Physical-Layer Security in the Finite Blocklength Regime over Fading Channels

Decoding with finite blocklength will inevitably reduce the secrecy capacity and some preliminary works have been devoted to analyzing the impact of finite blocklength on secrecy for the wiretap channel. For example, the authors in [17] derived an upper bound for the information leakage probability for a given target decoding error probability demonstrating the inherent trade-off between secrecy and reliability. The authors in [18] provided both upper and lower bounds for the maximal secrecy rate capturing the impact of finite blocklength, error probability, and information leakage in both degraded discrete-memoryless wiretap channels and Gaussian wiretap channels. The obtained bounds were shown to be tighter than existing ones from [19, 20]. The work in [18] was further extended by [21], in which the optimal second-order secrecy rate was derived for a semi-deterministic wiretap channel, and the optimal tradeoff between secrecy and reliability with finite blocklength was analytically characterized. It should be noted that, all the above works were aimed to uncover the fundamental limits of secrecy performance from the information theory point of view, whereas the design of practical signaling and transmission schemes were not investigated.

In practice, due to finite blocklength penalty for practical coding schemes, even a secrecy rate below the secrecy capacity cannot guarantee a perfectly successful and secure communication. In this sense, in addition to exploring and/or improving the fundamental limits of the maximal secrecy rate, optimizing secrecy throughput seems more important from the perspective of transmission efficiency, particularly for fading channels where the code rates can be adapted to the fading status. Herein, the secrecy throughput denotes the amount of successfully delivered secret information subject to certain reliability and secrecy constraints. In fact, the secrecy throughput has extensively been taken as an optimization objective for the design of secure transmissions in slow fading channels in the context of infinite blocklength [22]-[26]. Nevertheless, to optimize the secrecy throughput under the constraint of finite blocklength is difficult, and the results derived for infinite blocklength, e.g., [22]-[26], cannot be directly applied. Indeed, the blocklength itself is an optimization variable, and it couples with other variables in a sophisticated manner which makes the optimization problem intractable. For instance, the authors in a recent work [27] investigated the secrecy throughput of a relay-aided secure transmission with finite blocklength, where neither the instantaneous channel state information (CSI) with respect to (w.r.t.) the legitimate receiver nor the eavesdropper is available at the transmitter side. Numerical results were presented therein to show that there exists a critical value of the blocklength that maximizes the secrecy throughput.

Despite the above endeavors, there are some fundamental questions regarding the design of physical-layer security schemes with finite blocklength that have not been thoroughly addressed. First of all, a theoretical proof of the optimal blocklength and the corresponding secrecy rate for maximizing the secrecy throughput is of great significance for the practical design of secure transmissions, which however has not yet been reported by existing literature. Also, in many applications, the transmitter is capable to acquire the instantaneous CSI of the legitimate receiver in slow fading channels via training or feedback. Yet, the potential of exploiting the instantaneous CSI to alleviate the negative impact of finite blocklength on the performance of secure communications has not been exploited. Furthermore, only the single-antenna transmitter scenario has been considered, e.g., [17]-[21], [27], and the design of the optimal signaling and code rates for multi-antenna systems with finite blocklength is still an open issue. This research work aims to provide an analytical framework and design schemes to address the abovementioned problems.

This paper investigates the security issue between a pair of legitimate communicating parties in the presence of an eavesdropper, considering the impact of finite blocklength in secrecy coding. The secrecy throughput is thoroughly analyzed and optimized for both single- and multi-antenna transmitter scenarios. In particular, both adaptive and non-adaptive parameter design schemes are proposed for each scenario. The main contributions of this work are summarized as follows:

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  For the single-antenna transmitter scenario, the secrecy throughput is maximized by jointly optimizing the transmission policy, blocklength, as well as code rates. Closed-form bounds and approximations for the secrecy rate are provided to facilitate the practical design of code rates for achieving a close-to-optimal performance.

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  For the multi-antenna transmitter configuration, the optimality of the null-space artificial noise (AN) scheme in terms of secrecy throughput maximization is first investigated. Afterwards, the optimal transmission policy, blocklength, code rates, and power allocation between the information-bearing signal and the AN are derived. Particularly, the power allocation and the secrecy rate are designed via the alternating optimization method, and their impacts on the system performance are further revealed.

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  Numerous useful insights into the design of secure transmissions are provided with finite blocklength. For example, 1) increasing the blocklength can improve both reliability and secrecy, with properly exploiting the instantaneous CSI of the main channel and the statistical CSI of the wiretap channel, which has not been revealed by existing literature, e.g., [17]-[21]; 2) using the maximal blocklength is profitable for boosting the secrecy throughput, which is distinguished from the observation in [27]; 3) due to the finite blocklength penalty, there is a critical secrecy rate that can maximize the secrecy throughput even for the adaptive scheme, rather than always employing the maximal available secrecy rate, which is fundamentally different from the phenomenon with infinite blocklength, e.g., [22, 23].

The remainder of this paper is organized as below. Section II describes the system model and the underlying optimization problem. Sections III and IV detail the secrecy throughput maximization for both single- and multi-antenna transmitter scenarios. Section V draws a conclusion.

Notations: Bold lowercase letters denote column vectors. $|\cdot|$, $\|\cdot\|$, $(\cdot)^{\dagger}$, $(\cdot)^{\rm T}$, $\ln(\cdot)$, $\mathbb{P}\{\cdot\}$, $\mathbb{E}_{v}[\cdot]$ denote the absolute value, Euclidean norm, conjugate, transpose, natural logarithm, probability, and the expectation over a random variable $v$, respectively. $f_{v}(\cdot)$ and $\mathcal{F}_{v}(\cdot)$ denote the probability density function (PDF) and cumulative distribution function (CDF) of $v$, respectively. $F^{-1}(\cdot)$ denotes the inverse function of a function $F(\cdot)$. $\mathcal{CN}(\mu,\sigma^{2})$ denotes the circularly symmetric complex Gaussian distribution with mean $\mu$ and variance $\sigma^{2}$.

Consider a secure transmission from a transmitter (Alice) to a legitimate receiver (Bob) coexisting with an eavesdropper (Eve), as depicted in Fig. 1. Alice is equipped with $M\geq 1$ transmit antennas, whereas Bob and Eve are single-antenna devices. Quasi-static Rayleigh fading channels are considered, where the channel coherence time is on the order of the blocklength. More specifically, the fading coefficients are assumed to remain constant during the transmission of an entire codeword, but change independently and randomly between two codewords [2]. Denote the coefficients of the main and wiretap channels by $\bm{h}_{b}$ and $\bm{h}_{e}$, and each entry of $\bm{h}_{b}$ and $\bm{h}_{e}$ follow the Gaussian distribution $\mathcal{CN}(0,\sigma_{b}^{2})$ and $\mathcal{CN}(0,\sigma_{e}^{2})$, respectively.¹¹1The subscripts $b$ and $e$ are used to refer to Bob and Eve, respectively. A common hypothesis is adopted [22, 23], i.e., Bob and Eve know perfectly the instantaneous CSI of their individual channels $\bm{h}_{b}$ and $\bm{h}_{e}$, and Alice has the instantaneous CSI of Bob’s channel $\bm{h}_{b}$ but does not has the instantaneous CSI of Eve’s channel $\bm{h}_{e}$. Besides, the statistics of both channels $\bm{h}_{b}$ and $\bm{h}_{e}$ are available at Alice. Assume that $\bm{h}_{b}$, $\bm{h}_{e}$, and the receiver noise are mutually independent, where noise variances at Bob and Eve are denoted by $w_{b}^{2}$ and $w_{e}^{2}$, respectively. Alice adopts a constant transmit power $P$. For notational simplicity, define $P_{b}\triangleq\frac{P}{w_{b}^{2}}$ and $P_{e}\triangleq\frac{P}{w_{e}^{2}}$ as the normalized power for Bob and Eve, respectively.

To safeguard information confidentiality, secrecy coding should be employed to encode the secret information bits. Instead of investigating any explicit practical constructions of secrecy codes, the Wyner’s wiretap code [14], as a generic code structure, is employed in this paper. A synopsis of the state-of-the-art coding schemes for wiretap channels can be found in [28].

It is reported in [2] that the Wyner’s wiretap code possesses a binning structure, as illustrated in Fig. 1, where $2^{nR_{s}}$ messages are encoded to $2^{nR_{t}}$ codewords, and each message is mapped to a bin of $2^{nR_{e}}$ codewords. Here, $n$ denotes the blocklength (i.e., the codeword length or the number of channel uses), $R_{s}$ and $R_{t}$ (bits/s/Hz/channel) denote the secrecy rate and codeword rate, respectively. The binning codeword rate, i.e., the rate redundancy $R_{e}=R_{t}-R_{s}$, reflects the cost of providing secrecy.

It is well-known that, for an infinite blocklength with $n\rightarrow\infty$, as long as the codeword rate $R_{t}$ is not larger than Bob’s channel capacity, Bob can recover messages with an arbitrarily low decoding error probability. On the other hand, perfect secrecy cannot always be guaranteed due to the absence of Eve’s instantaneous CSI: once the rate redundancy $R_{e}$ falls below Eve’s channel capacity, perfect secrecy is compromised, and a secrecy outage event is said to have occurred. Nevertheless, in the finite blocklength regime which is restricted to a finite number of channel uses, no practical protocols can achieve perfectly reliable communications [29]. Hence, to capture the impact of finite blocklength, the maximal channel coding rate for sustaining a desired decoding error probability $\epsilon$ at a finite blocklength $n$ (e.g., $n\geq 100$) for a given signal-to-noise ratio (SNR) $\gamma$ was studied in [30] and can be approximated by

where $C(\gamma)\triangleq\log_{2}(1+\gamma)$ denotes the Shannon channel capacity, $V(\gamma)\triangleq\left(1-(1+\gamma)^{-2}\right)\log_{2}^{2}e$ denotes the channel dispersion [30], and $Q(x)$ is the $Q$-function defined as $Q(x)\triangleq\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-\frac{t^{2}}{2}}dt$. Equivalently, the decoding error probability for a given coding rate $R$ can be expressed as

For ease of notation, let $C_{i}\triangleq C(\gamma_{i})$ and $V_{i}\triangleq V(\gamma_{i})$ for $i\in\{b,e\}$, where $\gamma_{i}$ denotes the corresponding SNR. Define the successful decoding probability of Bob as the complement of its decoding error probability with the codeword rate $R_{t}$. Then, the successful decoding probability conditioned on the power gain of the main channel, i.e., $\eta\triangleq\|\bm{h}_{b}\|^{2}$, can be expressed as

The secrecy performance is characterized by the information leakage probability defined below:

Since Alice knows Bob’s instantaneous CSI perfectly, she is able to adapt the code rates to the instantaneous channel gain $\eta$, which implies that the code rates can be functions of $\eta$. This paper focuses on the metric named secrecy throughput (bits/s/Hz/channel), which measures the average successfully transmitted information bits per second per Hertz per channel use subject to a secrecy constraint $\mathcal{O}_{e}\leq\delta$, where $\delta\in[0,1]$ is a pre-established threshold for the information leakage probability. Formally, the secrecy throughput is defined as

which is averaged over $\eta$. Note that the introduction of finite blocklength leads to a different definition of secrecy throughput compared to the case of infinite blocklength which is $\mathcal{T}\triangleq\mathbb{E}_{\eta}\left[R_{s}(\eta)\right]$ [22, 23]. In addition, as will be shown later, in order to meet certain secrecy and reliability requirements during the transmission period, an on-off transmission policy is required;²²2The on-off policy was initially proposed for ergodic-fading channels [31], where a codeword experiences many channel realizations. It was later introduced to slow fading channels and well characterized the condition for secure transmissions [22]. i.e., the transmission should take place only when the channel gain $\eta$ exceeds some threshold $\mu>0$. With the on-off policy, $R_{s}(\eta)$ is set to zero for $\eta<\mu$.

This paper aims to maximize the secrecy throughput by designing the optimal on-off threshold, signaling, blocklength, as well as code rates. The following two sections will detail the optimization for single- and multi-antenna transmitter scenarios, respectively. For each scenario, both adaptive and non-adaptive design schemes are examined, where Alice adjusts the arguments based on the instantaneous and statistical CSI of the main channel, respectively.

In the adaptive scheme, the parameters $\mu$, $R_{s}$, $R_{e}$, and $n$ are designed based on $\eta$, i.e., they are adjusted in real time. A detailed optimization procedure is provided as follows.

This section devises a non-adaptive optimization scheme where the parameters $\mu$, $R_{s}$, $R_{e}$, and $n$ are designed based on the statistical CSI of the main channel and remain unchanged during the transmission period. Such a non-adaptive scheme can be computed off-line, which significantly lowers the complexity compared with an adaptive one.

Since all the parameters are independent of the channel gain $\eta$, the problem of maximizing the secrecy throughput in (5) can be recast as follows:

where $\bar{p}_{s}=\int_{P_{b}\mu}^{\infty}p_{s}f_{\gamma_{b}}(\gamma)d\gamma$ denotes the average successful decoding probability.

The above problem can be handled via similar steps for its adaptive counterpart in Sec. III-A. To begin with, in order to increase $p_{s}$ for a given $R_{s}$, a minimal rate redundancy $R_{e}$ should be chosen while satisfying the secrecy constraint $\mathcal{O}_{e}\leq\delta$. Hence, the optimal $R_{e}^{*}$ is given in (7). It can be inferred from (15) that a smaller transmission threshold $\mu$ can produce a larger $\mathcal{T}_{\rm N}$. Nonetheless, $C_{b}\geq\log_{2}(1+P_{b}\mu)\geq R_{s}+R_{e}^{*}$ must be ensured, since otherwise there would always exist a transmission initiated when $\eta>\mu$ while violating the reliability constraint (6b). Consequently, the optimal $\mu^{*}$ for a fixed $R_{s}$ is given by

Note that in order to support a constant secrecy rate $R_{s}$, the optimal on-off threshold $\mu^{*}$ for the non-adaptive scheme is generally larger than that of the adaptive one as given in (9). On the other hand, the optimal $\mu^{*}$ monotonically decreases with $\delta$ and $n$, which is similar to the adaptive case. That is to say, the transmission condition can be relaxed when facing a looser secrecy requirement or using a larger blocklength.

Substituting $R_{e}^{*}$ and $\mu^{*}$ into $\bar{p}_{s}$ and invoking the approximation of $Q$-function in (49) yields

where $\mathrm{(a)}$ is due to $\theta_{b}=\sqrt{P_{b}\mu^{*}}=\sqrt{2^{R_{s}+R_{e}^{*}}-1}$, $\beta=\frac{\sqrt{n}}{2\pi}$, and $\tau^{u}_{b}=\theta_{b}^{2}+\frac{\theta_{b}}{2\beta}$, and $\mathrm{(b)}$ stems from the use of partial integration. With (III-B), the problem of maximizing $\mathcal{T}_{\rm N}$ over $n$ and $R_{s}$ can be equivalently transformed as below:

Theorem 3 suggests that Alice should use the maximal blocklength to maximize the secrecy throughput for the non-adaptive scheme, regardless of other parameters, i.e., the globally optimal blocklength is $n^{*}=N$. More importantly, this conclusion holds for any distribution of $\gamma_{b}$.

Substituting the CDF $\mathcal{F}_{\gamma_{b}}(\gamma)=1-e^{-{\gamma}/{\Gamma_{b}}}$ into (III-B) yields

where $Y(\theta_{b})=\frac{2\beta\Gamma_{b}}{\theta_{b}}(1-e^{-\frac{\theta_{b}}{2\beta\Gamma_{b}}})>0$. The optimal $\theta_{b}^{*}$ that maximizes $\mathcal{T}_{\rm N}$ is provided below.

Based on Theorem 4, the optimal $\theta_{b}^{*}$ or secrecy rate $R_{s}^{*}=\log_{2}(1+(\theta_{b}^{*})^{2})-R_{e}^{*}$ can be efficiently calculated using a bisection search with $G(\theta_{b})=0$, and thus the maximal $\mathcal{T}_{\rm N}^{*}$ can be obtained from (18). The following corollaries demonstrate the behavior of $R_{s}^{*}$ w.r.t. to the average channel power gain $\sigma_{b}^{2}=\frac{\Gamma_{b}}{P_{b}}$ and provide a closed-form approximation of $R_{s}^{*}$ at the large $\sigma_{b}^{2}$ regime.

Corollary 3 suggests that a larger secrecy rate should be employed to boost the secrecy throughput when the quality of the main channel improves, despite the fact that it might deteriorate the decoding correctness at Bob.

Fig. 3 plots the secrecy throughput $\mathcal{T}_{\rm N}$ versus the secrecy rate $R_{s}$ for different values of the blocklength $N$ and the average channel gain $\sigma_{b}^{2}$. It can be seen that $\mathcal{T}_{\rm N}$ first increases and then decreases with $R_{s}$, which validates Theorem 4. The optimal $R_{s}^{*}$ maximizing $\mathcal{T}_{\rm N}$ increases with $\sigma_{b}^{2}$, which verifies Corollary 3 well, and the reason behind is similar to that for Corollary 2. It can also be observed that the optimal $R_{s}^{*}$ is almost impervious to different $N$. This is because, the optimal secrecy rate for the non-adaptive scheme only depends on the average successful decoding probability, and the averaging process softens the impact of the blocklength. Theorem 3 is also confirmed, where it is found that $\mathcal{T}_{\rm N}$ increases with $N$. In addition, the secrecy throughput with the approximate $R_{s}^{A}$ obtained in Corollary 4 is almost coincided with that of the optimal $R_{s}^{*}$, which demonstrates the practicability of the low-complexity approximation.

Fig. 4 compares the secrecy throughput for adaptive and non-adaptive schemes with different blocklength $N$. The left-hand-side figure depicts the maximal secrecy throughput $\mathcal{T}^{*}$, where $\mathcal{T}_{\rm A}^{*}$ for the adaptive case improves as $N$ increases whereas $\mathcal{T}_{\rm N}^{*}$ for the non-adaptive case almost remains unchanged. When the average channel gain $\sigma_{b}^{2}$ increases or the secrecy constraint becomes relaxed (i.e., a larger $\delta$), the maximal $\mathcal{T}^{*}$ for both schemes improves significantly, and the gap $\mathcal{T}_{\rm A}^{*}-\mathcal{T}_{\rm N}^{*}$ increases. The right-hand-side figure illustrates the relative throughput gain $\Delta\mathcal{T}\triangleq\frac{\mathcal{T}_{\rm A}^{*}-\mathcal{T}_{\rm N}^{*}}{\mathcal{T}_{\rm N}^{*}}$ which reflects the superiority of the adaptive scheme over its non-adaptive counterpart. It is shown that $\Delta\mathcal{T}$ grows dramatically with $N$ but decreases with $\sigma_{b}^{2}$ and $\delta$. This suggests that the adaptive scheme is more preferred for some unfavorable scenarios, e.g., with a large blocklengh (large delay), a poor channel quality, or a stringent secrecy requirement; otherwise, the non-adaptive scheme could be an alternative choice owing to its low implementation complexity.

This subsection optimizes the secrecy throughput $\mathcal{T}_{\rm A}$ by designing the parameters involved in problem (26) adaptively according to the instantaneous channel realization $\bm{h}_{b}$.