left\lbrace x_k,x^{\text{de}}_k,x^{\text{ds}}_k \right\rbrace} for Alice, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle Y\in\left\lbrace y_k,y^{\text{de}}_k,y^{\text{ds}}_k \right\rbrace} for Bob and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle Z\in\left\lbrace z_k,z^{\text{de}}_k,z^{\text{ds}}_k \right\rbrace} for Eve.

  1. Subfigures a) show the Pearson correlation (2.19) vs. geometrical distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \Delta_{BE}} between the three pairs (AliceFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \leftrightarrow} Bob Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_{XZ}} , AliceFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \leftrightarrow} Eve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_{XY}} , BobFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \leftrightarrow} Eve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_{YZ}} ).
  2. Subfigures b) zoom into the correlation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \Delta_{XY}} of AliceFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \leftrightarrow} Bob.
  3. Subfigures c) depict the three mutual information results (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle I(X; Y)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle I(X;Z)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle I(Y ;Z)} ) and

the secret-key rate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle R_{sk}} of (2.20) vs. geometrical distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \Delta_{BE}} .

Most of the practical key generation schemes use downsampling or decorrelation on the original observations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle v_k} . We introduce the Figs. 4.10, 4.11 and 4.12 in order to analyze whether downsampling and decorrelation obscure certain features of the channel that are important for the security evaluation of the system. We start with a comparison of the cross-correlation behavior between Alice and Bob, as well as to a potential attacker. By comparing Figure 4.10 (a-b) and Figure 4.11 (a-b) we see that after downsampling no significant differences in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_{XY}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_{XZ}} occur. (Further, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_{XZ}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_{YZ}} are almost identical due to channel reciprocity between Alice and Bob.) The high similarity is due to the fact that even the process vk does not exhibit much dependencies in time, as already hinted in Figure 3.21. As a consequence, the results obtained for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle v_k} expose a valid approximation of the cross-correlation. As it can be seen from Figure 4.11, in the case of downsampling the results are more noisy, since much fewer samples are available for the estimations. After decorrelation, the results (see Figure 4.12) show that (unlike in case of downsampling) the correlation decreases on average by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \approx 0.05} , which can have a significant negative impact on the performance of a potential quantization scheme, cf. [2]. Furthermore, the difference between the minimum and maximum value significantly decreases. Whereas in the original (and downsampled) signal the difference is 0.995−0.98 = 0.015, the difference is 0.97− 0.89 = 0.08 for the decorrelated signal. This probably stems from errors of the autocorrelation estimate (2.16), which is necessary for the linear forward prediction. Another reason might be the Pearson correlation where single outliers (e.g., strong peaks) significantly influence the result. Analyzing the impact of decorrelation techniques on the reciprocity and security in detail is left for future work.


Figure 4.10, 4.11, 4.12

By analyzing attacker’s opportunity, we observe a wavelength-dependent behavior of the correlation between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle z_k} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle x_k} (or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle y_k} ), as illustrated in Subfigures a). The following findings hold for all three processes: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle v_k, v_k^{ds}, v_k^{de}} . The correlation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle v_s} . distance function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_{XZ}} (and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_{YZ}} ) looks similar to the channel diversity function known from Jake’s model [3], which is a zero order Bessel function3 (cf. Figure 2.2). However, the highest correlation is not at distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \Delta_{BE} = 0} , where the correlation is only 0.2. The highest cross-correlation is given at a distance of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \Delta_{BE} \approx 12.5 cm} , which is the wavelength of the 2.4 GHz carrier. The first correlation of zero is given at a distance of 4 cm.

Note that the cross-correlation behavior of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle x_k} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle y_k} is not independent of Eve’s antenna position. Figure 4.10(b) illustrates the correlation behavior in detail. The correlation has an “oscillating” behavior with a wavelength of approximately 11 cm, whereby at a distance of 5 cm the curve decreases rapidly to the lowest level of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \approx 0.98} . The reason for that might be the non-perfect uniformly distributed scatterers in the environment, which are the basis of Jake’s model. The oscillating behavior in Alice’s and Bob’s original observation is also given in the downsampled and decorrelated versions, cf. Figure 4.11(b) and Figure 4.12(b). This behavior is contradictory to theoretical approaches based on Jake’s Doppler spectrum [25]. The reason might be because the narrow band fading models do not include coupling and near-field effects between both antennas for the spatial evaluation of autocorrelation, cross-correlation, and power spectral density (cf. [72, Chapter 3.2]).

The boundary B between the near field zone and the far field zone can usually be determined by the following relationship: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle B\geq \frac{2D^2}{\lambda}} , where D is the largest antenna size [4]. We estimated the size of our antenna to be 6 cm. Therefore, the boundary is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \approx 5.7 cm } . Analyzing near field boundaries in detail is left for future work.

Compared to the cross-correlation behavior between the i.i.d. samples Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle x_k^{ds}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle y_k^{ds}} (after downsampling), both mutual information I(X;Y) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle R_{sk}} have very similar oscillating behavior, shown in Subfigures c). The (minimum, maximum) values of the correlation are (0.980, 0.995) and the ones of the mutual information are (2.1, 2.75). By analyzing Eve’s observation, we see only a slight similarity between the mutual information I(X;Z) (and I(Y ;Z)) to the correlation behavior of her observation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_{XZ}} (and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_{YZ}} ). The similarity can be found by comparing the maximum absolute values. For instance, the highest correlation occurs at 10 cm with a value of 0.5, and corresponds to the highest mutual information of 0.5 bits per sample.

However, the Bessel-like behavior is not evident. Attention should be paid to the fact that the attackers observation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle z_k} does not significantly impact Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle R_{sk}} . Our results show that Rsk is mainly dependent on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle x_k} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle y_k} . However, Eve’s antenna affects Alice’s and Bob’s observation and, therefore, affects Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle R_{sk}} . Table 4.3 summarizes our results.

Conclusion

In this work, we have provided an important pillar to bridge the gap between theory and practice-oriented approaches for CRKG. Our experimental study helps to provide a better understanding of channel statistics in wireless environments for security applications. We presented reproducible results based on a relevant environment which justifies the joint stationarity of a random process. We showed results of cross-correlation, mutual information, and secret-key rates, which are dependent on attacker’s (or third device’s) position. As a result, we discovered that the observer effect occurs, which most probably originates from near field distortions. We believe the effect needs to be considered in the future. Common channel models like Jake’s model for channel diversity need to be extended in order to be valid for key generation setups. Furthermore, it might be pertinent, for instance, to detect the proximity of Eve. Basing on our results two bidirectionally communicating nodes might recognize a third device, its relative position, and its motion in the proximity. Further studies might use complex-valued channel profiles to analyze third party positioning based and motion based influences.

References

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  4. 47