Feature Extraction for Channel Reciprocity Based Secret Key Generation Methods

I. INTRODUCTION

THE evolution towards the 5G-Internet of Things (IoT) enables conventional systems to integrate with various technologies and develop related innovative applications. Millimeter Wave (mm-Wave) wireless communication addresses the requirements of these applications concerning high energy efficiency and low latency 1], [2]. Hence, industrial IoT applications deployed in densely populated environments and application scenarios with a very high level of network connectivity benefit from mm-Wave wireless communication, as it provides them with an ultra-high speed and reliable communications over relatively short distances [3], [4]. Despite enabling easy integration of devices, the IoT poses significant challenges to communications security due to the nature of the edge devices used. Due to current constraints related to energy and resource demands necessary for cryptography implementation, the secure exchange of sensitive data over wireless channels requires relatively powerful sensors and devices with larger batteries [5]. One common solution for securing communication is through the use of symmetric cryptography. However, asymmetric cryptography, such as the Diffie-Hellman key agreement protocol, often facilitates the necessary key exchange, which introduces significant computational overhead and typically requires a trusted third party to manage public keys [6]. Furthermore, the emergence of quantum computing poses a significant threat to the security of asymmetric cryptography, as it provides a substantial boost in computational power [7]. PLS is a promising strategy for overcoming these limitations, harnessing the reciprocal nature of wireless transmission channels among trusted entities to create secure keys. This approach relies on channel reciprocity, which indicates the consistency of multi-path fading between uplink and downlink signals within the channel’s coherence time. This similarity ensures that the transceivers can obtain identical channel measurements. The techniques that generate such keys are called CRKG [8]–[10].

The idea of CRKG was presented by Maurer [11] as well as Ahlswede and Csiszar in 1993 [12]. They proposed a security scheme to derive common shared secret keys from correlated observations of randomness, while a third party cannot infer the derived secrets from its correlated observations. This is because within the coherence time of the channel, the twotransceiver links are reciprocal and the observation of two legitimate partners is highly correlated. In recent literature, various CRKG-based approaches have been proposed for PLS, most of which are inspired by Claude Shannon’s definition of information-theoretic security to establish a secure transmission model in communication systems, followed by the idea of the wiretap channel model proposed by Wyner [13]. Ever since the main idea of CRKG was revealed in the 90s [12], various significant improvements have been made theoretically and practically in this area. These contain comprehensive theoretical foundations that prove both feasibility and theoretical security of information [8].

A. Related Works

One of the initial approaches in the field of CRKG involved utilizing the Received Signal Strength Indicator (RSSI) as a common characteristic of wireless channels, as proposed by Mathur et al. [14]. They showed that by using off-theshelf 802.11a cards to collect coarse RSSI measurements, secret bits with a noticeable rate can be derived in a real, indoor wireless environment. Jana et al. [15] introduced an additional RSSI-based method with a focus on the quantization part. They put forth a distinct quantization approach named Adaptive Secret Bit Generation (ASBG), which applied block-wise processing to create the necessary statistical parts for effective quantization. Their strategies incorporated the Cascade scheme [16] for information reconciliation, a technique that found adoption in upcoming endeavors. The scope of their work later expanded to encompass Multiple Input Multiple Output (MIMO) setups and technologies like IEEE 802.15.4 [15] and IEEE 802.15.1 (Bluetooth) [17]. In a subsequent study [18], researchers delved into the analysis of RSSI values across multiple channels. They introduced a robust multilevel quantization system based on block-wise statistics. Additionally, they proposed an innovative information reconciliation scheme where the offset of the current block’s mean is communicated to the partner. By subsequently adjusting their measurements based on this information, they achieved a remarkable reduction in Bit Error Rate (BER), resulting in highly accurate data transmission.

Although systems relying on RSSI are common for practical implementations, there are significant limitations associated with this approach. One major issue is the lack of consideration for the complex multi-path propagation in wireless channels. Furthermore, while the channel maintains reciprocity, there is a lack of standardized methods for calculating RSSI values. The calculation approach for reciprocity in wireless channels varies among different hardware providers, leading to the potential for discrepancies between measured values. This lack of standardized calculations can hinder the accurate reflection of the inherent reciprocity of the channel. Therefore, an alternative solution is to apply Channel State Information (CSI), which can be acquired by Channel Impulse Response (CIR) measurements and Channel Frequency Response (CFR) as a source of randomness, where all the available information in the physical channel can be exploited [5], [19].

The CSI constitutes a comprehensive parameter offering detailed insights into channel characteristics. CSI-based systems exhibit a remarkable Key Generation Rate (KGR) and have demonstrated resilience against predictable channel attacks. In this context, CSI primarily encompasses CIR and CFR. CIR captures both amplitude and phase information of multipath channels, which are pivotal for key generation. However, phase information is susceptible to noise and other disruptions, constraining its practical implementation. Notably, the amplitude of CIR, especially in Ultra Wide Band (UWB) systems, can be effectively estimated [5]. Nevertheless, the power of CIR diminishes with delay, presenting challenges for key generations. These challenges can be mitigated through various techniques such as employing peak CIR or adaptive quantization algorithms. CFR delineates channel characteristics in the frequency domain and finds common usage in Orthogonal Frequency-Division Multiplexing (OFDM) systems, particularly in IEEE 802.11 OFDM systems [1]. Practical implementations often utilize only the amplitude of CFR due to the complexities associated with phase estimation arising from time and frequency offsets. CFR-based systems benefit from the uniform power distribution across all frequencies in uncorrelated scattering environments, thereby contributing to the augmentation of KGR. While CFR is more prevalently utilized for Secret Key Generation (SKG) in OFDM systems, there exist studies reporting CIR-based secret key generation [1]. While CFR is commonly preferred for SKG in OFDM systems, certain studies have showcased the effectiveness of CIR-based secret key generation methods. Both CIR and CFR provide crucial insights into channel characteristics, although they encounter challenges such as susceptibility to noise and offset disturbances. Nonetheless, they are indispensable for attaining high KGR in wireless communication systems.

Ye et al. [20] were among the pioneers in the theoretical exploration of CSI for CRKG, proposing the use of direct quantization to derive bits from Gaussian random variables. They introduced an Low-Density Parity-Check (LDPC) code for information reconciliation, while their primary contribution involved analyzing the theoretical secret key rate relative to the Signal-to-Noise Ratio (SNR) of the channel. Kitaura et al. [21] also proposed a system based on time delays of simulated UWB multi-path components. They used syndromebased decoding and achieved perfect agreement between Alice and Bob while maintaining at least a 2-bit difference from Eve (with a 30-bit key length). However, this method needs robust detection of multi-path clusters, which remains a challenge in real-world measurements. The method was adopted by Huang et al. [22], who effectively employed Rake Receivers to extract time delays between different multi-path clusters. Subsequently, they applied the cascade protocol for information reconciliation without incorporating privacy amplification. The assessment primarily focused on the BER, ranging from 1% to 10% depending on the simulated SNR for Line of Sight (LOS) and Non-Line of Sight (NLOS) channels. Marino et al. [23] carried out real-world UWB CIR measurements to compare various quantization schemes with a focus on the importance of time synchronization for raw CIRs. The evaluation, in terms of BER, revealed that the guard band-based quantization achieved mismatches ranging from 9% to 41% for legitimate partners, and from 46% to 52% for potential eavesdroppers. Additionally, correlations of the no-cut material were reported, with values between 0.89 and 0.97 for legitimate partners, and between 0.03 and 0.43 for potential eavesdroppers.

Zhang et al. [24] presented a method for generating secure key bits from the channel responses of OFDM subcarriers, which quantizes key bits across all subcarriers. The method quantizes key bits from individual subcarrier responses, enabling modeling as Wide Sense Stationary (WSS) random processes. To validate their approach, the authors implement a time-variant multi-path fading channel and an IEEE 802.11 OFDM transceiver. They explore correlations between adjacent measurements and the randomness of generated key sequences, introducing the X% coherence time concept to quantify correlations. Notably, the paper challenges the conventional 50% coherence time assumption for ensuring key randomness.

In [25], real-world experimental findings using 802.11n devices are presented. They revealed that near subcarriers exhibit similar physical characteristics, potentially leading to correlated CSI measurement values and compromised key entropy. This work used hash functions to ensure bitstream consistency between communicating parties and generated secret bits via key recombination from multiple subcarriers to enhance efficiency against correlated bitstreams.

Meanwhile, to the best of our knowledge, there has not been a comprehensive investigation into the impact of CIR features on the security of key generation in terms of BER, Mutual Information (MI), and entropy within the context of CRKG.

B. Main Contributions

The primary goal of this paper is to extract and analyze relevant features from CIR measurements for input into the SKG procedure, as opposed to utilizing the raw measurements. We adopt the IEEE 802.15a channel specifications for UWB systems, to minimize BER and enhance entropy within the resulting key generation process in such a system. We evaluate the security performance of the intended method based on different metrics, such as Shannon Entropy, MI, and BER. For this purpose, we utilize time and frequency domain features in the key generation data processing step. Then we demonstrate that the proposed feature set is highly beneficial as an input to the SKG procedure compared to applying the raw CIR’s measurement applied in conventional methods. This research aims to leverage the fundamental attributes of the wireless channel, such as path loss exponent, Doppler spread, and coherence bandwidth, to identify features that accurately represent the CIR or are influenced by it. To capture the static characteristics of the wireless channel, well-known statistical measures like [TeX:] $$n^{\text {th }}$$ order moments are also employed. Our inspiration for this approach stems from prior studies such as [26]–[28] and [29]–[31], which have successfully selected specific CIR features for applications like signal fingerprinting and localization.

C. Outline

The remaining content of the paper is organized as follows. In Section II, the functionality of CRKG, the system model, and the technical background are described, and the main assumptions and performance metrics required for our analyses are introduced. Then, features and their extraction for the time and frequency domains are explained in Section III. Simulation results are presented in Section IV. Finally, Section V gives an overview of the results and conclusions.

II. BACKGROUND

Measuring the channel properties through multi-path propagation in wireless channels is difficult for attackers due to the spatial decorrelation. The reciprocal channel assumptions can be fulfilled when the spatial distance between Alice and Eve is larger than half a wavelength of the modulated signal [32]. The legitimate partners can exploit this by generating a common secret key through exchanging messages over an authenticated channel. The messages are exchanged to fulfill four essential steps: Randomness sharing, advantage distillation [33], information reconciliation, and privacy amplification [34].

Random sharing is an indispensable part of the key generation process. This step is pivotal in ensuring that the CRKG meets its requirements by generating keys that are independent identically distributed (i.i.d) and possess high entropy [35]. During the advantage distillation step, continuously measured channel profiles are quantized into bit vectors to obtain initial preliminary key material. Then, the same fixed-length key strings can be achieved in the step of information reconciliation. This is achieved by sharing information through a secure, authenticated channel to fix errors, often utilizing errorcorrecting codes for support. Finally, privacy amplification is applied to the reconciled key strings to reduce the effect of leaked information throughout the information reconciliation phase. This paper focuses only on the randomness sharing and advantage distillation phases.

A. System Model and Assumptions

This paper considers a half-duplex mode within an intended UWB communication system, in which the CIR as a source of randomness contains all channel properties. Impulse Radio UWB channels have the advantage of utilizing a wide frequency spectrum, allowing them to capture a substantial number of multipath components effectively. Taking into account the time-variant characteristics of the complex channel, according to the model proposed in [36], by analyzing the variances between successive measurements, the model can be expressed as [34]

where [TeX:] $$\delta(\cdot)$$ signifies the Dirac delta function, while N(t) denotes the count of resolvable multipath components at time t. Each individual path, indexed by n, is characterized by attenuation [TeX:] $$\alpha_n(t),$$ phase shifts [TeX:] $$\phi_n(t),$$ and propagation delays [TeX:] $$\tau_n(t)$$ These parameters collectively shape the channel’s characteristics and describe CIR multipath components, being considered as inputs for CRKG in some literature [37], [38].

In our model, Alice sends an initial probing message to Bob and then receives Bob’s message in the same way as shown in Fig. 1.

In this model, there is a difference between Alice and Bob’s transmission channel, i.e., [TeX:] $$h_{A B} \text { and } h_{B A}$$, in comparison with the channel between Alice and Eve [TeX:] $$\left(h_{A E}\right)$$. The assumption of channel quality between legitimate communication partners over the attacker arises from various channel propagation characteristics, for example, fading, interference, or spatial variations and SNR, some of which are useful for security purposes. However, the broadcast nature of the wireless channels allows Eve to listen to the message exchange and record her channel measurements. Despite knowing the key generation algorithms between Alice and Bob, Eve cannot modify the protocol messages, as the communication is considered over an authenticated wireless channel, which is time-variant due to noise, interference, movement of objects and terminals. To account for the assumption of a static channel, we collect our observations on both the Alice and Bob sides within a single time coherence interval, which can be represented as

where [TeX:] $$h_{A B}$$ stands for the estimated reciprocal channel on Bob’s side, which is followed by Bob transmitting a similar probing message in a half-duplex transmission setup. Since the channel coefficient changes due to movement speed, the coherence time interval of the channel depends on the Doppler frequency and transmission wavelength [39]. On the other hand, channel properties should change sufficiently to gain fresh entropy for new keys, which could be achievable due to the dynamic nature of the channel implied by terminal motion or objects passing through the channel’s LOS. Additionally, enough measurements should be taken to detect noticeable changes in the CIR components to meet the fresh entropy. This is attainable if the CIR components are sampled at a frequency [TeX:] $$f_s$$ of at least twice the maximum Doppler frequency [40], i.e. [TeX:] $$f_s \geq 2 f_D.$$ It should be noted that after the acquisition of CIR measurements in each terminal, h can be expressed as a vector of complex values that represents the channel properties by the two legitimate partners. It can then be applied to signalprocessing steps in the time and frequency domains.

To analyze and evaluate the performance of each feature extracted from CIRs to be applied for the CRKG procedure, the following performance metrics are used [41].

· Entropy is a metric to express the amount of uncertainty of the quantized features, which can be expressed as

where p(x) denotes the probability of the variable x within the context of a specific bit sequence [TeX:] $$\mathrm{x},$$ which could represent the bit quantization output of Alice, Bob, or Eve. Additionally, [TeX:] $$\log _2$$ is utilized since the quantization output is binary, thus expressing entropy in bits. To prevent an attacker from guessing the generated key, it is essential to maximize entropy, considering its range between 0 and 1. It should be noted that entropy achieves its maximum when the probability distribution of the final key bit string is uniform across both 0 and 1.

· Mutual Information (MI) is a valuable metric that quantifies the mutual dependence between two random variables, which reflects the extent to which the knowledge of one variable [TeX:] $$\mathrm{y}$$ enhances our understanding or prediction of the other variable [TeX:] $$\mathrm{y}.$$ In the context of key generation, it is desirable to have a high amount of MI between Alice and Bob and a low amount of MI between Eve and Alice/Bob.

where the probabilities [TeX:] $$p_{\mathbf{x}, \mathbf{y}}(x, y), p_{\mathbf{x}}(x), \text { and } p_{\mathbf{y}}(y)$$ are the joint and marginal probability distributions of the variables [TeX:] $$\mathrm{x}, \text { and } \mathrm{y}$$ in the bit sequence space, respectively.

· Bit Error Rate (BER) is a metric that quantifies the proportion of bits in the resulting key from Alice and Bob’s protocol that do not align. This metric is applied to the quantized extracted feature and describes how well the results of two quantized feature series match. This measure can also be assessed from Eve’s perspective, ideally aiming for around 50% to indicate optimal security [42].

[TeX:] $$x \oplus y$$ denotes the XOR operation between the bit strings x and y, weight([TeX:] $$x \oplus y$$) is the number of 1s in the XOR result (i.e., the number of errors), and length(x) represents the length of the bit strings.

III. PROPOSED CRKG METHOD VIA FEATURE EXTRACTION TECHNIQUE

This section introduces a novel feature extraction scheme in the time and frequency domains to eventually get robust and secure key material from CIR measurements.

Typically, hardware introduces complex channel properties. Despite the challenges associated with estimating the phase of the CIR owing to its time-variant nature, the attenuation parameter [TeX:] $$\alpha_n(t)$$ stands out as a robust source of key derivation material, as reported in prior research [43]. In CRKG, emphasis is often placed on the amplitude of the CIRs, with the phase receiving comparatively less attention in most scenarios. This preference arises for several reasons. Firstly, capturing the amplitude information of CIRs is generally simpler and demands less complex hardware and resources, potentially leading to streamlined mathematical operations for key generation. This can reduce the processing power required in resourceconstrained sensor devices, in contrast to the more intricate task of precisely measuring phase information [44]. Moreover, measuring the amplitude of CIRs in real-world channels tends to be more straightforward and reliable than measuring phase. Phase measurements can be sensitive to factors such as antenna imperfections and synchronization issues between transmitters and receivers, introducing additional phase shifts and rendering them less dependable for key generation. Despite focusing solely on amplitude, satisfactory randomness and key generation rates can often still be achieved for many applications. Additionally, the coherence time of the wireless channel, which dictates the correlation between uplink and downlink channels, permits the use of amplitude-based key generation techniques [45]. Furthermore, radio channels inherently exhibit variability, with phase fluctuating rapidly due to factors like multipath propagation, ultimately diminishing the cross-correlation between uplink and downlink in our TDD half-duplex communication setup. While both amplitude and phase theoretically offer avenues for key generation, phase information may be more susceptible to interception or manipulation by attackers, who could introduce pre-intended noise to distort the signal. Therefore, when extracting features in the time domain, we prioritize the absolute values of the CIRs.

Therefore, the amplitude vector [TeX:] $$\mathbf{h}_{\mathcal{A}}$$ for Alice, Bob and Eve can be defined as

Considering potential hardware imperfections and synchronization challenges that may introduce phase discrepancies in the CIR, thereby potentially undermining reciprocity between legitimate partners, we extend our analysis. In addition to evaluating features derived from the original CIR [TeX:] $$\mathbf{h}_{\mathcal{C}}$$, we also compute and assess features in the frequency domain based on the Discrete Fourier Transform (DFT) over [TeX:] $$\mathbf{h}_{\mathcal{C}}$$ and its amplitude. Therefore, the frequency representation of CIR measurements can be obtained by applying the DFT to the CIRs, as demonstratedbelow.

where N denotes the number of components in each estimated CIR, [TeX:] $$k \in[0, N-1]$$ represents the frequency index. Furthermore, mirroring the procedure applied to [TeX:] $$\left|h_{\mathcal{C}}[n]\right|,$$ we derive the features from the absolute values of [TeX:] $$H_{\mathcal{C}}[k] \text { and } H_{\mathcal{A}}[k] .$$ Subsequently, the vectors of absolute values are formally defined as

In this scenario, the LOS component or the first dominant component between Alice and Bob is considered, which is beneficial for finding a strong starting data point for exploiting the features of the surrounding samples. To avoid considering unwanted samples and to reduce the computational complexity, we set the cut length of size L and apply it to the initial received signal with length N. This approach allows us to compute CIR features throughout L points, beginning with the most dominant component. Consequently, we employ 128 sample points from the signal component, referred to as the “Cut” signal, and contrast the outcomes with features extracted from the “No-Cut” version of the CIR signal.

Finally, a quantization step as an indispensable part of CRKG, which transforms continuous values of extracted features into bits, is applied to the features’ vector. In this scheme, we use a multi-bit quantization method where a window with a size of B is applied to the feature vector to detect slow changes in values (more details in Sec. IV). Subsequently, the quantized bins within the window range are assigned to a Gray code [46].

In general, feature extraction is the process of transforming raw data into numerical features to reduce the computational complexity of big data. This avoids information redundancy and implies that data can be processed while keeping the information of the original data set. Following feature extraction, feature selection can also determine which subset of the feature vector contains more important information about the initial data set. Then, it can be replaced with the initial features vector without losing important information.

Hence, by analyzing features extracted from CSI, we can gain insights into channel properties like path loss, attenuation, delay spread, Doppler spread, and noise. These features, along with the inherent randomness from noise and interference, are incorporated into key generation. This diversity makes the key unique to the specific channel and difficult to predict or replicate in different scenarios, enhancing security. Additionally, raw CIR directly reflects the channel but lacks inherent randomness and is predictable using channel models. Therefore, feature extraction overcomes this limitation by capturing diverse and unpredictable aspects of the channel, leading to more secure keys. Here, the rationale behind selecting specific features based on their correspondence to channel properties and their contribution to the security of the generated keys is explained. These features capture the inherent randomness and variations within the channel, making it difficult to predict the CIR and enhancing key security.

· Path Loss, Attenuation, and Noise: These properties affect the overall signal strength and introduce randomness in the received CIR. Features like peak amplitude, received signal energy, in both domains, decay time index, peak decay exponent, variance, and mean absolute deviation in the time domain, and Frequency Spectrum Integration Area (FSIA) capture channel properties and randomness, thereby potentially enhancing security against eavesdropping.

· Delay Spread: This characterizes the time delay between the arrival of different signal copies. In the time domain, some features represent this property. Features such as the power ratio between the first and second peaks, where a larger ratio implies greater spread; the ratio between the strongest component and the total components indicates less spread with a higher ratio. Variance reflects a wider arrival time distribution, and the number of points above or below the mean denotes delayed components. In addition, the number of peaks directly correlates with the number of delayed paths contributing to the signal, while the longest strikes above or below the mean suggest a larger spread. Moreover, mean excess delay directly measures the average delay spread.

· Doppler Frequency: This frequency shift due to relative motion between transmitter and receiver affects the temporal energy distribution within the CIR. Features like the number of peaks, longest strike above/below the mean, decay time index, and peak decay exponent capture these temporal variations, contributing to key uniqueness based on specific channel dynamics.

By capturing the unique characteristics of the communication channel, these features contribute to key unpredictability and improved security metrics like BER, MI, and key entropy. The categorization based on channel properties provides a framework for understanding how specific features enhance key generation security.

Therefore, we investigate and analyze the robust features in the time and frequency domain of the signal, which have also been used in some related works, to find the final feature set that contains a strong representation of the raw CIR measurements [28], [47]. Below is an overview of the features:

A. Time and Frequency Features

There are several key features to consider when examining the features that can be extracted in both the time and frequency domains. These features provide valuable insights into the characteristics of signals and how they vary over time and across different frequencies. From this point forward, we refer to [TeX:] $$h_{\mathcal{C}} \text { as } h, H_{\mathcal{A}} \text { and } H_{\mathcal{C}} \text { as } H,$$ while omitting the subscripts denoting Complex ([TeX:] $$\mathcal{C}$$) and Amplitude ([TeX:] $$\mathcal{A}$$) for simplicity. Here are some important features to consider:

1) Peak Amplitude represents the maximum value of the CIR envelope in both the time and frequency domains. In the time domain, if there is a LOS component, it will typically have the dominant value. The calculation of peak amplitude in both domains can be expressed as follows.

where [TeX:] $$\mathbf{h} \text{ and } \mathbf{H}$$ denote the representations of the CIR’s components in the time and frequency domains, respectively, as derived from equations (6) and (8).

It should be noted that finding a feature like “peak amplitude” directly in complex numbers is not straightforward due to the lack of an inherent ordering in the complex domain. Hence, we extract features in the frequency domain using the following methods. In amplitude-based, we compute the absolute values of complex CIRs, apply DFT, then find the peak by taking the absolute values of the DFT results. In complex-based, we apply DFT directly to the complex CIRs, then find the peak by taking the absolute values of the DFT coefficients. Both rely on the absolute values of complex numbers to ensure meaningful and consistent feature extraction.

2) Received Signal Energy refers to the total energy of the received signal.

It should be noted that as the most amount of signal energy received from the LOS component, there is expected to not be a noticeable difference between the no-cut and cut performance of the CIRs.

3) Power Ratio between the first and second strongest components, and the power ratio between the strongest and total components, are two features derived from the received signal energy and peak amplitude of the received CIR’s components. A high power ratio indicates a dominant path or potential LOS, with potentially less interference from other paths, carrying a significant portion of the received signal’s power. These features aid in characterizing the dominance of specific multi-path components within the CIR.

where [TeX:] $$\mu_{\mathbf{h}} \text { and } \mu_{\mathbf{H}}$$ are the mean absolute value of the CIR’s time and frequency representations, respectively. In the context of signal amplitudes, variance indicates the variability within the channel environment. A higher variance signifies greater fluctuations in signal amplitudes, often associated with complex channel conditions that involve multiple significant multi-path components. In such scenarios, the channel environment can be more challenging due to the presence of various signal paths with different strengths and delays. Conversely, a lower variance suggests that the received signal exhibits relatively stable amplitudes. This implies that the channel environment is less prone to significant fluctuations or variations in signal strength. In such cases, the received signal is more consistent and predictable in terms of its amplitude behavior. By considering the variance of signal amplitudes, we can gain insights into the stability and complexity of the channel environment.

5) Fano Factor (FaF) is a metric that quantifies signal strength variability among the multi-path components in both the time and frequency domains. It is defined as the ratio of the variance to the mean of the CSI measurements. The FaF can be calculated in both domains as

By applying the FaF to CIRs, we can assess signal strength variability among the multi-path components. This parameter provides insights into the distribution and dispersion of CIR components. A FaF greater than 1 indicates greater variance than the mean, suggesting randomness in the signal strength of the multi-path components. Conversely, an FaF less than 1 implies a lower variance than the mean, indicating a more regular pattern in the signal strength of the multi-path components. The FaF is a useful tool for analyzing the variability and patterns in signal strength. It can be applied in various fields, including wireless communication, channel modeling, and signal processing. By understanding the FaF, we can gain valuable insights into the characteristics of the channel and make informed decisions in signal analysis.

6) The number of points above and below the mean provides insights into the distribution of CIR components and the symmetry or skewness of their distribution. By analyzing the distribution of points above and below the mean, we can gain insights into the signal propagation in the channel. If there are more points above the mean, it suggests that the distribution may be skewed to the right (positively skewed). On the other hand, if more points are below the mean, the distribution may be skewed to the left (negatively skewed). Skewness refers to the degree of asymmetry observed in a probability distribution. A positively skewed distribution has a longer right tail, while a negatively skewed distribution has a longer left tail. Skewness can be used to describe the shape and characteristics of a dataset. It is important to note that the number of points above and below the mean is just one aspect of assessing the distribution and skewness of data. Other statistical measures, such as skewness and kurtosis, can provide a more comprehensive understanding of the data distribution and its characteristics.

7) The number of peaks indicates the complexity of the channel environment and the presence of various dominant propagation paths at different time intervals. This count is determined using an auxiliary parameter, which is adjustable to the desired measuring time period, allowing for improved performance. For a point to qualify as a peak, it must represent the highest point within the vicinity of its neighboring peaks, with no data point possessing a greater value between them. To illustrate, if we establish a minimum distance of 2 data points, only a point meeting the criterion of having at least 2 lower points preceding and succeeding it, while lacking any higher points in between, will be recognized as a peak. Fig. 2 depicts the implementation of this regulation with a minimum distance of 2, and it becomes evident that 3 distinct peaks can be identified.

8) The longest strikes above and below mean help to understand the persistence of certain data trends and determine the duration of strong signal paths and fading intervals. This feature is extracted by applying a mean line to the dataset, and then one can count the number of subsequent data points above and below the mean value. By understanding the persistence of certain data trends, we can assess the stability and reliability of the wireless channel. If there are more points above the mean, it suggests a positively skewed distribution, indicating a higher occurrence of strong signal paths. Conversely, if more points are below the mean, it suggests a negatively skewed distribution, indicating a higher occurrence of fading intervals. Fig. 3 shows the longest strike above and below being marked.

In the analysis of CIR, we derived various features in both the time and frequency domains. These features provided valuable insights into the characteristics and behavior of the wireless channel. In the following subsections, we introduce different features in each domain separately.

B. Time Domain Features

Here, we present several features derived from the CIR among legitimate partners.

1) Decay Time Index (DTI) describes the time at which a certain percentage (ρ) of the CIR’s energy remains over time. This parameter captures how quickly subsequent components of CIRs attenuate and can provide insights into the temporal characteristics of the channel’s fading behavior. A higher DTI value indicates a faster decay, while a lower value indicates a slowerdecay.

where the adjustable parameter ρ is determined based on the analysis of performance metrics.

2) Peak Decay Exponent (PDE) refers to the rate at which a quantity decreases in an exponential decay process, being useful for understanding the severity of multi-path fading and the effects of multi-path propagation, absorption, and scattering. It helps in modeling and analyzing processes that exhibit exponential decay, such as CIR attenuation.

It has been proven that the amplitude of CIR decreases exponentially [48], which provides the necessary conditions to extract the Characteristic Exponential Function (CEF) from CIR. The CEF is obtained by identifying significant peaks and then using least-squares fitting [48]. Therefore, the slope of the exponential function defined as PDE can be derived from CEF. [TeX:] $$A_p$$ is also defined as the first peak amplitude of CIR.

3) Slope Sign Changes (SSC) can provide information about the rate of fluctuation in the signal amplitude, which is relevant for understanding the rapid changes and variations in the received signal. It is increased by one iff the following inequality holds and corresponds to a threshold of [TeX:] $$\epsilon .$$ The inequality defines the minimum difference [TeX:] $$\epsilon$$ between two successive components of the CIR to reduce noise-induced counts [29]. This threshold is an adjustable parameter based on analyses of different metrics.

4) Waveform Length (WL) represents the complexity of the CIR measurement that can provide insights into the overall “rockiness” or “roughness” of the CIR’s amplitude profile. A higher WL value indicates more pronounced variations in the signal amplitude, while a lower value suggests relatively smoother changes.

It is identified as the cumulative length of the waveform over the time range [30].

5) Willison Amplitude (WA) identifies rapid amplitude variations that might correspond to multi-path components arriving at different delays, sudden signal reflection or scattering changes, and other abrupt phenomena in the channel. This parameter counts the number of times that the change in CIR’s amplitude exceeds a predefined threshold [29]. A higher Willison Amplitude value indicates more significant changes, while a lower value means smoother amplitude variations.

This threshold [TeX:] $$\epsilon$$ needs to be adjusted based on different metrics analyses.

6) Logarithmic Detector (LD) is suitable for high dynamic range power measurements and used in a wide variety of automatic gain control and pulse detection applications [29]. This parameter helps to emphasize the presence of weaker signal components by compressing the amplitude range by taking the logarithm of the CIR’s amplitude values. This can make it easier to detect and analyze smaller amplitude components that stronger components might otherwise obscure.

7) Mean Absolute Value (MAV) is defined as follows.

8) Mean Absolute Deviation (MAD) is defined as the average distance between each data point and the mean value. It’s a straightforward way to assess the distribution of the signal’s strength:

In the LOS condition, MAD diminishes the effect of the dominant CIR exponent, which results in high entropy.

9) Mean Excess Delay (MED) is useful to understand the average time it takes for multi-path components to arrive after the first significant peak. This parameter helps characterize the spread and distribution of signal arrivals. A higher MED indicates that the average delay of multipath components is farther from the first peak, suggesting a more dispersed channel.

where [TeX:] $$E_\mathbf{h}$$ is calculated from equation (10) [27].

10) Time Skewness (TS) is relevant for understanding the temporal characteristics of the channel and how signal components are dispersed in time. This parameter is the 3^rd order moment of CIR components and represents the degree of asymmetry in the timing distribution.

where [TeX:] $$\mu_{\mathbf{h}} \text { and } \sigma_{\mathbf{h}}$$ are defined as the mean and standard deviation of the CIR measurements, respectively. A positive skewness indicates that there are more latearriving components, while a negative skewness indicates more early-arriving components. A skewness value of zero suggests a symmetric distribution.

11) Kurtosis (Ku) is defined as the 4th order moment of CIR components and represents the shape of the amplitude distribution. High kurtosis indicates more outliers and a heavier-tailed distribution, potentially indicating the presence of more significant peaks or fluctuations.

It should be noted that the moment orders of the CIR component strongly indicate the dynamic characteristics of the CIR, which provide us with a high amount of entropy.

12) Form Factor (FoF) can be useful in understanding the overall amplitude profile of the CIR and can provide insights into the degree of fluctuations and variability in the received signal. It characterizes the shape of the CIR waveform by comparing the RMS amplitude to the mean of the CIR [27].

where [TeX:] $$RMS_{\mathbf{h}}$$ is defined as the root mean square of CIR components as [TeX:] $$R M S_{\mathrm{h}}=\sqrt{\frac{1}{N} \sum_{n=1}^N|h[n]|^2}.$$

A higher FoF indicates more pronounced variations from the mean, suggesting a more complex waveform, while A lower FoF suggests a smoother waveform.

13) Crest Factor (CF) explains how peaked the waveform is in comparison to its average level. A higher CF indicates more pronounced peaks, suggesting the potential for rapid signal fluctuations or transient events.

where [TeX:] $$h_{\max }$$ is the maximum value of the CIR. A lower CF suggests a relatively more consistent waveform. It is relevant for assessing the potential for signal distortion in communication systems, as high CFs can lead to clipping or overdriving in certain system components.

C. Frequency Domain Features

From the frequency representation of the CIR, we extract various features as presented below. It is important to highlight that we compute these features utilizing the DFT of both the complex and amplitude domains of the CIRs, and subsequently evaluate their performance.

1) FSIA represents the total energy or power across different frequencies in the frequency domain and shows how the signal energy or power is distributed across various frequency components. It is defined as the sum of frequency components H[k] in the desired frequency range as follows.

2) Spectral Centroid (SC) is a weighted mean of frequency components of the CIR defined as

3) Spectrum Roll-off (SR) shows the frequency range that contains ρ% of the frequency spectrum integration area [27] and can be calculated as

It helps identify the frequency spectrum portion that carries most of the power. A lower Spectral Roll-off indicates the power is concentrated at lower frequencies, while a higher value suggests a broader distribution across higher frequencies. The adjustable parameter ρ can be set based on different metrics analyses.

4) Spectral Fluctuations (SF) represents the variations or changes in the magnitudes of frequency components in the frequency domain and can be valuable for characterizing the frequency-selective fading behavior of the channel. This parameter is defined as the sum of squared differences between frequency components.

5) Spectral Entropy (SE) is related to the level of randomness in the frequency distribution of CIRs’ spectrum and can provide insights into the frequency-selective fading behavior of the channel.

6) Spectral Flatness (SF), also known as Wiener entropy of the CIR, characterizes the uniformity of the energy or power distribution in the frequency domain [49]. It can be measured as

A higher spectral flatness indicates a more even distribution of energy across frequency components, suggesting a flatter spectrum, while a lower spectral flatness suggests that energy is concentrated in fewer frequency components, indicating a more peaked spectrum.

7) Spectral Spread (SSP) is a measurement that depicts how the spectrum is distributed around the centroid.

where SC is calculated from (27). A higher SSP indicates a wider energy distribution, suggesting that the signal occupies a broader frequency range. A lower SSP suggests the energy is concentrated within a narrower frequency range.

8) Spectral Skewness (SSK) measures the amount of symmetry of the frequency spectrum around the center.

where 2SCM is the second Spectral Central Moment (SCM) and SC is calculated from (27). A positive spectral skewness indicates that the distribution is skewed to the right (more energy on the right side), while a negative value suggests a skew to the left. A value close to zero suggests a relatively symmetric distribution.

9) Spectral Kurtosis (SKU) depicts the tailedness of the CIR’s spectrum defined as follows.

where 2SCM is the second Spectral Central Moment (SCM) and SC is calculated from (27). A higher spectral kurtosis indicates a distribution with heavier tails, suggesting the presence of more significant peaks or fluctuations. A lower spectral kurtosis indicates a distribution with lighter tails, suggesting a more Gaussian distribution.

10) j^th Spectral Moment (jSM) captures the distribution of the signal’s power around all frequency components. Different values of j provide different perspectives on energy distribution. It is defined as

where the degree of moments j is an adjustable parameter based on different metrics.

11) j^th Spectral Central Moment (jSCM) provides insights into how the energy distribution is spread around a central reference frequency. Different values of j and the choice of reference frequency impact the interpretation. The jSCM can be calculated as

where the adjustable degree of the moments is set the same as the j^th Spectral Moment (SM) and SC is calculated from (27). It should be noted that the jSM measures the distribution of power across frequencies, while the jSCM measures the distribution of power around the central reference frequency.

With these features introduced, the subsequent section will focus on the quantization process within the framework of SKG.

IV. MULTI-BIT QUANTIZATION

The quantization process plays a crucial role in converting the continuous output of the feature extraction step into a discrete bit sequence. To make it adaptive to slow changes in the feature extraction results, algorithm 1 applies a window mechanism with a window size of B. From this window, the maximum value, minimum value, and consequently the range value of the window are determined. After that, equal-sized quantization bins in the range of the window are created, and each gets a Gray Code to enhance accuracy and decrease BER. The Gray Code is assigned to data points according to their positions within the quantization range. In this work, a bit number of 2 is applied, leading to four quantization zones, and a window size of 10 captures temporal trends effectively. This approach improves accuracy and adaptability regarding the quantization of the extracted features.

V. EVALUATION RESULTS

In this section, we leverage our dataset introduced and published in [50] to evaluate the performance of the proposed feature set. This dataset provides a rich collection of realworld measurements capturing channel characteristics in various usage scenarios, aimed at supporting the examination of Physical Layer Security assumptions and theoretical models with concrete data. The dataset is centered around an indoor office room measuring 5.9m in length and 3.05m in width. Regarding the system model, it is designed for physical layer key derivation schemes, where two legitimate parties, Alice and Bob, exchange messages back and forth to assess the characteristics of their shared channel. A passive eavesdropper, Eve, monitors their communication. The measurement setup utilizes DecaWave EVB1000 evaluation boards with UWB transceivers and the Lego Mindstorms Education EV3 kit. The DW1000 is a low-power, single-chip CMOS radio transceiver that adheres to the IEEE 802.15.4-2011 UWB standard, operating at a center frequency of 3.9936 GHz with a bandwidth of 499.2 MHz.

These measurements were conducted across diverse scenarios, including scenarios involving the movement of a robot within the space as follows:

· Symmetric scenario: Basic setting with symmetric anchor placements.

· Asymmetric scenario: Similar to symmetric, with a variation in one anchor’s position.

· Varying speed scenario: Robot’s speed gradually reduced from 50% to 5% of its maximum.

· Reflector scenario: Introducing a moving reflector to induce interference.

· No movement scenario: Robot placed manually at specific spots without autonomous movement.

It should be mentioned that the effect of the hardwarespecific characteristics, asymmetric interference, and synchronization offset on the reciprocity of the channel and the performance of channel reciprocity-based key generation should not be underestimated. Asymmetric interference in CRKG introduces disparities in communication quality between legitimate partners and eavesdroppers, potentially compromising security, reducing key generation efficiency, and posing challenges in establishing a common basis for key agreement. In scenarios where time synchronization is lacking and reciprocal CIRs are not synchronous, the processing post-quantization is significantly hindered due to large mismatches between the quantized reciprocal bit strings and requires preprocessing to synchronize the measurements. While our dataset assumes synchrony by aligning to peak correlations for simplicity, various synchronization methods such as cross-correlation metrics and modified EM clustering algorithms offer potential solutions [51], [52]. We investigated several criteria within the dataset to ensure the validity of the reciprocity assumption and the security criteria for generating secret keys. To verify the reciprocity assumption, we measured cross-correlation between legitimate partners and eavesdropper CIRs, and examined the effect of time changes to ensure fresh keys under different conditions. In the analysis of raw measurement data, the cross-correlation between Alice, Bob, and Eve’s CIRs was assessed to determine if the fundamental assumption of CRKG holds, finding that legitimate partners exhibit significantly higher correlations (mean: 0.9783 ± 0.0159, median: 0.9828) compared to the eavesdropper (mean: 0.8231 ± 0.0503, median: 0.8277), implying an advantage for legitimate communication partners on the shared channel (Table 5.1 [34]). The analysis of temporal decorrelation involves comparing correlations between successive measurements with time-shifted variants, demonstrating a rapid decline in correlation for legitimate partners over time, with average correlation dropping to 0.89 at 240 ms and further to 0.86 at 1 second, while eavesdroppers’ correlation remains consistently low regardless of time offset (Table 5.1 [34]). This means that the measurement dataset has the potential to generate fresh keys over time.

To investigate the performance of each feature, we evaluate them individually compared to the bit string generated directly from the raw CIR’s measurements based on the introduced performance metrics (see Sec. III). Then, we selected those with the most appropriate performance efficiency and measured the correlation between them in the time and frequency domains to avoid the redundancy that correlated features may cause. It should be noted that features were extracted from no-cut CIRs and their cut versions to investigate the effect of noise on their performance. In addition, the adjustable parameters of some features are considered as follows:

· ρ = 0.85 for decay time index

· peaknum = 2 for the number of peaks (in time )

· threshold [TeX:] $$\epsilon$$ = 100 for slope sign changes

· threshold [TeX:] $$\epsilon$$ = 200 for Willison amplitude

· peaknum = 3 for the number of peaks (in frequency)

· ρ = 0.5 for spectral roll-off

· j = 5 for jSM

· j = 5 for jSCM

Notably, these parameters were derived through an extensive investigation and trade-off between the above performance metrics for Alice, Bob, and Eve. More precisely, for example, the adjustable parameter of ρ for the time decay index is chosen based on the considerations below:

For the entropy, a minimum value of ρ = 25% ensures an entropy of more than 0.95. The results of the MI suggest a minimum value of ρ = 20% to have higher MI for Alice and Bob than for Eve. For values greater than ρ = 85% the MI for Eve increases again, which is why these values should also be avoided. Regarding the BER, a value of ρ = 70% and higher leads to a strong increase in the error rate for Eve. These analyses lead to choosing ρ = 85% as a parameter for the decay time index.

We use candlestick charts to visualize the performance of features as they offer several advantages and can help to gain insights and make informed decisions. This chart provides a comprehensive view of the performance of multiple features simultaneously. This allows us to quickly compare and contrast how different features are performing and identify patterns or trends across them. Moreover, this is beneficial to identify which features are consistently strong performers and which ones might need improvement. The length and shape of candlesticks provide an intuitive representation of the variation and volatility in feature performance. This identifies which features exhibit more consistent behavior and which ones have greater fluctuations regarding different types of data. In addition, to highlight the performance result for raw CIRs (referred to as ’raw data’ in the figures) and make them more comparable with other features’ performance, we use a red dashed line. This visual aid helps differentiate the raw data from other features in the figures.

A. Entropy

As is shown in Fig. 4 and 5, the entropy of bits in the time and frequency domains can be achieved to the maximum of one bit, and over 0.96 bits respectively, which means that the features provide a high amount of uncertainty in generated bit strings. It should be noted that the entropy directly depends on the quantization algorithm, so finding an optimal algorithm can improve the performance of the entropy increase. Furthermore, as depicted in the figures, features exhibiting long shadows imply greater variability in their performance across different scenarios. Consequently, features characterized by higher density are deemed more dependable when confronted with diverse scenarios. In addition, the entropy of generated bit strings from the ’raw-data’ indicates lower entropy with higher variation between its maximum and minimum values, meaning that it shows various performance for different scenarios. Moreover, a considerable number of outliers in the candle of raw CIRs make features more reliable than the raw CIRs for secret key generation in different scenarios.

B. Mutual Information

The MI figures provide valuable information in the time domain, showing that the MI for Alice and Eve is significantly low for all features (see Fig. 6), with negligible upper outliers below 0.02. This is while peak amplitude, received signal energy, mean absolute value, variance, logarithmic detector (for no-cut signal), mean absolute deviation (for no-cut signal), longest strike above mean, number of peaks, slope sign changes, and Willison amplitude are discarded to provide MI at least 0.1 between Alice and Bob (see Fig. 7).

As is shown in Fig. 8 MI between Alice and Eve in the frequency domain is also considerably low. However, the low amount of under 0.1 for MI of generated bit strings between Alice and Bob leads to a removal of several features from the final feature set. These features include received signal energy, FSIA, and the number of peaks from both complex and amplitude versions of the frequency domain. Furthermore, features such as peak amplitude, variance, FaF, and spectral flatness from the amplitude version of frequency representation of the CIRs are removed, as illustrated in Fig. 9.

Regarding the MI for generated bit strings from raw CIRs, as shown, it has an amount of below 0.02 and lower than 0.1 with considerable outliers for Alice and Bob and Alice and Eve, respectively. This suggests that raw CIRs exhibit lower performance and reliability across different scenarios compared to features extracted from both domains. This indicates that using raw CIRs directly might be less effective than using features extracted from them in various situations.

C. Bit Error Rate

The analysis of BER for Alice and Eve in both domains reveals an average value of 0.5, indicating that Eve’s ability to guess the key materials is significantly limited (Fig. 10 and 11). Additionally, BER values for the legitimate partners in both domains mostly remain quite low, which is similar to the substantial MI shared between Alice and Bob for these features. Consequently, as can be seen in Fig. 12, the subsequent features extracted from both the no-cut and cut versions of the CIRs are discarded due to a BER above 0.3: peak amplitude, received signal energy, mean absolute value, variance, mean absolute deviation, the longest strike above the mean, number of peaks, slope sign changes, Willison amplitude, and logarithmic detector for the no-cut version of the signal.

Regarding frequency domain features, Fig. 13 represents the features derived from both versions of CIRs with high BER, including received signal energy, FSIA, variance, spectral flatness, and the number of peaks are removed.

Furthermore, it is observed that the following features derived from the transformed amplitude of the CIRs exhibit a BER exceeding 0.3 and are subsequently excluded: peak amplitude, FaF, spectral centroid, spectral fluctuation, spectral entropy, spectral moment, and spectral central moment. Most of these features are inherently sensitive to the environment’s noise and interference. As is shown in Fig. 10 and Fig. 12 performance of generated bit strings from raw CIRs is lower than most of the features in both domains with a considerable amount of outliers and more extended candle in the plot, which shows a high variation of the BER regarding different scenarios. This means that, compared to the features, raw measurements are more dependent on the scenarios, and features are more reliable for multiple scenarios.

D. Correlation of the Features

After removing those features with low performance according to the metrics mentioned above, we measure the correlation between them to find the individuality of the features in both domains. This can be beneficial to avoid excessive computation, leading directly to an appropriate feature set for the key generation. Hence, we use the absolute value of the correlation, mainly because strong positive and negative correlations mean the same, and both are undesirable. The lower bound for the correlation is set to 0.9, which means the MI between the two features is high enough. Hence, we can choose the one with higher performance or lower computational complexity.

Fig. 14 depicts the correlation of the features in the time and frequency domain for the cut CIRs version. The feature named “the number of peaks below and above mean” has full correlation, because of the statistical nature of the mean standing between these points, which led us to choose only the number of peaks below the mean for the final feature set with the same performance and computational complexity. In addition, the CF is not completely correlated with kurtosis, but it is significant at above the value of 0.9. Moreover, the high correlation of kurtosis and skewness due to their same static nature and the distribution of CIR’s components led us to remove kurtosis with the same performance and higher computational complexity than skewness.

Fig. 15 presents the correlation between frequency domain features. Similar to the time domain, a full correlation can be seen between the number of points above and below the mean. With the same performance, we discard one of them. The feature spectral centroid highly correlates with spectral skewness, roll-off, and central moment. Since the centroid represents the center of a CIR, it affects skewness (indicating the bias of energy towards certain frequencies), roll-off (indicating where most energy lies), and central moment (indicating the distribution around the centroid). When the centroid shifts, these other features can change accordingly. The performance of the spectral central moment is lower than the other, leading to its discard from the feature set, and as the spectral centroid has lower computational complexity, we choose it among spectral skewness and spectral roll-off for the final feature set.

E. Final Feature Set

There was no dominant correlation between the time and frequency features. Therefore, they can be calculated and derived from the CIRs and afterwards be fed to SKG independently.

To summarize our investigations, we identify the final feature set based on those with higher performance, lower computational complexity, and lower correlation between each feature in the set. It should be noted that features in the time domain are extracted from a cut of 128 CIR values to avoid the unwanted effect of the noise, considering the LOS components compared to the no-cut CIR materials. In the frequency domain, features are calculated from the complex measurements of the CIRs, resulting in more specific feature values than calculating them from the signal’s amplitude. The final feature set is chosen, shown in a way that satisfies the following goals for generated bit strings:

· Mean entropy of at least 0.96 bits

· BER for Alice and Bob less than 0.3 and for Alice and Eve at least 0.49

· MI between Alice and Bob of at least 0.1 bits and between Alice and Eve less than 0.0025 bits.

Table I displays all features along with their selection status and the reasons for their rejection. Ultimately, a final feature set comprising 13 and 11 features from the cut and no-cut versions in the time domain, and 14 and 7 features derived from DFT of complex and amplitude versions of the signal in the frequency domain, is presented. This final feature set fulfills the requirements for a key generation process, offering both efficiency and high security.

VI. CONCLUSION

In this paper, we presented a new approach for SKG using a feature extraction scheme on CIR measurement data. We examined many features in both time and frequency domains and evaluated them through different performance metrics.

To evaluate the efficacy of the proposed feature-based key generation approach, leveraging our UWB dataset, we employed several information-theoretic performance metrics, including entropy, MI, and BER. We identified features with high entropy, MI, and low BER between the two authenticated parties while exhibiting low MI and high BER concerning an eavesdropper.

The performance of the features extracted from the cut signal and the no-cut CIR measurement to investigate the effect of noise in the presence of the dominant CIR component of LOS. The result of the cut materials shows higher performance compared to the no-cut measurement data. In addition, we evaluated the feature set by transforming the complex representation of the time domain into the frequency domain. Moreover, we conducted a comparative analysis of the performance of the proposed feature set against the conventional SKG method, which utilizes raw CIR measurements. The results consistently showed that the proposed features outperformed those derived from raw CIR data across all performance metrics. This enhancement signifies higher security and confidentiality for the generated bit string. Afterward, we calculated the correlation between each of these features. The main reason for this is high correlations between various features that create excessive redundancy.

It is worth mentioning that there is a trade-off between performance and computational complexity of features that include higher-order moments of the CIR’s component. Nevertheless, the proposed feature set provides many features that can be tailored to different application scenarios.