In physical-layer key generation (PLKG), researchers often assess key security using randomness tests or secret key capacity analysis. However, randomness tests can be misleading, as low-entropy inputs may still pass when obscured by hash functions or extractors. While key capacity captures the theoretical upper limit of secret bits from raw channels, it overlooks the impact of practical preprocessing. Recent studies have explored min-entropy as a more practical security metric. However, commonly used min-entropy estimators, such as those in NIST SP 800-90B (90B), may yield inaccurate results for complex sources, even overestimating entropy and thereby undermining security. Their applicability to PLKG also remains unverified. To address the above research gap, this paper focuses on modeling the received signal envelope, a common feature extracted for key generation. Leveraging typical wireless fading scenarios considered in key capacity analysis, we adopt the more general Weibull distribution to characterize the envelope statistics, which generalizes the Rayleigh model and better captures diverse propagation conditions. We first derive an explicit expression for the min-entropy in the absence of preprocessing, and then systematically analyze the impact of typical preprocessing operations on the statistical properties of the observed channel. Building on this foundation, we further establish theoretical upper and lower bounds on the min-entropy under representative preprocessing strategies. We further evaluate all 90B entropy estimators using both simulations and real-world channel measurements, comparing their estimates with our analytical results. While some estimators work well on raw data, they often fail or overestimate entropy after preprocessing, compromising security. In contrast, our conservative lower bound remains robust across all tested scenarios and preprocessing conditions. It offers a reliable fallback when standard estimators break down, providing a stronger theoretical foundation for secure key extraction in PLKG systems.
In physical-layer key generation (PLKG), researchers often assess key security using randomness tests or secret key capacity analysis. However, randomness tests can be misleading, as low-entropy inputs may still pass when obscured by hash functions or extractors. While key capacity captures the theoretical upper limit of secret bits from raw channels, it overlooks the impact of practical preprocessing. Recent studies have explored min-entropy as a more practical security metric. However, commonly used min-entropy estimators, such as those in NIST SP 800-90B (90B), may yield inaccurate results for complex sources, even overestimating entropy and thereby undermining security. Their applicability to PLKG also remains unverified. To address the above research gap, this paper focuses on modeling the received signal envelope, a common feature extracted for key generation. Leveraging typical wireless fading scenarios considered in key capacity analysis, we adopt the more general Weibull distribution to characterize the envelope statistics, which generalizes the Rayleigh model and better captures diverse propagation conditions. We first derive an explicit expression for the min-entropy in the absence of preprocessing, and then systematically analyze the impact of typical preprocessing operations on the statistical properties of the observed channel. Building on this foundation, we further establish theoretical upper and lower bounds on the min-entropy under representative preprocessing strategies. We further evaluate all 90B entropy estimators using both simulations and real-world channel measurements, comparing their estimates with our analytical results. While some estimators work well on raw data, they often fail or overestimate entropy after preprocessing, compromising security. In contrast, our conservative lower bound remains robust across all tested scenarios and preprocessing conditions. It offers a reliable fallback when standard estimators break down, providing a stronger theoretical foundation for secure key extraction in PLKG systems.
Wireless physical-layer security is recently proposed for several types of cutting-edge networks, such as 5G and 6G. In wireless physical-layer security, it is usually suggested in literature that eavesdroppers need to locate themselves near legitimate transmitters or receivers if they want to steal some secret key information. By means of scenario analysis, this paper shows that if the eavesdropper knows the channel statistics better than the legitimate stations, e.g., know the K-factor estimates of Rician fading channel, they can exploit these knowledges to derive some secret key information. We define a concept called “improper quantization decision lossk” as the entropy loss in the generated key that can be obtained by the eavesdropper. We show step-by-step how the eavesdropper can obtain such information. We also propose mathematical expressions ofnand derive some results based on empirical values of channel parameters available in literature. From our analytical result for an urban area case, the loss is as large as almost 4/5 of key information, thus requiring a privacy amplification process that yields about 1/5 of the original key length. With our new concept, the new upper and lower bounds of vulnerable key bits are derived.
Intrigued by the rapid growth and expand of wireless devices, data security is increasingly playing a significant role in our daily transactions and interactions with different entities. Possible examples, including e-healthcare information and online shopping, are becoming vulnerable due to the intrinsic nature of wireless transmission medium and the widespread open access of wireless links. Traditionally, the communication security is mainly regarded as the tasks at the upper layers of layered protocol stack, security techniques, including personal access control, password protection, and end-to-end encryption, have been widely studied in the open literature. More recently, plenty of research interests have been drawn to the physical layer forms of secrecy. As a new but appealing paradigm at physical layer, physical layer security is based on two pioneering works: (i) Shannon’s information-theoretic formulation and (ii) Wyner’s wiretap formulation. \n \nOn account of the fundamental of physical layer security and the different nature of various wireless network, this dissertation is supposed to further fill the lacking of the existing research outcomes. To be specific, the contributions of this dissertation can be summarized as three-fold:(i) exploration of secrecy metrics to more general fading channels; (ii) characterization a new fading channel model and its reliability and security analysis in digital communication systems; and (iii) investigation of physical layer security over the random multiple-input multiple-output (MIMO) α −μ fading channels. \n \nTaking into account the classic Alice-Bob-Eve wiretap model, the first contribution can be divided into four aspects: (i) we have investigated the secrecy performance over single-input single-output (SISO) α −μ fading channels. The probability of non-zero (PNZ) secrecy capacity and the lower bound of secrecy outage probability (SOP) are derived for the special case when the main channel and wiretap channel undergo the same non-linearity fading parameter, i.e., α. Later on, for the purpose of filling the gap of lacking closed-form expression of SOP in the open literature and extending the obtained results in chapter 2 to the single-input multiple-output (SIMO) α − μ wiretap fading channels, utilizing the fact that the received signal-tonoise ratios (SNRs) at the legitimate receiver and eavesdropper can be approximated as new α −μ distributed random variables (RVs), the SOP metric is therefore derived, and given in terms of the bivariate Fox’s H-function; (ii) the secrecy performance over the Fisher-Snedecor F wiretap fading channels is initially considered. The SOP, PNZ, and ASC are finalized in terms of Meijer’s G-function; (iii) in order to generalize the obtained results over α −μ and Fisher-Snedecor F wiretap fading channels, a more flexible and general fading channel, i.e., Fox’s H-function fading model, are taken into consideration. Both the exact and asymptotic analysis of SOP, PNZ, and average secrecy capacity (ASC), are developed with closed-form expressions; and (iv) finally, motivated by the fact that the mixture gamma (MG) distribution is an appealing tool, which can be used to model the received instantaneous SNRs over wireless fading channels, the secrecy metrics over wiretap fading channels are derived based on the MG approach. \n \nDue to the limited transmission power and communication range, cooperative relays or multi-hop wireless networks are usually regarded as two promising means to address these concerns. Inspired by the obtained results in Chapters 2 and 3, the second main contribution is to propose a novel but simple fading channel model, namely, the cascaded α −μ. This new distribution is advantageous since it encompasses the existing cascaded Rayleigh, cascaded Nakagami-m, and cascaded Weibull with ease. Based on this, both the reliability and secrecy performance of a digital system over cascaded α −μ fading channels are further evaluated. Closed-form expressions of reliability metrics (including amount of fading (AF), outage probability, average channel capacity, and average symbol error probability (ABEP).) and secrecy metrics (including SOP, PNZ, and ASC) are respectively provided. Besides, their asymptotic behaviors are also performed and compared with the exact results. \n \nConsidering the impacts of users’ densities, spatial distribution, and the path-loss exponent on secrecy issue, the third aspect of this thesis is detailed in Chapter 8 as the secrecy investigation of stochastic MIMO system over α −μ wiretap fading channels. Both the stochastic geometry and conventional space-time transmission (STT) scheme are used in the system configuration. The secrecy issue is mathematically evaluated by three metrics, i.e., connection outage, the probability of non-zero secrecy capacity and the ergodic secrecy capacity. Those three metrics are later on derived regarding two ordering scheme, and further compared with Monte-Carlo simulations.
Abstract: Physical layer key generation, leveraging the reciprocity and randomness of wireless fading channels, has garnered significant research interest in recent years. While theoretical studies have demonstrated its potential to generate informationtheoretically secure keys, challenges remain in translating theory into practice. This paper provides an overview of the physical layer key generation process and discusses its practical challenges. Various passive and active attacks are analyzed and evaluated through numerical studies. A new key generation scheme using random probing signals and combining usergenerated randomness with channel randomness is introduced to counteract active attacks. Numerical results demonstrate that the proposed scheme achieves higher security strength than existing schemes using constant probing signals under active attacks. Secret-key generation relying on wireless channel reciprocity is an intriguing solution as it can be efficiently implemented at the physical layer of emerging wireless communication networks, while providing information-theoretic security guarantees. In this paper, we investigate and compare the secret-key capacity based on the sampling of the entire complex channel state information (CSI) or only its envelope, the received signal strength (RSS).
In much recent discussion on wireless physical layer security there is a strong tendency to take for granted that the randomness provided by the wireless channel is adequate for cryptographic applications. One typically assumes that multipath fading, interferences and changes in the environment lead to sufficiently random channel measurements. We show that this does not hold for the majority of practical implementations where the underlying channel response is estimated based on the Received Signal Strength (RSS) values, and we identify the statistical defect that is primarily responsible for the statistical bias in the RSS values and discuss the source of this defect. Our observation implies that RSS-based physical layer key agreement schemes require a dedicated post-processing of the channel measurements together with an on-line statistical test suite. To address the latter, we define requirements for statistical testing in the context of physical layer key agreement. Based on these requirements, we select some well-known low complexity statistical tests to form a reduced test suite that requires less than 10,000 bit of input data. The selected tests have low complexity allowing to execute the test suite at the run-time even on severely resource-constrained devices. Our results show that it is possible to identify statistical defects with high reliability.
A few years ago, physical layer based techniques have started to be considered as a way to improve security in wireless communications. A well known problem is the management of ciphering keys, both regarding the generation and distribution of these keys. A way to alleviate such difficulties is to use a common source of randomness for the legitimate terminals, not accessible to an eavesdropper. This is the case of the fading propagation channel, when exact or approximate reciprocity applies. Although this principle has been known for long, not so many works have evaluated the effect of radio channel properties in practical environments on the degree of randomness of the generated keys. To this end, we here investigate indoor radio channel measurements in different environments and settings at either 2.4625 GHz or 5.4 GHz band, of particular interest for WIFI related standards. Key bits are extracted by quantizing the complex channel coefficients and their randomness is evaluated using the NIST test suite. We then look at the impact of the carrier frequency, the channel variability in the space, time, and frequency degrees of freedom used to construct a long secret key, in relation to the nature of the radio environment such as the LOS/NLOS character.
Symmetric cryptography relies on pairs of identical secret keys shared by the legitimate communicating parties. To implement a symmetric-key algorithm for cryptography, a major concern is to develop secure methods for distribution of the secret key. In securing the wireless fading channels by symmetric-key algorithms, the physical layer properties of the channel can be exploited for distribution of the secret keys. In this approach, the channel state provides a common randomness which is shared by the legitimate users but is mostly unknown to an eavesdropper. by means of signal processing techniques, this common randomness is extracted into random secret keys. This paper establishes an information theoretic upper bound on the rate at which the secret keys can be extracted. Instead of the conventional approach that relies on mathematical models for the wireless channel, this paper adopts an experimental approach to estimate this bound from empirical data. A set of signal processing techniques is developed here to numerically estimate this bound for a pair of received signal strength (RSS) recorded by indoor commercial radios.