Predictor-corrector models for lightweight massive machine-type communications in Industry 4.0

Abstract

Future Industry 4.0 scenarios are characterized by seamless integration between computational and physical processes. To achieve this objective, dense platforms made of small sensing nodes and other resource constraint devices are ubiquitously deployed. All these devices have a limited number of computational resources, just enough to perform the simple operation they are in charge of. The remaining operations are delegated to powerful gateways that manage sensing nodes, but resources are never unlimited, and as more and more devices are deployed on Industry 4.0 platforms, gateways present more problems to handle massive machine-type communications. Although the problems are diverse, those related to security are especially critical. To enable sensing nodes to establish secure communications, several semiconductor companies are currently promoting a new generation of devices based on Physical Unclonable Functions, whose usage grows every year in many real industrial scenarios. Those hardware devices do not consume any computational resource but force the gateway to keep large key-value catalogues for each individual node. In this context, memory usage is not scalable and processing delays increase exponentially with each new node on the platform. In this paper, we address this challenge through predictor-corrector models, representing the key-value catalogues. Models are mathematically complex, but we argue that they consume less computational resources than current approaches. The lightweight models are based on complex functions managed as Laurent series, cubic spline interpolations, and Boolean functions also developed as series. Unknown parameters in these models are predicted, and eventually corrected to calculate the output value for each given key. The initial parameters are based on the Kane Yee formula. An experimental analysis and a performance evaluation are provided in the experimental section, showing that the proposed approach causes a significant reduction in the resource consumption.

Keywords

Predictor-corrector methods differential models Industry 4.0 massive machine-type communications Physical Unclonable Functions lightweight communications

1. Introduction

Industry 4.0 [1] is typically defined as the next industrial revolution, characterized by the seamless integration of computational and physical processes. Paradigms such as Cyber-Physical Systems [2] or Artificial Intelligence [3] are usually referred to enabling technologies in future Industry 4.0 scenarios. However, current challenges related to Industry 4.0 technologies are closer to the hardware level than to knowledge or data management. Actually, how this seamless integration between the cyber and physical world may be achieved through currently existing sensors, actuators, programming languages and recognition technologies [35] is a major open question [4].

In general, ubiquitous platforms including thousands of hardware devices per square kilometer are deployed to monitor and control physical processes in a tight and real-time manner [61]. These devices are usually manufactured as sensing nodes whose computational resources are sparse and only enough to capture data and transmit it to a central gateway using point-to-point communications [5]. Feedback control algorithms, orchestration and resource allocation mechanisms, communication management protocols and other similar computationally heavy technologies are delegated to gateways, placed at the central point of a star topology network, and where more resources and available [6]. Also, with such several devices, human intervention is totally inefficient, and devices must operate autonomously. This also affects communications, which must follow machine-type communication (MTC) [7] standards and solutions.

One of the key problems in MTC is the way in which communications are secured. Although several powerful security technologies are currently available (elliptic curves, trust management, etc.), they are usually too computationally heavy for Industry 4.0 nodes [8]. In this context, hardware solutions are employed and, nowadays, Physical Unclonable Functions (PUF) are the most promising and popular solution among all existing proposals. PUF are hardware devices with unpredictable behavior, so its output (known as ‘response’) is unique for any given input (known as ‘challenge’) [9]. So, this output is a fingerprint that enables authentication and cryptographic applications with no computational cost. As this random behavior is inherited from uncontrolled factors during manufacturing, by now, responses cannot be predicted or replicated.

Many different PUF implementations have been reported, but currently, those based on silicon circuits are the ones more commonly employed in major industrial systems. For example, SRAM PUF chips are now commercialized by several leader semiconductor companies for edge-to-cloud securization in factories [36] and legacy SCADA systems [37]. In SRAM PUF, security solutions take advantage of the random differences in the threshold voltages of transistors making up the static RAM memories to create unique identifiers. When powered up, each memory cell will take its preferred state (zero or one), creating a unique, random, and unclonable binary pattern.

As, by now, PUF responses to challenges cannot be calculated or estimated, the only way to enable gateways to learn about the nodes’ PUF responses is a registration process [10]. During this registration process, PUF is tested, and all challenge-response pairs are stored in a key-value database accessed by the gateway. Then, gateways may look for the appropriate response for any given challenge.

However, Industry 4.0 is becoming increasingly massive, and paradigms such as 5G envision up to one million devices per square kilometer [11]. Although the resources are usually abundant, for such massive platforms, the memory usage is not scalable and information access and processing delays grows exponentially. Thus, massive key-value databases become very resource consuming, and can even produce congestion and unavailability of gateway [12] (because of the lack of memory for other computational processes and/or the huge delays in accessing such a massive database). Considering that gateways are placed at the central point of a start topology network, this finally causes Industry 4.0 to shut down. Then, new lightweight approaches are required to allow Industry 4.0 platforms to deal with secure massive machine-type communications.

Therefore, the objective of this paper is to develop a lightweight solution for secure massive machine-type communications (mMTC) in Industry 4.0 scenarios. The solution is based on predictor-corrector differential models, representing the challenge-response functions in PUF. Models are mathematically complex, but we argue that they consume less computational resources than current approaches. The lightweight models are based on complex functions managed as Laurent series, cubic spline interpolations, and Boolean functions also developed as series. Unknown parameters in those models are predicted, and eventually corrected, to calculate the output value for each given key. The initial parameters are based on the Kane Yee formula [31].

Briefly, the main novelties and advantages of this new approach based on algebraic methods are:

•
Reduced computational resource consumption (memory and processing delay), This new algebraic technology can be implemented in several different scenarios, including those where edge devices have a (very) limited computational power.
•
Improved scalability, enabling even future envisioned applications with up to one million devices per square kilometer.
•
Enhanced flexibility and multi-PUF support. The proposed predictor-corrector model is very flexible and can be applied to all silicon-based PUF with no required software update or adaptation. Even different PUF technologies can coexists using the same software process, and PUF may change dynamically.

The remainder of the paper is organized as follows. Section 2 presents the state of the art in PUF modeling and mMTC management techniques. Section 3 presents the proposed solution, including the predictor-corrector method. Section 4 describes the experimental evaluation; Section 5 discusses the obtained results; and Section 6 concludes the paper.
2. State of the art

This section presents an overview of the currently reported mechanisms for PUF modeling and massive machine-type communication management, with a special interest in security solutions.

2.1 Massive machine-type communications management

Among all the challenges introduced by 5G mobile networks and Industry 4.0 systems (both have similar requirements), mMTC is probably the least studied paradigm.

With up to one million devices per square kilometer, different works have identified four basic challenges to be addressed [22]: Quality-of-Service provision, physical access, transmission scheduling (including spectrum issues, transmission schemes, resource management, traffic characterization and low-power communications), and congestion management. In this work we describe an innovative transmission scheme for secure communications, considering resource consumption and management aspects too.

Anyway, all these issues are related to communication protocols or media resources. Thus, solutions such as new MAC protocols where access is much more multiplexes to allow all devices to communicate [23] have been reported.

Focusing on the security issues, works are sparse. Many different proposals on secure (not massive) machine type communications were reported in the past for 4G cellular communications [38, 39]. However, currently, authors are working on making next generation mMTC feasible at physical level (spectrum and energy management, traffic conformation, etc.) [40], and security technologies are not understood as a priority yet [41]. This is mainly because other primary challenges in mMTC are still open and prevent security solutions to operate correctly in real scenarios (for example, the mitigation of co-channel interferences, the accurate channel information capture, or the dynamic device management). Besides, reported security mechanisms are very low-level. Some works report new link-level protocols for massive access, where headers are calculated through physical security techniques [42]. While others propose innovative carrier selection algorithms, modulation schemes and array antenna designs [43] to enhance precision in the power radiation and protect massive communications against physical intruders.

But all these schemes present two basic problems. First, attacks at network or application level are not addressed, and mMTC implementing these physical security technologies are still vulnerable against network sniffing or spoofing attacks (among others). And second, physical security is very rigid, and requires the entire communication stack to get adapted to the characteristics and capabilities of the security layer. This is very inefficient and costly, while flexible security solutions being able to adapt to different scenarios, architectures and user equipment are preferred.

In this paper, on the other hand, we describe a new secure communication scheme which may be applied at several different levels (including link, network and application levels), with high flexibility, as it can adapt (even dynamically) to many hardware technologies (such as different types of microprocessors, sensors, actuators, single-board computers, etc.) and system architectures (from distributed computing schemes, to centralized star-topologized solutions and from isolated local infrastructures to cloud deployments).

But actually, these problems have been already studied by some authors. Thus, some lightweight authentication schemes based on random numbers can be found [28]. However, the results show that this approach is only feasible for applications where latency is not a critical parameter (500 ms is the expected value), and no evidence about the resource consumption in these mechanisms is provided. Similarly, device-type authentication mechanisms where unique devices’ fingerprints are obtained from spectrum patterns have been reported [44]. But in this approach, gateways must store every single device fingerprint, which introduces severe limitations in scalability (although this fact is not deeply studied). Moreover, cryptographic solutions for secure mMTC have also been reported [29], but results show that resource consumption still grows exponentially with the number of devices, although at a lower rate.

In fact, the impact of security schemes on infrastructure cost or size; or the resources these new approaches consume are not addressed. Only exploratory works on these issues, proposing some challenges and research opportunities, can be found for some specific scenarios (such as heterogenous networks) [24].

In that way, currently reported authentication and cryptographic technologies require large amounts of memory resources, and present scalability problems which may prevent them to be implemented in the most extreme and dense mMTC scenarios (i.e., one million devices per square kilometer). On the contrary, the new technology reported in this paper highly reduces the required memory consumption (up to 60%) and improves scalability achieving a linear complexity (with the number of user devices) instead of the traditional exponential evolution.

Finally, resource consumption reduction, currently, is only addressed from the data engineering perspective. Solutions for data compaction and aggregation in mMTC [25], cooperative techniques to manage massive data efficiently [26], and machine learning techniques to find relationships among all received information [27] and remove redundancies have been reported. Nevertheless, all these approaches cannot be applied to security solutions.

2.2 Physical Unclonable Functions modeling

In the last ten years, many different types of PUF have been reported. From optical resonant rings [9] to resonant structures [10]. However, all these PUF are very complex and must be complemented with heavy-measure instruments. On the contrary, PUF based on time domain or memory are supported by silicon circuits and are easy to integrate in Industry 4.0 sensing nodes [9]. This section is focused on those kinds of PUF.

Silicon-based PUF take advantages of small and random variations within materials, which cause circuits to response slightly different to the same excitation at electrical and physical level [46]. Signals may propagate at different speeds through the material, or thresholds voltages may change. Although these variations are usually very small to be noticeable, when an enough number of silicon-based devices are connected, the global effect can be easily detected. For example, in a common arbiter PUF one hundred and twenty-eight two-to-one multiplexers are connected in cascade to make noticeable the different speed propagation through the two channels [45].

In general, PUF are characterized by three basic properties: reliability, uniqueness and randomness [13]. Reliability forces a strong PUF to generate the same response to the same challenge every time that it is posed in equivalent conditions. Thus, a deterministic equivalence function could be defined between challenges and responses. However, uniqueness implies that every PUF circuit has a totally different behavior. So, those equivalence functions cannot be generic, and randomness causes PUF to be erratic (and obviously non-linear), so close challenges cannot generate similar responses. Actually, all responses in a strong PUF should be statistically independent.

However, practical implementations of PUF meet these characteristics at different levels. The intra-chip distance (in bits) measures how different are responses when the same PUF is excited several times with the same challenge. While inter-chip distance (also in bits) measures how different are the responses when different PUF are excited several times with the same challenge. Considering a PUF implementation with a 256-bit response, the maximum inter-chip distance is 150 bits [47]. So different PUF, even in the best situation, have around 100 bits (almost 40%) in common. On average inter-chip distance is 128 bits (50%), but it can be as small as 100 bits (so different PUF may share up to 60% of bits). However, intra-chip distance has a much better behavior [47]. The average distance is 20 bits (around 8% of the key length), and the maximum is only 30 bits (less than 12%), while the minimum value for intra-chip distance is very close to zero. Then, the challenge-response relation can be considered strong and deterministic.

This context, where intra-chip distances are reduced, but inter-chip distances (although higher) allow different PUF to share 50% of bits (on average) makes mathematical models for PUF feasible [16]. Different authors have addressed this open issue from different perspectives, although works on this topic are still sparse.

Table 1
State of the art in PUF modeling

Reference	Approach	Model type	Supporting technology	Comments
[14]	Attack	Generic	Machine learning	High computational cost
[15]	Attack	Generic	Machine learning	High computational cost
[18]	Attack	Generic	Machine learning	High computational cost
[19]	Attack	Generic	Machine learning	High computational cost
[20]	Attack	Specific	Numerical model	Valid only for some PUF
[13]	Analysis	Specific	Numerical model	Valid only for some PUF
[16]	Attack	Generic	Algebraic framework	No challenge-response function provided
[17]	Attack	Generic	Algebraic framework	No challenge-response function provided

Considering the potential existence of a strong and deterministic mathematical relation between challenges and responses, but its non-obvious analytical expression, several authors have proposed supervised learning technologies to capture and replicate the behavior of PUF. Models for bistable ring (BR) PUF [14] or twisted bistable ring (TBR) PUF [15] (both PUF based on feedback loops made of multiplexers and NOR gates which must achieve a stable state – the key-) have been reported. Typical solutions are based on support vector machines (SVM) [18]. Some authors have even analyzed different types of algorithms (SVM, genetic algorithms, etc.) to identify the best learning technique to replicate the behavior of PUF [19].

However, these models present some problems. First, they are useful (accuracy above 95%) as attacking technologies [20], where they must only replicate the behavior of one specific PUF circuit. But when they are employed to model the behavior of an entire PUF technology (several different circuits), accuracy highly decreases and SVM models show an error slightly below 50% (47%, approximately, to be precise) [15]. Even, for some authors, that means the PUF technology is resistant to machine learning models [21]. Second, these modeling technologies are not flexible, and different techniques and models must be deployed to precisely replicate different PUF behavior. Then, in mMTC scenarios, major software updates and algorithm trainings are needed when new PUF are deployed. This fact makes it difficult the dynamic evolution of hardware and the coexistence of different PUF technologies.

On the contrary, the proposed prediction-corrector model in this paper shows an average accuracy up to 50% better than SVM-models (depends on the configuration). Besides, the proposed solution is very flexible and can be applied to all kinds of silicon-based PUF. The model, thanks to the correction stage, is continuously refined with no additional effort, so no software update or new training is required.

When models fail because of a reduced accuracy, many authors propose the creation of large key-value databases [48] where the challenge-response pairs are maintained, but this approach is not scalable. In very dense mMTC scenarios (up to one million devices per square kilometer) those databases would be massive and searching times too high to be compatible with communication streams. In this approach, actually, memory consumption and processing delays increase exponentially with the number of devices [49], contrary to the new algebraic model proposed in this work where a linear evolution is achieved.

A second group of PUF models are based on physical laws. In this case, models of the physical phenomena supporting the PUF behavior are proposed. Typically, models describe the behavior of transistors and other silicon-based elements [13]. These models are considerably lighter than SVM, but they require a deep knowledge of PUF implementation (voltage thresholds with an extreme precision, conductivity, magnetic permeability, etc.). That may not be realistic in most massive scenarios, where different hardware technologies coexist and mMTC users cannot operate an electronic laboratory to acquire all this information from every single device. In commercial and engineering applications this approach is not feasible.

Finally, some authors are reporting exploratory work on how generic models could be achieved [16]. In this approach, no analytic mathematical model is proposed, but a general algebraic framework that describes the properties that these models should meet and how feasible they are [17]. In general, the results prove that these models are achievable, although they do not clearly show how they can be identified or managed. In this paper we explore and advance in this direction. We propose a general model, based on different lightweight mathematical expressions. It cannot generate responses with full precision, but it can be automatically adjusted through a prediction-correction process.

Table 1 summarizes previous discussion.

3. A predictor corrector model for mMTC

In this Section, we propose a predictor-corrector model to enable lightweight secure mMTC. The mathematical model and algorithm described are supported by Industry 4.0 architecture in Fig. 1.

In our scenario, Industry 4.0 systems consist of up to one million devices per square kilometer. Thousands of devices are connected through low-power communications to one gateway. Devices (or sensing nodes) are resource-constrained and employ Physical Unclonable Functions to generate keys and protect communications. On the other hand, gateways have a high computational power and access to great data storage capabilities. Using those resources, gateways can execute complex mathematical operations and maintain in a key-value database some challenge-response pairs (or, eventually, the full catalogue) of the PUF of nodes.

The approach proposed in this paper requires minor and negligible changes in the sensing nodes, so their resources are not affected. While, in gateways, we highly reduce the memory consumption and processing time, by replacing full catalogues in databases by complex predictor-corrector mathematical models.

In general, PUF can be analyzed at three different abstraction levels. First, as silicon-based materials under an electromagnetic excitation [13]. Second, as binary (Boolean) logical circuits managing binary words [16]. And third, as numerical mathematical functions [15] (this last one is the most common approach in machine learning models and other similar techniques). However, all these approaches face the same challenge: it is extremely complicated to accurately model the entire PUF behavior, and all the variables that affect it, using only techniques at one abstraction level. Some phenomena, such as voltage thresholds or signal filtering, are very easily described using electromagnetic and signal propagation laws. Nevertheless, some other, such as the cascade connections of logical devices, are more precisely analyzed using binary circuits. Moreover, for example, the inference of PUF responses for similar challenges shows a lower error when using mathematical functions (because of the extremely high precision required to physical models to distinguish two very close challenges and responses).

Figure 1.

Industry 4.0 architecture.

This problem, at the end, is the cause of the poor performance of these models (accuracy is around 50% [15] when they operate independently). As well as the reason why massive key-value databases, with an exhaustive description of all challenge-response pairs, are the common feasible solution for real mMTC scenarios (despite their resource consumption and scalability problems).

As a possible solution, in this paper we propose a combined model, where each PUF item is analyzed in parallel using these three abstraction levels. First, models operate independently and, later, results are aggregated so they can compensate the inaccuracies of each other. This enables us to reduce the complexity and computational cost of models, as they do not have to describe all the variables affecting the PUF behavior. For example, we can use algebraic models instead of machine learning techniques, which are much more flexible and allow the coexistence of different PUF technologies in the same scenario.

But the actual accuracy and significance of every model is unknown a priori. Then, a predictor-corrector scheme is deployed in order to refine models and weights continuously according to real exchanged challenge-response pairs between user devices and gateways. This approach allows the solution to learn from the hardware platform and adapt to it dynamically, without heavy training processes or major software updates.

Although each model could manage different precisions (word lengths or information bits), in this work we are assuming all three models produce responses or results with the same number of true information bits (in real applications, and considering the state of the art, 256 bits would be a usual number). Later, using different weights during the aggregation, the number of significant figures in each model may be limited if needed (never increased). In any case, the final combined result will have the same number of true information bits than every independent model (for example, 256-bit responses in a common real mMTC application).

Figure 2.

Proposed predictor-corrector model.

Figure 2 shows the general scheme for the proposed predictor-corrector model. In our model, PUF responses to challenges can be predicted by combining three different models: an analog model, a Boolean model, and a numerical model. In the analog model, PUF are understood as a physical material media. Analog signals that excite the material are represented through the complex envelope, whereas the PUF response is estimated through an unknown complex function. In the Boolean model, PUF are represented by multidimensional binary inputs (response), and Boolean unknown functions generating a binary response. Finally, numerical models understand PUF as a complex real function. Challenges and responses are represented using real numbers, and responses are estimated through an interpolation scheme.

All unknown functions include a collection of unknown variables which may be initially estimated using the Kane Yee formula (prediction) [31] and later corrected and refined through the Mean Square Error (MSE) algorithm (correction).

These three models, in order to generate the final predicted responses, are combined in a weighted average where the reliability, randomness, and uniqueness of each model are taken into account.

Finally, the prediction-correction process is not unilateral and requires the participation of sensing nodes. These nodes must negotiate to identify whether the gateways’ predictions are good or should be corrected according to their hardware PUF. A negotiation protocol is proposed to address this final issue. In this protocol, devices (nodes) will perform just two simple mathematical operations, to determine if the predicted value is acceptable or not.

The next subsections describe all details about each one of these three components.

3.1 Analog model: PUF as a propagation media

Every time-domain or memory-based PUF consists of an electronic silicon circuit. This circuit may be analyzed from different perspectives, but at a physical level it is a silicon material under an electromagnetic excitation. This electromagnetic excitation $c_{i}\left(t\right)$ in the $i-th$ node is called the challenge. As challenges are physical signals, they are continuous and differentiable for all degrees of differentiation Eq. (1). Challenges, in general, are passband temporal signals restricted to the interval $\left[{0,T}\right]$ , being $T_{ex}$ the duration of the excitation. A square window signal $W\left(t\right)$ represents this restriction Eq. (2).

$\displaystyle c_{i}\left(t\right)\in C^{\infty}$ (1) $\displaystyle W\left(t\right)=\left\{\begin{array}[]{ll}1&\ \ \textrm{if}\ \ % \ 0\leqslant t\leqslant T_{ex}\\ 0&\ \ \textrm{otherwise}\\ \end{array}\right.$ (2)

The material will generate a response $r_{i}\left(t\right)$ to this challenge. In this paper, we are assuming only reliable PUFs are deployed in Industry 4.0 scenarios. In practical situations, reliable PUF are those whose different responses to the same challenge differentiate, on average, less than 1% from each other [57]. Although it is a very restrictive criterion, currently many silicon-based PUFs fulfill this requirement [58].

Reliable PUF are time invariant and then, the relation between the challenge and the response is a function $f_{i}\left(\cdot\right)$ , only dependent on the challenge. Besides, we assume that, in three-dimensional silicon materials, signals propagate the same in all directions. Then, it is enough to study only one direction and unidimensional functions may be employed Eq. (3). This function is, in general, unknown.

$\displaystyle r_{i}=f_{i}\left({c_{i}\cdot W}\right)$ (3)

However, passband signals include two types of information: frequency information and amplitude information. Thus, every passband function may be represented through two new complex subsignals Eq. (4): the complex envelope $c_{i}^{a}\left(t\right)$ containing the amplitude information Eq. (5), and the instantaneous carrier frequency $c_{i}^{f}\left(t\right)$ containing the frequency information Eq. (6).

$\displaystyle c_{i}\left(t\right)={\cal R}e\left\{{c_{i}^{a}\left(t\right)% \cdot e^{j\cdot c_{i}^{f}\left(t\right)\cdot t}}\right\}=c_{i}^{ph}\left(t% \right)\cos\left({\omega_{i}^{c}\left(t\right)t}\right)-c_{i}^{Q}\left(t\right% )\cdot\sin\left({\omega_{i}^{c}\left(t\right)\cdot t}\right)$ (4) $\displaystyle c_{i}^{a}\left(t\right)=c_{i}^{ph}\left(t\right)+jc_{i}^{Q}\left% (t\right)$ (5) $\displaystyle c_{i}^{f}\left(t\right)=\omega_{i}^{c}\left(t\right)$ (6)

Figure 3.

In-phase signal calculation scheme.

To calculate the complex envelope two baseband signals only with amplitude information must be obtained: the in-phase signal $c_{i}^{ph}\left(t\right)$ and the quadrature signal $c_{i}^{Q}\left(t\right)$ . These signals may be obtained Eq. (3.1) from the original passband challenge $c_{i}\left(t\right)$ by multiplying it with a sinusoidal signal wit frequency $\omega_{s}$ , and removing components at high frequency (i.e. frequencies $\omega_{i}^{c}+\omega_{s}$ ) using a simple lowpass filter (see Fig. 3).

$\displaystyle c_{i}^{\textit{cos}}=c_{i}\left(t\right)\cdot\cos\left({\omega_{% s}\cdot t}\right)=\frac{1}{2}c_{i}^{ph}\left(t\right)+\frac{1}{2}\left[c_{i}^{% ph}\left(t\right)\cdot\cos\left({\left[{\omega_{i}^{c}+\omega_{s}}\right]\cdot t% }\right)\right.\left.-c_{i}^{Q}\left(t\right)\cdot\sin\left({\left[{\omega_{i}% ^{c}+\omega_{s}}\right]\cdot t}\right)\right]$ $\displaystyle c_{i}^{\textit{sin}}=c_{i}\left(t\right)\cdot\sin\left({\omega_{% s}\cdot t}\right)=-\frac{1}{2}c_{i}^{Q}\left(t\right)+\frac{1}{2}\left[c_{i}^{% ph}\left(t\right)\cdot\cos\left({\left[{\omega_{i}^{c}+\omega_{s}}\right]\cdot t% }\right)\right.\left.-c_{i}^{Q}\left(t\right)\cdot\sin\left({\left[{\omega_{i}% ^{c}+\omega_{s}}\right]\cdot t}\right)\right]$ (7)

Many different filters could be used, but we are looking for a precise response estimation, so we are using a Butterworth filter as it does not show distortion in the pass band. The bandwidth will be $B$ Hertz, and filter’s gain $G$ is set to two to compensate attenuations in signal processing. The filter order is noted as $K$ . The transfer function of this filter $H\left(s\right)$ in the Laplace domain Eq. (8) is easy to represent using the Butterworth polynomial $B_{K}\left(a\right)$ Eq. (3.1), where $s$ is the complex frequency in the Laplace domain.

$\displaystyle H\left(s\right)=\frac{G}{B_{K}\left({\frac{s}{B}}\right)}$ (8) $\displaystyle B_{K}\left(a\right)=\mathop{\sum}\limits_{m=0}^{K}a_{m}\cdot s^{m}$ $\displaystyle a_{0}=1$ $\displaystyle a_{m}=\mathop{\prod}\limits_{\mu=1}^{m}\frac{\cos\left({\left[{% \mu-1}\right]\cdot\frac{\pi}{2K}}\right)}{\sin\left({\mu\cdot\frac{\pi}{2K}}% \right)}$ (9)

If required or desired, different filter types (Chebyshev, elliptical, etc.) could be employed too. Although distortion may appear in the pass and/or the stop band, and that might affect the model’s precision (i.e., the distance between two PUF responses in one PUF item associated to two very close challenges), weights during the aggregation stage (see Section 3.4) will mitigate this impact. As we’ll explain later (see Section 3.4) weights are proportional (exponentially) to the inter-challenge distance (37), and accurate models are highly strengthened. So, the impact of any distortion in the combined final result (prediction) would be negligible.

Using the convolution and the Laplace inverse transform the in-phase and quadrature signals may be calculated Eq. (3.1).

$\displaystyle c_{i}^{ph}\left(t\right)=c_{i}^{\textit{cos}}\left(t\right)\ast h% _{K}\left(t\right)=\int_{-\infty}^{\infty}c_{i}^{\textit{cos}}\left(\tau\right% )\cdot h_{K}\left({t-\tau}\right)d\tau$ $\displaystyle c_{i}^{Q}\left(t\right)=c_{i}^{\textit{sin}}\left(t\right)\ast h% _{K}\left(t\right)=\int_{-\infty}^{\infty}c_{i}^{\textit{sin}}\left(\tau\right% )\cdot h_{K}\left({t-\tau}\right)d\tau h_{K}\left(t\right)={\cal L}^{-1}\left% \{{H\left(s\right)}\right\}$ (10)

On the other hand, the frequency signal $c_{i}^{f}\left(t\right)$ can be calculated through the Fourier transform. The instant carrier frequency $\omega$ is the one for which the power spectrum $C_{i}\left({\omega,t}\right)$ of the original challenge is maximum Eq. (11). This spectrum is directly obtained from the Fourier transform Eq. (12).

$\displaystyle c_{i}^{f}\left(t\right)=\omega_{i}^{c}\left(t\right)=\textit{arg% }\mathop{\max}\limits_{\omega}\left\{{C_{i}\left({\omega,t}\right)}\right\}$ (11) $\displaystyle C_{i}\left({\omega,t}\right)=\left|{\frac{1}{\sqrt{2\pi}}\int_{-% \infty}^{\infty}c_{i}\left(t\right)\cdot e^{-j\omega t}dt}\right|^{2}$ (12)

The same process may be applied to response signal $r_{i}\left(t\right)$ and function $f_{i}\left(t\right)$ Eq. (3.1).

$\displaystyle r_{i}^{a}\left(t\right)=f_{i}^{a}\left({c_{i}^{a}}\right)$ $\displaystyle r_{i}^{f}\left(t\right)=f_{i}^{f}\left({c_{i}^{f}}\right)$ (13)

With this approach, function $f_{i}^{a}\left(\cdot\right)$ is complex, while $f_{i}^{f}\left(\cdot\right)$ is still real, but their behavior is easier to discover, as they are only managing one parameter each (either amplitude, or frequency). Thus, more precise predictions may be made in a faster way.

At this point, functions $f_{i}^{a}\left(\cdot\right)$ and $f_{i}^{f}\left(\cdot\right)$ are unknown, but since they are differentiable for all degrees of differentiation, they can be developed as series. Laurent series will be employed for complex function $f_{i}^{a}\left(\cdot\right)$ . If we define an anulus $A_{z_{0}}$ Eq. (14) centered in $z_{0}$ complex point, with minor radius $R_{1}$ and major radius $R_{2}$ , and we name $\gamma_{1}$ the interior border of the anulus Eq. (15), and $\gamma_{2}$ the exterior border Eq. (16), function $f_{i}^{a}\left(\cdot\right)$ may be written as a polynomial Eq. (3.1). This canonical expression can be manipulated, putting together all numerical coefficients for each power Eq. (18).

$\displaystyle A_{z_{0}}=\left\{{z\vdots R_{2}\leqslant\left|{z-z_{0}}\right|% \leqslant R_{1}}\right\}$ (14) $\displaystyle\gamma_{1}=\left\{{z\vdots\left|{z-z_{0}}\right|=R_{1}}\right\}$ (15) $\displaystyle\gamma_{2}=\left\{{z\vdots\left|{z-z_{0}}\right|=R_{2}}\right\}$ (16)

$\displaystyle f_{i}^{a}\left(z\right)=\mathop{\sum}\limits_{m=0}^{\infty}a_{i}% ^{m}\cdot\left({z-z_{0}}\right)^{m}+\mathop{\sum}\limits_{m=1}^{\infty}\frac{l% _{i}^{m}}{\left({z-z_{0}}\right)^{m}}$ $\displaystyle a_{i}^{m}=\frac{1}{2\pi j}\cdot\int_{\gamma_{2}}\frac{f_{i}^{a}% \left(\tau\right)}{\left({\tau-z_{0}}\right)^{m+1}}d\tau$ $\displaystyle{b}_{i}^{m}=\frac{1}{2\pi j}\cdot\int_{\gamma_{1}}\frac{f_{i}^{a}% \left(\tau\right)}{\left({\tau-z_{0}}\right)^{-m+1}}d\tau$ (17)

$\displaystyle f_{i}^{a}\left(z\right)=\mathop{\sum}\limits_{m=0}^{\infty}\xi_{% i}^{m}\cdot z^{m}+\mathop{\sum}\limits_{m=1}^{\infty}\eta_{i}^{m}\cdot z^{-m}$ (18)

Thus, unknown function $f_{i}^{a}\left(\cdot\right)$ may be described as a polynomial with unknown coefficients $\xi_{i}^{m}$ and $\eta_{i}^{m}$ . Those coefficients will be initially estimated through the Kane Yee formula and later improved through the proposed correction process (see Section 3.5).

On the other hand, Taylor series will be employed to develop real function $f_{i}^{f}\left(\cdot\right)$ as a polynomial around point $c_{0}$ , in a very similar way to the process described above Eq. (19). Equally, unknown coefficients $\beta_{i}^{m}$ will be initially estimated through the Kane Yee formula and later improved through the proposed correction process.

$\displaystyle f_{i}^{f}\left(c\right)=\sum_{m=0}^{\infty}\frac{f_{i}^{f\left(m% \right)}\left({c_{0}}\right)}{m!}\left({c-c_{0}}\right)^{m}=\sum_{m=0}^{\infty% }\beta_{i}^{m}\cdot c^{m}$ (19)

The previous model is adequate for analog signals. However, the challenges in gateways are not managed as electromagnetic waves, but as a binary number $b_{i}^{c}$ with $N$ bits. Then, binary numbers are transformed into signals in the digital-to-analog converter (DAC). However, this mapping function Eq. (20) ${\cal D}$ is, in all situations, a known function. It is also known the transformation ${\cal E}$ to obtain binary numbers $b_{i}^{r}$ with $M$ bits from analog responses Eq. (21), carried out by the Analog-to-Digital converter (ADC). Those functions will be equally applied to predictions, corrections and real PUF challenges or responses.

$\displaystyle{\cal D}\left({b_{i}^{c}}\right)=c_{i}\left(t\right)$ (20) $\displaystyle{\cal E}\left({r_{i}\left(t\right)}\right)=b_{i}^{r}$ (21)

3.2 Boolean model: PUF as a binary function

As time-domain or memory-based PUF consists of a digital electronic silicon circuit, it can be modeled through Boolean functions and relations. In this scenario, challenges are represented by binary numbers $b_{i}^{c}$ with $N$ bits, and responses are binary numbers $b_{i}^{r}$ with $M$ bits. Then, both binary numbers, in the $i$ -th node, are related through and unknown function ${\Gamma}_{i}\left(\cdot\right)$ , where $S_{N}$ is the set of natural numbers that may be represented with $N$ bits Eq. (3.2). Besides, we can understand function ${\Gamma}\left(\cdot\right)$ as a vector function where each component ${\Gamma}^{n}\left(\cdot\right)$ obtains the $n$ -th bit $y_{n}$ in the response $b_{i}^{r}$ from the challenge $b_{i}^{c}$ Eq. (3.2).

$\displaystyle{\Gamma}_{i}:S_{N}\to S_{M}$ $\displaystyle{\Gamma}_{i}\left({b_{i}^{c}}\right)=b_{i}^{r}$ (22) $\displaystyle{\Gamma}\left({b_{i}^{c}}\right)$ $\displaystyle\ \ =\left({{\Gamma}_{i}^{1}\left({b_{i}^{c}}\right),{\Gamma}_{i}% ^{2}\left({b_{i}^{c}}\right),\ldots,{\Gamma}_{i}^{n}\left({b_{i}^{c}}\right),% \ldots,{\Gamma}_{i}^{M}\left({b_{i}^{c}}\right)}\right)$ $\displaystyle=\left({y_{1},y_{2},\ldots,y_{n},\ldots,y_{M}}\right)=b_{i}^{r}$ (23)

To define our mathematical model, we are developing the function ${\Gamma}_{i}\left(\cdot\right)$ as a series or as a binary polynomial [30]. Properly, ${\Gamma}_{i}\left(\cdot\right)$ is not a Boolean function, as its image set is not ${\mathbb{Z}}_{2}=\left\{{0,1}\right\}$ but $S_{N}$ . However, each component ${\Gamma}_{i}^{n}\left(\cdot\right)$ is a Boolean function. So hereafter we are working with function ${\Gamma}_{i}^{n}\left(\cdot\right)$ .

We consider the definition of positive cofactor ${\Gamma}_{x_{j}}^{n,i}$ and negative cofactor ${\Gamma}_{\overline{x_{j}}}^{n,i}$ done in the Reed–Muller expansion Eq. (24), being $x_{j}$ the $j$ -th bit in binary number $b_{i}^{c}$ . And then, the Boolean derivation of function ${\Gamma}_{i}^{n}\left(\cdot\right)$ may be expressed as a combination of these cofactors and the exclusive binary addition (XOR), $\oplus$ Eq. (25).

$\displaystyle\hskip-8.535827pt{\Gamma}_{x_{j}}^{n,i}\left({b_{i}^{c}}\right)={% \Gamma}_{i}^{n}\!\left({x_{1},\ldots,x_{j-1},1,x_{j+1},\ldots,x_{N}}\right)$ (24) $\displaystyle\hskip-8.535827pt{\Gamma}_{\overline{x_{j}}}^{n,i}\left({b_{i}^{c% }}\right)={\Gamma}_{i}^{n}\left({x_{1},\ldots,x_{j-1},0,x_{j+1},\ldots,x_{N}}\right)$ $\displaystyle\hskip-8.535827pt\frac{\partial{\Gamma}_{i}^{n}}{\partial x_{j}}% \left({b_{i}^{c}}\right)={\Gamma}_{x_{j}}^{n,i}\left({b_{i}^{c}}\right)\oplus{% \Gamma}_{\overline{x_{j}}}^{n,i}\left({b_{i}^{c}}\right)$ (25)

Equally, any derivation ${\Gamma}_{i}^{n\left(r\right)}$ of superior order $r$ may be calculated using an extended version of this Eq. (3.2). In this context we name as $r_{j}$ the $j$ -th bit in $r$ .

At this point and considering the Reed-Muller expansion Eq. (27), we can propose an expansion as a binary polynomial for function ${\Gamma}_{i}^{n}\left(\cdot\right)$ Eq. (3.2) around binary point $d_{i}^{n}$ ( $N$ bits) from $S_{N}$ . Through this expansion, any Boolean (binary) function can be represented with full precision. We name $w_{j}$ the $j$ -th bit in $d_{i}^{n}$ . In this case, as done for the analog model, unknown coefficients may be put together, so an explicit polynomial expression with unknown coefficients $\rho_{i}^{n,m}$ is obtained for each individual PUF item Eq. (3.2).

$\displaystyle{\Gamma}_{i}^{n\left(r\right)}\left({b_{i}^{c}}\right)=\frac{% \partial^{r}{\Gamma}_{i}^{n}}{\partial x_{1}^{r_{1}}\cdot\ldots\cdot\partial x% _{j}^{r_{j}}\cdot\ldots\cdot\partial x_{N}^{r_{N}}}\left({b_{i}^{c}}\right)$ $\displaystyle=\frac{\partial{\Gamma}_{i}^{n}}{\partial x_{1}^{r_{1}}}\left({% \frac{\partial^{r}{\Gamma}_{i}^{n}}{\partial x_{2}^{r_{2}}\cdot\ldots\cdot% \partial x_{j}^{r_{j}}\cdot\ldots\cdot\partial x_{N}^{r_{N}}}}\right)\left({b_% {i}^{c}}\right)$ $\displaystyle=\ldots=\frac{\partial{\Gamma}_{i}^{n}}{\partial x_{1}^{r_{1}}}% \ldots\left({\frac{\partial{\Gamma}_{i}^{n}}{\partial x_{j}^{r_{j}}}\ldots% \left({\frac{\partial{\Gamma}_{i}^{n}}{\partial x_{N}^{r_{N}}}}\right)}\right)% \left({b_{i}^{c}}\right)$ (26) $\displaystyle\textrm{where}\ \partial x_{j}^{r_{j}}=\left\{{{\begin{array}[]{% ll}1&\textrm{if}\ \ r_{j}=0\\ \partial x_{j}&\textrm{if}\ \ r_{j}=1\\ \end{array}}}\right.$

$\displaystyle{\Gamma}^{n}\left({b_{i}^{c}}\right)={\Gamma}_{\overline{x_{j}}}^% {n}\left({b_{i}^{c}}\right)\oplus x_{j}\cdot\frac{\partial{\Gamma}^{n}}{% \partial x_{j}}\left({b_{i}^{c}}\right)$ (27)

$\displaystyle{\Gamma}_{i}^{n}\left({b_{i}^{c}}\right)={\begin{array}[]{*{20}c}% {\textit{max}\left\{{S_{N}}\right\}}\hfill\\ \bigoplus\hfill\\ {r=0}\hfill\\ \end{array}}$ $\displaystyle\left[{{\Gamma}_{i}^{n\left(r\right)}\left({d_{i}^{n}}\right)% \cdot\mathop{\prod}\limits_{j=1}^{N}\left({x_{j}\bigoplus w_{j}}\right)^{r_{j}% }}\right]$ (28)

$\displaystyle{\Gamma}_{i}^{n}\left({b_{i}^{c}}\right)=$ $\displaystyle\bigoplus\limits_{\begin{subarray}{c}{r\in S_{2N}}\\ {r_{j}\neq r_{j+N}}\\ {\forall j\in\left[{1,N}\right]}\end{subarray}}\left[{\rho_{i}^{n,r}\mathop{% \prod}\limits_{j=1}^{N}\left({x_{j}}\right)^{r_{j}}\cdot\left({\overline{x_{j}% }}\right)^{r_{j+N}}}\right]$ $\displaystyle\rho_{i}^{n,r}\in{\mathbb{Z}}_{2}=\left\{{0,1}\right\}$ (29)

In this case we must remember that coefficients $\rho_{i}^{n,r}$ are binary Eq. (3.2). This fact simplifies their calculation process and makes the model computationally lighter, but it is associated with a more limited precision and more intense fluctuations in predictions while the model converges to a stable situation. Therefore, we cannot employ this model independently, but combined with other that improve their limitations.

Those coefficients will initially be estimated through the Kane Yee formula (see Section 3.5) and the transformation functions ${\cal D}$ and ${\cal E}$ .

3.3 Numerical model: Cubic spline interpolation

Boolean models (see Section 3.2) are lightweight and very easy to manipulate and compute. However, they may cause great transitory fluctuations in the PUF response predictions. As the number of bits representing the challenge, $N$ , reduces; the weight of each bit increases in the final result. While converging, the predictor-corrector model may change several parameters with each iteration, so in models with a limited precision fluctuations may be relevant.

To reduce this effect, in our proposal, the Boolean model is combined with a numerical model. In this approach, binary numbers representing the challenge $b_{i}^{c}$ and the PUF response $b_{i}^{r}$ are managed as natural, integer or real numbers (all approaches are considered in our model), not as vectors of bits.

A similar approach to those described in Sections 3.1 and 3.2 could be employed, but then the obtained model will suffer similar problems to those described ones. The aim of this numerical model is to compensate these inefficiencies. To do that we are using an interpolation technique based on cubic splines, which is characterized by its smooth behavior.

A spline is a differentiable curve $P_{i}\left(v\right)$ defined in different parts by polynomials $P_{i}^{m}\left(v\right)$ (30). In our case, we are using cubic polynomials (31). Points $c_{m}$ are challenges for which the response $r_{m}$ is known (through an initial configuration and/or the correction process). As the challenges in this model are understood as numbers (natural, integer or real), they can be ordered and define the interval $\left[{c_{0},c_{Q}}\right]$ . Where $Q$ is the number of known challenges, and coefficients $\gamma_{m}^{1},\gamma_{m}^{2},\gamma_{m}^{3}$ and $\gamma_{m}^{4}$ are unknown.

$\displaystyle P_{i}\left(v\right)=\left\{\begin{array}[]{ll}P_{i}^{0}\left(v% \right)&\text{if}\ \ v\in[{c_{0},c_{1}})\\ \ldots\\ P_{i}^{m}\left(v\right)&\text{if}\ \ v\in[{c_{m-1},c_{m}})\\ \ldots\\ P_{i}^{Q-1}\left(v\right)&\text{if}\ \ v\in[c_{Q-1},c_{Q})\\ \end{array}\right.$ (30)

$\displaystyle P_{i}^{m}\left(v\right)=\gamma_{m}^{1}\cdot v^{3}+\gamma_{m}^{2}% \cdot v^{2}+\gamma_{m}^{3}\cdot v+\gamma_{m}^{4}$ (31)

But this curve $P_{i}\left(v\right)$ must be continuous and derivable for all order or derivation (see Section 3.1), so an analytic expression of each polynomial $P_{i}^{m}\left(v\right)$ may be deducted after applying those restrictions Eq. (32) and grouping unknown coefficients to reduce the problem’s complexity. Coefficients $\alpha_{i}^{m}$ are unknown.

$\displaystyle P_{i}^{m}\left(v\right)=\frac{\alpha_{i}^{m}}{6h_{i}^{m}}\left({% c_{m+1}-v}\right)^{3}+\frac{\alpha_{i}^{m+1}}{6h_{i}^{m}}\left({v-c_{m}}\right% )^{3}+\left({\frac{r_{m+1}}{h_{i}^{m}}+\frac{\alpha_{i}^{m+1}h_{i}^{m}}{6}}% \right)\cdot\left({v-c_{m}}\right)+\left({\frac{r_{m}}{h_{i}^{m}}-\frac{\alpha% _{i}^{m}h_{i}^{m}}{6}}\right)\cdot\left({c_{m+1}-v}\right)\textit{being}\ h_{i% }^{m}=c_{m+1}-c_{m}m\in\left[{0,Q}\right]$ (32)

In this case, all these unknown variables may be calculated using the same restrictions described above (continuity and derivable). They define a system with $Q+1$ unknown variables but $Q-1$ Eq. (3.3). In that way, for two unknown coefficients $\alpha_{j}$ arbitrary values must be provided. In our model those coefficients will be initially estimated through the Kane Yee formula (see Section 3.5) and transformation functions ${\cal D}$ and ${\cal E}$ .

$\displaystyle h_{i}^{m-1}\alpha_{i}^{m-1}+2\left({h_{i}^{m}+h_{i}^{m+1}}\right% )\alpha_{i}^{m}+h_{i}^{m}\alpha_{i}^{m+1}$ $\displaystyle=\frac{6}{h_{i}^{m-1}}\left({r_{m+1}-r_{m}}\right)-\frac{6}{h_{i}% ^{m-1}}\left({r_{m}-r_{m-1}}\right)$ $\displaystyle m\in\left[{1,Q-1}\right]$ (33)

On the other hand, in order to correct and improve the model, the value of $Q$ parameter is increased each time a new actual PUF response $r_{m}$ for a given challenge $c_{m}$ is received. The entire problem must be solved again when that happens. Algorithm 1 shows the calculation process.

Algorithm 1: Spline solving algorithm
Input Known $Q$ challenge-response pairs $\left({c_{m},r_{m}}\right)$
Parameters $\alpha_{i}^{0}$ , $\alpha_{i}^{Q}$
Output Unknown parameters $\alpha_{m}$
for $integerm\in\left[{0,Q-1}\right]$ do
$h_{i}^{m}\leftarrow c_{m+1}-c_{m}$
$u_{m}\leftarrow\frac{6}{h_{i}^{m-1}}\left({r_{m+1}-r_{m}}\right)$
end for
$g_{1}\leftarrow 2\left({h_{i}^{0}+h_{i}^{1}}\right)$
$v_{1}\leftarrow 2\left({u_{1}-u_{0}}\right)$
for $\textit{integer}\ m\in\left[{2,Q-1}\right]$ do
$u_{m}\leftarrow 2\left({h_{i}^{m}+h_{i}^{m-1}}\right)-\frac{\left({h_{i}^{m-1}% }\right)^{2}}{g_{m-1}}$
$v_{m}\leftarrow\left({r_{m}-r_{m-1}}\right)-\frac{h_{i}^{m-1}v_{m-1}}{g_{m-1}}$
end for
for $\textit{integer}\ m\in\left[{Q-1,1}\right]$ do
$\alpha_{i}^{m}\leftarrow\frac{v_{m}-h_{i}^{m}\alpha_{i}^{m+1}}{g_{m}}$
end for

When the spline $P_{i}\left(v\right)$ is fully defined, it is simple to generate a new prediction $r_{i}$ for challenge $c_{i}$ just applying the function Eq. (34).

$\displaystyle r_{\textit{pred}}=P_{i}\left({c_{\textit{pred}}}\right)$ (34)

3.4 Prediction algorithm

At this point, when a new prediction must be made, three different responses are obtained: the one from the analog model $r_{i}^{\textit{an}}$ , the one from the Boolean model $r_{i}^{\textit{bol}}$ , and the one from the numerical model $r_{i}^{\textit{num}}$ . At this point, all these responses are binary numbers, which may be understood as sequences of bits, natural, integer or real numbers as convenience.

For each PUF item, a set of three different models (and responses) is generated; but as all three models refer to the same device, they are not independent but moderately correlated. This correlation, besides, increases, as the prediction-correction mechanism refines the models and all of them get closer to the real PUF item’s behavior. Therefore, the combined result of these three responses does not vary randomly but converges to a final value where models can compensate and mitigate the inaccuracies of each other.

Thus, before generating the final and global response $r_{i}^{\textit{glo}}$ , all three partial responses must be combined. In this work we are using an arithmetic weighted average Eq. (35).

$\displaystyle r_{i}^{\textit{{glo}}}=\lambda_{i}^{\textit{an}}\cdot r_{i}^{% \textit{an}}+\lambda_{i}^{\textit{bol}}\cdot r_{i}^{\textit{bol}}+\lambda_{i}^% {\textit{num}}\cdot r_{i}^{\textit{num}}$ (35)

Weights $\lambda_{i}^{\textit{an}}$ , $\lambda_{i}^{\textit{bol}}$ and $\lambda_{i}^{\textit{num}}$ are specific for each PUF and may vary along time. To calculate those weights, we are considering three basic quality parameters of any strong PUF: reliability, randomness and uniqueness. Those parameters, however, do not have a clear mathematical definition in the PUF context. Thus, in this paper, we are considering three probabilistic parameters, that are directly related to how reliable, random and unique is a PUF, they do have a clear mathematical definition: PUF noise $P_{\textit{noise}}$ , inter-challenge distance $P_{\textit{inter}}$ and PUF stability $P_{\textit{stb}}$ .

PUF noise $P_{\textit{noise}}$ Eq. (36) represents the precision of the model. It is the probability of a predicted response $r_{i}$ to be correct, given a challenge $c_{i}$ . This probability is calculated as the ratio between the number of correctly predicted responses $co_{i}$ over the total number of predictions $to_{i}$ .

$\displaystyle P_{\textit{noise}}=\frac{co_{i}}{to_{i}}$ (36)

The inter-challenge distance $P_{\textit{inter}}$ Eq. (37) represents the overfitting of models, so they converge to an equilibrium point, numerically correct, but where the differences of between different challenges and responses are not well represented. It is the probability of different challenges $c_{i},c_{j}$ to generate different responses $r_{i},r_{j}$ . This probability is obtained by generating $L$ different challenge-response pairs using the corresponding model and analyzing the distance between responses and challenges. In this case, we are using the Hamming distance $d_{H}\left({\cdot,\cdot}\right)$ . In this definition, distance is equal to the minimum number of binary substitutions required to make equal both responses or challenges (i.e., the number of places or bits where both vectors are different).

Besides, $M$ is the number of bits employed to represent the response, and $N$ the number of bits to represent PUF challenges (see Section 3.1).

$\displaystyle P_{\textit{inter}}=$ (37) $\displaystyle\ \ \frac{2^{-N+1}}{L\left({L-1}\right)}\mathop{\sum}\limits_{j=1% }^{L-1}\mathop{\sum}\limits_{i=j+1}^{L}\left[{2^{\frac{N}{d_{H}\left({C_{i},C_% {j}}\right)}}\cdot\frac{d_{H}\left({r_{i},r_{j}}\right)}{M}}\right]$

In general, the probability of two independent responses $r_{i},r_{j}$ to be different may be calculated through the Hamming distance $d_{H}\left({r_{i},r_{j}}\right)$ ; but in real PUF, the key problem is one unique item generating very different responses from very close challenges (but generating the same response every time is excited with the same challenge), while two different PUF items must generate very different responses from the same challenge. Factor $2^{\frac{N}{d_{H}\left({C_{i},C_{j}}\right)}}$ represents this fact, so models generating different responses for similar challenges are much probable to preserve the inter-challenge distance. The remaining constants are employed to normalize the probability and ensure it takes values in the interval $\left[{0,1}\right]$ .

Finally, the PUF stability $P_{\textit{stb}}$ Eq. (38) represents the ability of each model to apply corrections to wrong predictions but keeping correct the predictions done without error. As the models are based on numeric manipulations, small changes are allowed; but models showing great fluctuations and not converging uniformly are penalized because of their worse behavior. It is the probability of a correctly predicted response $r_{i}$ to keep the same value (or a very close one) as the model evolves. This probability is calculated by storing $L$ predicted responses $r_{j}$ for challenge $c_{j}$ , obtained after the correct prediction $r_{i}$ was calculated in the first place. To analyze how different the predicted responses are, we are using the Hamming distance $d_{H}\left({\cdot,\cdot}\right)$ . The remaining constants are just included in the expression to ensure the probability takes values in the interval $\left[{0,1}\right]$ .

$\displaystyle P_{\textit{stb}}=1-\frac{1}{L}\mathop{\sum}\limits_{j=1}^{L}% \frac{d_{H}\left({r_{j},r_{i}}\right)}{M}$ (38)

Now, in order to calculate each weight $\lambda_{i}^{\textit{an}}$ , $\lambda_{i}^{\textit{bol}}$ and $\lambda_{i}^{\textit{num}}$ from probabilities $P_{\textit{noise}}$ , $P_{\textit{inter}}$ and $P_{\textit{stb}}$ a multivariable mapping function is required.

This function must be monotonically non-decreasing in the interval $\left[{0,1}\right]$ . Besides, it must take values in the same interval to be coherent with the definition of weight. On the other hand, no weight may be null as all models must be considered. Therefore, the abscissa axis is an asymptote. Finally, the function must preserve the magnitude of the predicted responses, so the properties of the model are not modified Eq. (39).

$\displaystyle\lambda_{i}^{\textit{an}}+\lambda_{i}^{\textit{bol}}+\lambda_{i}^% {\textit{num}}=1$ (39)

With these requirements, we propose an exponential function as the mapping function Eq. (40). In this function we are considering three real parameters $\tau_{\textit{noise}}$ , $\tau_{\textit{inter}}$ and $\tau_{\textit{stb}}$ representing the increasing speed of mapping function with respect to probabilities $P_{\textit{noise}}$ , $P_{\textit{inter}}$ and $P_{\textit{stb}}$ . These parameters are employed to control de sensitivity of weights. As these parameters are reduced, the variations of the probabilities are mapped into greater variations of the mapping function and weights.

$\displaystyle\sigma_{i}^{s}=$ (40) $\displaystyle\exp\left\{{\frac{\left({P_{\textit{noise}}^{s}-1}\right)}{\tau_{% \textit{noise}}^{s}}+\frac{\left({P_{\textit{inter}}^{s}-1}\right)}{\tau_{% \textit{inter}}^{s}}+\frac{\left({P_{\textit{stb}}^{s}-1}\right)}{\tau_{% \textit{stb}}^{s}}}\right\}$ $\displaystyle\lambda_{i}^{s}=\frac{\sigma_{i}^{s}}{\mathop{\sum}\nolimits_{s}% \sigma_{i}^{s}}$ (41) $\displaystyle s=\left\{{\textit{an, bol, num}}\right\}$ (42)

3.5 Correction process

Figure 4 shows a flow chart for the correction process in the proposed predictor-corrector model.

This process starts with an initial configuration stage. At this stage, probably carried out before node deployment, a real and physical challenge-response pair $\left({c_{j},r_{j}}\right)$ is captured from the node (PUF). This is the only pair that is preconfigured and must be stored by the gateway in our approach. This initial pair is required to trigger the system operation and enable an encrypted and secure mMTC from the very beginning. Immediately, response $r_{j}$ is set as main secret key (probably through a key generation algorithm or a similar instrument) in the node and the gateway.

After that and before any transmission is allowed, the predictor model must be defined. To do that, parameters $\xi_{i}^{m}$ , $\eta_{i}^{m}$ and $\beta_{i}^{m}$ in the analog model; parameters $\rho_{i}^{n,r}$ in the Boolean model, and parameters $\alpha_{i}^{m}$ in the numerical model must be obtained. The approach is similar for these three models and for all parameters. A set of $Q$ challenge-response pairs $\left({c_{i},r_{i}}\right)$ must be obtained. Using these pairs, parameters $\alpha_{i}^{m}$ may be obtained using Algorithm 1. Parameters $\xi_{i}^{m}$ and $\eta_{i}^{m}$ may be obtained by replacing (18) all $Q$ challenge-response pairs $\left({c_{i},r_{i}}\right)$ and applying an optimization algorithm (in this case, Mean Square Error) to find the best set of parameters fulfilling the $Q$ equations (one for each pair). The same approach is applied to parameters $\beta_{i}^{m}$ Eq. (19) and parameters $\rho_{i}^{n,r}$ Eq. (3.2).

These initial $Q$ challenge-response pairs do not have to be stored or preconfigured. In our proposal, they are obtained by solving the underlying electromagnetic problem controlling the PUF behavior. This can be done through the Maxwell’s equations [54] where the initial conditions are considered to be null, as PUF are not excited in any way until a new challenge is applied. On the other hand, the boundary conditions are selected according to the PUF hardware structure.

However, the complexity of the hardware structure in most PUF make quite impossible to solve this problem in an analytical manner. Therefore, numerical methods are needed. In our predictor-corrector model we are using the Kane Yee formula.

The Kane Yee formula [31] is a numerical method for initial boundary value problems based on finite differences in the time domain (FDTD). It was specifically designed for solving the Maxwell equations, by replacing spatial and temporal derivatives using central finite differences.

Finally, as said before, to calculate parameters $\xi_{i}^{m}$ , $\eta_{i}^{m}$ and $\beta_{i}^{m}$ in the analog model and parameters $\rho_{i}^{n,r}$ in the Boolean model we are using an optimization algorithm. In this case we are using the Mean Square Error (MSE) algorithm.

Regarding functions $f_{i}^{a}$ and $f_{i}^{f}$ , given a set of challenge-response pairs $\left({c_{i},r_{i}}\right)$ , we are looking for the best set of parameters $\xi_{i}^{m}$ , $\eta_{i}^{m}$ and $\beta_{i}^{m}$ , so when introduced in the model, the predicted responses $r_{i}^{\textit{an}}$ they generate for challenges $c_{i}$ are, on average, the best approximation (with minimum mean error) to original responses $r_{i}$ Eq. (3.5). An equivalent process may be employed with function ${\Gamma}\left(\cdot\right)$ and coefficients $\rho_{i}^{n,r}$ Eq. (44).

$\displaystyle\left\{{\left\{{\xi_{i}^{m}}\right\},\left\{{\eta_{i}^{m}}\right% \},\left\{{\beta_{i}^{m}}\right\}}\right\}=$ $\displaystyle\quad\textit{argmin}\ \left\{{\frac{\mathop{\sum}\nolimits_{i=1}^% {Q}\left({r_{i}^{an}-r_{i}}\right)^{2}}{Q}}\right\}$ (43) $\displaystyle\left\{{\rho_{i}^{n,r}}\right\}=\textit{argmin}\ \left\{{\frac{% \mathop{\sum}\nolimits_{i=1}^{Q}\left({r_{i}^{\textit{bol}}-r_{i}}\right)^{2}}% {Q}}\right\}$ (44)

This approach is very scalable. Different strategies could be employed to solve this optimization problem Eqs (3.5–44), but even classic methods can achieve a result in a few tenths of a second [55]. In that way, only one gateway could manage up to ten thousand devices with a key renovation period $T$ around fifty (50) minutes (enough for most Industry 4.0 and mMTC applications). However, in practical scenarios, gateways manage a much lower number of devices: because of limitations in the communication protocols, the packet delivery ratio goes below 50% for any gateway managing more than five thousand devices [56]. In fact, in very dense scenarios (up to one million of devices per square kilometer) several gateways would be deployed. In conclusion, our approach does not hinder mMTC scalability, and it even enables a 100% increasing in the number of devices controlled by each single gateway.

Anyway, MSE (minimization) is a generic algorithm, but the proposed predictor-corrector scheme allows the integration of other optimization solutions, adapted to the characteristic of specific Industry 4.0 scenarios. For example, the Harmony search algorithm [53] might be implemented in scenarios where the pitch adjustment and/or the harmony memory consideration rules make sense.

Once all these coefficients are obtained and the models are set up for the predictions, the system may initiate mMTC. To maintain a high security level, secret keys are renovated each $T$ second. Different values for $T$ may be proposed, according to how critical the Industry 4.0 scenario under study is. During this time, correction algorithm stays in a “waiting point” (the “open for transmissions” state), which is also the natural end of the algorithm’s flow for gateways. When the timer expires, the gateways generate a new challenge $c_{j}$ and the corresponding predicted response $r_{j}^{\textit{glo}}$ . Then, they send the challenge to the sensor nodes, so they can recalculate the secret key through the actual response to the challenge $r_{j}$ generated by PUF.

Figure 4.

Correction algorithm.

Figure 5.

Proposed data formats for the test message and the correction message.

Before applying any change, the sensor node waits for a gateway test message. The “open for transmissions” state is the waiting point (and the end of the natural flow of the proposed correction algorithm for user devices too). See Fig. 4. This message is encrypted using the predicted response $r_{j}^{glo}$ . This test message is a standard control message (depending on the network protocol, if IP protocol is available, an ICMP message is employed), but with a new header where a code to indicate the type of message being exchanged is included. Figure 5 represents the data format for this message.

If the control message is decrypted by the node, the predicted response is correct, and the secret key is updated. If control message is unintelligible, thus means that the predictor model has made a mistake and it must be corrected. A correction message is sent to the gateway. This message contains the correct response $r_{j}$ but encrypted with the previous key. So, all communications are safe. Other information could be added as needed in the Industry 4.0 scenario under study. Figure 5 shows the data format for the control message.

After receiving this message, the gateway recalculates all parameters in the analog, Boolean, and numerical models, by refining the calculation of models’ parameters Eqs (3.5–44) and the weights $\sigma_{i}^{s}$ Eq. (40) for the $i$ -th node. The Mean Square Error algorithm is employed as described above, together with the spline solving algorithm (Algorithm 1). In this case, however, only $Q-1$ challenge-response $\left({c_{j},r_{j}}\right)$ pairs are estimated using the Kane Yee formula, as the corrected pair by the node is added too to the calculation process, so coefficients are improved with new information. Also, this new challenge-response pair is added to the boundary and initial conditions in the Kane Yee formula, so estimated pairs $\left({c_{j},r_{j}}\right)$ are also closer to the actual hardware implementation.

The secret key is finally updated to the new PUF response and mMTC are available with the new security configuration.

4. Experimental validation

In order to evaluate the proposed technologies for lightweight mMTC, an experimental validation was carried out. The experimental validation was mainly based on simulation scenarios. Using simulation tools, it is possible to evaluate very precisely the performance of the proposed predictor-corrector scheme, guaranteeing exogenous phenomena, such as noise, delays, or transmission errors, do not affect the results. Besides, using simulations, it is possible to run and evaluate all algorithms in their original implementation, without any modification to extract measurements, communicate to the data repository, etc. Simulation tools can monitor the entire scenario behavior unobtrusively.

In order to guarantee that simulation results represent accurately the performance of a real deployment, hardware devices and sensor data are not computational models but replay processes fed by repositories with real physical information coming from real deployments (more details are provided below).

The proposed validation consisted of two different phases: The first one is focused on a numerical analysis of the proposed solution, while the second one was based on a performance analysis.

All simulations were supported by a scenario that describes an Industry 4.0 application, where mMTC must be used. Three different sensor nodes were considered in the simulation: temperature sensors, quality air sensors and humidity sensors. The scenario represented a large infrastructure, such as a production center with several production lines, ovens, etc. The simulated scenario had an area of 4000 square meters, and different device densities were considered in different experiments (see Section 4.2). Twenty gateways were considered in this scenario, each one managing all devices deployed in a region of 200 square meters. Regions covered by different gateways did not overlap.

All sensor nodes and gateways were based on a model that represents a real device. The sensor node model represented an ESP-32 device with different peripherals. Some nodes were provided with the LM35 temperature sensor. This is an analog sensor, with high precision, to measure temperature using the Celsius scale. Other nodes were provided with the DHT-11 humidity sensor. This is a digital sensor that measures the amount (percentage) of vapor in the air and uses an eight-bit word to codify the result. Finally, some other nodes were provided with the CCS811 air quality sensor. This is a digital sensor with an I2C interface, which monitors the Volatile Organic Compounds (VOC) and the equivalent carbon dioxide reading (eCO2). On the other hand, the model for gateways is based on a Linux server with 50 GB RAM and 10TB of long-term storage capacity.

In order to ensure that the simulation scenario represent accurately a real physical deployment, sensor data are not generated by a computational model, but they are “reproduced” by a replay process using real information coming from a real Industry 4.0 deployment implementing the same sensors and technologies [59].

Regarding PUF, three different implementations, based on different technologies, were randomly associated to sensor nodes. Those technologies were: a multiplexer-based arbiter PUF [32], the butterfly PUF based on D flipflops [33] and a magnetic PUF based on exchange bias effect [10]. In this case, to preserve the random, reliable and unique behavior of original PUF, nodes were provided with a catalogue of challenge-response pairs taken from real measurements over physical PUF implementations.

To perform the experiments, the simulation scenario was implemented and executed using MATLAB 2017a software. All simulations were performed using a Linux architecture (Ubuntu 20.04 LTS) with the following hardware characteristics: Dell R540 Rack 2U, 96 GB RAM, two processors Intel Xeon Silver 4114 2.2G, HD 2TB SATA 7,2K rpm. All simulations represented an operation time of seventy-two (72) hours. Each simulation was repeated twelve times, and final results were obtained as the average of all partial results. Deviations or mean errors are not discussed unless they go above the standard limits for this kind of experiments and their value, then, turns relevant. Typically, deviations up to 10% around the average value are not significant and can be associated to common experimental errors.

Some variables and parameters in the proposed model were employed as independent variables during the experiments, but some other had a fixed value for the entire experimental phase. Table 2 shows the configuration parameters and their values.

The proposed values were obtained through a heuristic process, where different parameter sets were tested until the entire predictor-corrector scheme achieved its optimum operation point (although we cannot guarantee it in its global optimum configuration). All these parameters are not correlated and can be changed independently.

Variables not included in Table 2 will be employed as control parameter in the experiments.

Table 2
Configuration parameters

Parameter	Value	Parameter	Value
$M$	10 bits	$B$	1 MHz
$N$	10 bits	$T$	0.1 hour
$\tau_{\textit{noise}}^{s}\forall s$	5	$T_{\textit{ex}}$	10 seconds
$\tau_{\textit{inter}}^{s}\forall s$	5	$K$	12
$\tau_{\textit{stb}}^{s}\forall s$	5	$L$	15

4.1 Numerical analysis: Methodology

In order to study the behavior of the proposed solution from a numerical point of view, the quality of predicted responses was monitored, as well as the quality of the correction process. To do that, we monitored four variables: non-corrected correct predictions, corrected correct predictions, non-corrected wrong predictions and corrected wrong predictions. In addition, for all these groups and predicted responses, the normalized Hamming distance between the real response and the predicted one is also measured.

These variables were monitored both as a global parameter for the entire simulation scenario, but partial studies for different types of PUF were also done. The experiment was repeated for different values of $Q$ parameter.

All those variables were calculated offline, as the proposed predictor-corrector algorithm does not have information about which predictions are truly correct or not while running. Even less about corrected correct predictions and non-corrected wrong predictions. For the correction algorithm (see Section 3.5) the only knowledge available is which predicted responses can actually decrypt the correction message coming from the $i$ -th node (so they are considered correct), and which ones cannot (so they are considered wrong predictions and the correction and refining mechanism is triggered). However, for the experimental validations, as challenge-response pairs in the simulation come from a repository with real measurements, it is computationally simple to identify non-corrected correct predictions, corrected correct predictions, non-corrected wrong predictions and corrected wrong predictions offline, just comparing information in the repository to information generated during the simulation.

Using those variables, the following statistics were calculated to validate the behavior of the proposed corrector scheme: sensitivity, specificity, precision, and accuracy. On the other hand, in order to evaluate the behavior of the predictor model, we employed the mean normalized Hamming distance between real and predicted responses and the correct prediction rate.

Ideally, these results should be compared to other previous models (SVM models, for example), but that was not feasible in this case. Mainly, because machine learning models require large datasets to be trained, and we lacked that resource for the PUF items and technologies under consideration. Other datasets could be employed, but then two critical problems emerge. On the one hand, comparison would not be scientifically relevant. And, on the other hand, previous works typically employ datasets describing only one PUF technology; while in our approach we would like to evaluate and highlight its flexibility to operate with several PUF technologies at the same time. We faced a similar situation with physical models, as we could not determine the value of some very relevant parameters in those models such as the voltage threshold of transistors.

Finally, the probability density function (PDF) of PUF responses is also calculated for all three PUF technologies and the proposed predictor-corrector model. PDF is obtained calculating the normalized histogram for all the challenge-response pairs generated during the simulation. Overlapping and comparing the PDF for the real physical PUF, and the PDF for the proposed predictor-corrector algorithm, we can identify biases or patterns to be removed from the proposed models in order to increase and improve their accuracy and performance.

4.2 Performance analysis: Methodology

The main characteristic of the proposed solution is its computationally lightweight implementation, and, besides, its scalability, which must fit the requirements of mMTC.

Thus, in this second phase, we evaluate two different variables: the memory usage of the proposed predictor-corrector model in the gateways and the computation delay in the gateways. We are also analyzing their evolution with the number of devices in the scenario in order to study the scalability. The experiment was repeated for different values of $Q$ parameter.

On the other hand, we will compare the results obtained with the processing delay and memory usage of the traditional approach, based on key-value databases were all the challenge-response pairs are stored.

Finally, we are proving some results from real devices. The proposed corrector algorithm was implemented on a hardware ESP-32 device. The memory usage and the processing time required by the proposed scheme are monitored to ensure it matches the reduced capacities of sensor nodes.

5. Results and discussions

In this section we present and discuss the results from the experiments described in Section 4. Section 5.1 introduces and analyzes the main results, while Section 5.2 discusses the limitations of the proposed technology, considering the previous results.

Figure 6.

Correct prediction rate.

Figure 7.

Hamming distance.

5.1 Experimental results

First, we discuss the results from the numerical analysis. Figure 6 shows the correct prediction rate for the different PUF technologies and the global results for the entire scenario.

As can be seen, the correct prediction rate increases as the value for $Q$ parameter goes up. This behavior is coherent. In general, as the order of a model increases, its correctness also improves until the intrinsic maximum for the model is reached. For the proposed model, as can be seen in Fig. 6, this value is reached, approximately, for $Q=$ 35.

Figure 8.

Sensitivity.

The long-term correct prediction rate for the proposed model is around 78%. This includes all the historical predictions, but it is expected the correct prediction rate to be lower at the first operation moments and, as time passes, achieve this value. However, this rate is not the same for all PUF technologies. Magnetic PUF shows a correct prediction rate very high, around 94% at long-term. That’s because Maxwell’s laws with very simple boundary conditions (materials are usually parallelepiped) are enough to model this type of PUF technologies in a very exact manner. While the Butterfly PUF model, as this PUF is an unstable digital circuit based on feedback oscillating loops, is not able to achieve that rate, and the long-term correct prediction rate is around 63%. Finally, the MUX-based arbitrator PUF is a balance. Their behavior is stable, but the initial and boundary conditions are complex. In general, the model for MUX-based PUF shows a correct prediction rate (77%) very close to the aggregated global results for all PUF technologies. In conclusion, the proposed predictor model has a good behavior. Most of the predicted responses are correct, so the correction process must be triggered only in approximately 22% of cases. This will help our approach to be lightweight, more scalable, and much more efficient than traditional mechanisms for secure mMTC.

Similar results and conclusions are obtained if we go through the Hamming distance (see Fig. 7). As can be seen, the distance tends to reduce as the value for $Q$ parameter reduces. This is consistent with an increasing correct prediction rate. In this case, magnetic PUF are the ones showing a faster decreasing and the minimum long-term distance (we can consider it zero). Mux-based PUF have slightly worse behavior, and around 5% of bits are wrong in predicted responses. This value increases for Butterfly PUF up to 15%. Globally, for all PUF, the expected Hamming distance is 0.07, i.e., on average 7% of bits in predicted responses are wrong. This value is below the traditional 10% error that is considered acceptable in most scenarios.

Figure 8 shows the sensitivity of the proposed predictor model. As sensitivity is greater than 80%, it means that, in the worst case, only 20% of correct predictions are considered wrong predictions. As communications in Industry 4.0 are envisioned to be ultrareliable, this problem is not quite relevant, but it is still present. Actually, if we focus on a more realistic situation ( $Q=$ 35), the sensitivity is above 92% for all PUF technologies.

Contrary to the correct prediction rate, in this case, MUX-based PUF is the technology showing the lowest sensitivity. This may be explained by the nature of these PUF. As it is based on delays, responses are not fully reliable for long periods: as materials degrade, the temporal response changes. Therefore, although the predicted response may be correct, the sensor node may not be able to decrypt the test message using the physical PUF response, triggering the correction process. This effect, in any case, is present in any infrastructure based on arbiter PUF and must not be identified as a weakness for the proposed solution.

Figure 9.

Specificity.

On the other hand, in Fig. 9 we display the results obtained for specificity. One relevant observation is the constant value we obtain in specificity for Butterfly PUF. In this case, the specificity is around 55%, regardless of the value for $Q$ parameter. The reason for this observation is the limited distance between responses in butterfly PUF. Although the predicted response may be wrong, the difference is so small that the encrypted test message may be almost fully correct, so the correction process is not triggered. This phenomenon is does not depend on the $Q$ parameter.

Regarding magnetic and MUX-based PUF, specificity does evolve with changes in $Q$ parameter. Values of around 96% and 90% (respectively) are reached for a good enough model ( $Q=$ 35). These values are very high, and non-corrected wrong predictions may be associated to common numerical errors.

Figure 10.

Precision.

Figure 10 shows the precision for the proposed predictor-corrector mode. As can be seen, magnetic and MUX-based PUF have very high precision values (up to 98% for $Q=$ 35 , or even 99,5% for $Q=$ 50), and common numerical errors may explain why 100% is not fully achieved. On the other hand, the precision for butterfly PUF is 20% lower, and only reaches 83% for $Q=$ 35. This observation is consistent with the analysis done in Fig. 6, where we already explained that the unstable behavior of butterfly PUF reduces the correct prediction rate, decreasing the model’s precision.

As the precision for the global scenario includes all technologies previously analyzed, the precision value is a balance and reaches 95% for a good enough model ( $Q=$ 35).

All reported values are very high. The predictor model is precise enough to allow for a sparse execution of the correction process. So, the lightweight character of the proposed schemes and its scheme is preserved in mMTC scenarios.

Figure 11.

Accuracy.

Figure 12.

Probability density function.

Finally, Fig. 11 shows the accuracy of the model. This represents the percentage of non-corrected correct predictions and the corrected wrong predictions (i.e., percentage of times the corrector model did a correct classification). The global value for all PUF is close to 90% for $Q=$ 35, although this is the balance between magnetic PUF (97% for $Q=$ 35) and butterfly PUF (82% for $Q=$ 35). Differences and particular values are explained by the combination of all previously discussed phenomena (corrected correct predictions and non-corrected wrong predictions).

In these hybrid scenarios, where different PUF technologies coexist, the proposed predictor-corrector model (and its accuracy) could be enhanced using swarm intelligence techniques [50]. Solving the optimization problem (3.5–44) through this technique (instead the initially proposed MSE in this paper), new and optimum values for the models’ parameters could be found.

Finally, Fig. 12 shows the PDF for all three PUF technologies and the proposed predictor-corrector scheme. For this experiment, we are considering a configuration were $Q=$ 50, because it is the situation for which the proposed technology shows the best behavior.

As can be seen, butterfly PUF tend to generate responses with a large majority of bits one. This phenomenon has been reported in the literature before [60], and one possible explanation is because the “high state” is more stable than the “low state” for most flip-flops. However, the proposed predictor-corrector method is not designed to generate similar or identical responses within a reduced interval to different challenges, but just the opposite (different challenges must generate different responses). Then, there is a very low but still non-zero probability associated to responses in other regions, whose aggregation is close to 40% of all generated responses.

On the other hand, magnetic PUF has a totally different behavior. In this case, responses distribute homogenously within the possible variation range (although probability seems to be slightly higher in the central region). And the proposed model is able to replicate this behavior accurately, as unique challenge-response pairs are more easily modeled through algebraic frameworks.

Finally, MUX-based PUF show a PDF function somehow in the middle of the two previous PUF technologies. Although this third PDF is not as narrow as the Butterfly PUF’s one, probability is still mostly concentrated around the central region with a small deviation to the higher values. The proposed predictor-corrector scheme can replicate this behavior, but its PDF is still wider, and around 10% of the aggregated probability is spread in regions where the real physical PUF do not generate responses.

The second phase of the experimental validation consisted of a performance evaluation. Figure 13 shows the memory usage (normalized). As the absolute memory consumption depends on the selected operating system, its version, the employed hardware and, in our simulation scenario, on the MATLAB software (all of them external causes to the scenario under study), we are only studying the memory consumption scalability. Therefore, the results of memory usage are normalized to study the tendencies.

Figure 13.

Memory usage.

In this second experimental phase, results from the proposed predictor-corrector scheme are compared to the traditional approach (see Section 4.2). This consists of an architecture where challenge-response pairs are not calculated through a computational model but stored in an exhaustive key-value database describing the entire PUF behavior for every single PUF item.

For these analyses, three different values for $Q$ parameter are considered. The first one ( $Q=$ 20) represents a situation where accuracy (as well as the other numerical parameters) achieves a good value (between 70% and 90%, depending on the PUF technology and the indicator under consideration), but it can still increase significantly. The second one ( $Q=$ 35) represents a situation where accuracy and the other indicators stabilized (with global values above 90%). They increase much slower, but they cannot be considered high precision. Finally, the third value ( $Q=$ 50) represents a high precision performance where indicators (accuracy, sensitivity, etc.) take global values above 95% and near 100% in some circumstances.

Figure 14.

Processing delay.

As can be seen, memory consumption evolves linearly with the number of devices in all cases (Fig. 13 is a logarithm on the horizontal axis). However, consumption evolution varies among the approaches described. All approaches have a similar memory consumption for low device densities (10000 devices per square kilometer). However, when the device density goes up from 100000 devices/km ${}^{2}$ to 1000000 devices/km ${}^{2}$ our approach shows a very good behavior. The memory usage of our proposed predictor model, when $Q=$ 20, is just 30% of the traditional approach. It is true that this configuration is not the most accurate, sensitive, etc. as we discussed before; but, even for a better configuration such as $Q=$ 35, the memory usage is only 60% of the required memory consumption in the traditional approach. Other configurations ( $Q=$ 50), where much more parameters and operations must be performed to operate with the proposed predictor-corrector model are closer the traditional PUF and mMTC management strategy. Nevertheless, the memory usage is lower (90% of the one required in a traditional scheme).

Table 3

Resource consumption in sensor nodes: results

Use of RAM	Use of program space	Mathematical operations	Processing delay
2%	3%	452	987 $\mu$ s

Additionally, thanks to these improvement in memory scalability, Industry 4.0 systems will be prepared for future upcoming scenarios without needing an enlargement of the network infrastructure (which may be costly, may increase the network management tasks, …). For example, 6G technologies are envisioned to operate in massive scenarios with up to ten million devices per square kilometers [34] (ten times more than the current 5G networks). Systems implementing our approach could increase their capacity and integrate such enormous number of hardware nodes, without requiring additional gateways or processing (or edge) servers.

Figure 14 shows the results for the processing delay. As can be seen, there is a huge difference. Although the proposed solution presents a linear evolution, the traditional approach shows an exponential increase (Fig. 14 is logarithmic on the vertical axis). This is caused by the double search required in the traditional PUF and mMTC management strategy: first, the gateway must look for table or database where the challenge-response pairs for a specific PUF are stored, later, the gateway must look for the specific pair to be employed.

As can be seen in Fig. 13, the processing delay is one million higher in the traditional approach than in any configuration for the proposed predictor-corrector solution. Although variations are also observed for different values of the $Q$ parameter, they are negligible compared to the large improvement that is achieved over the traditional approach.

In manufacturing logistics, the balance between accuracy and computational performance is critical. Therefore, it is essential to jointly analyze results from both experimental phases. Differences in the processing delay (Fig. 14) are not significant, and then the most accurate configuration would be preferred ( $Q=$ 50). However, memory consumption (Fig. 13) shows relevant differences. Thus, the configuration where $Q=$ 35 is preferred, as accuracy goes above 90%, and memory usage is around 60% in the densest scenario. Only if high precision is required, and enough memory is available, the configuration where $Q=$ 50 should be selected, because memory consumption grows to 90%. On the other hand, for mMTC applications where memory is very sparse, configuration where $Q=$ 20 can be employed. In this case, memory consumption is only 30% but accuracy also reduces and goes below 90%.

Finally, and considering these results (Figs 13 and 14), we can conclude that the proposed predictor-corrector model is computationally lightweight and scalable enough for mMTC scenarios and Industry 4.0 applications.

Although the focus of the proposed predictor-corrector model is the gateway managing a large number of sensor nodes, it is also important to ensure that the proposed corrector procedure may be implemented in resource-constrained nodes. Table 3 shows the computational resources required from a real ESP-32 when implementing the proposed predictor-corrector model.

As can be seen, consumptions are very reduced, and only 2% of the RAM memory and 3% of the program space are consumed. Mathematical operations are consistent with traditional decryption algorithms (only one cell as the test message is no longer than one hundred bits). Besides, the processing delay is similar to other algorithms reported for resource constrained nodes [6].

5.2 Limitations

The main limitation of the proposed technology is its inability to model optical PUF and other similar non silicon-based PUF technologies. Mainly because only silicon-based PUF can be successfully modeled as binary systems or functions. Besides, technologies such as resonant optical rings cannot be modeled as a transmission medium, so the analog model is not valid either. Although, currently, silicon-based PUF is the common (and almost unique) technology in Industry 4.0 and mMTC scenarios, the proposed predictor-corrector scheme cannot be extended to applications supported by optical or hybrid PUF techniques.

On the other hand, there are limitations to the key space, depending on the considered PUF technology. As seen before, those PUF supported by unstable phenomena (resonances or oscillations, for example) show a lower accuracy and specificity. Thus, valid keys must keep a higher distance among them, in order to ensure a correct detection, and the key space (for a given key length, in bits) is then reduced. This also makes systems more vulnerable against brute force attacks, among others. Mitigation actions could be needed.

Moreover, the proposed scheme requires gateways to have computational capabilities. So, this predictor-corrector scheme is not appropriate for those scenarios where gateways are just brokers, routers or switches to concentrate information and deliver it to cloud servers (centralized architectures). Edge computing architectures and other similar distributed architectures are the suitable scenario for the proposed approach.

Additionally, further analyses are required to guarantee the feasibility of this new predictor-corrector scheme in 6G communication scenarios, where up to ten million devices moving with speeds up to one thousand kilometers per hour are expected [34]. As several message exchanges are needed to complete the proposed corrector algorithm and the key selection procedure, the full flow could not be completed under those extreme mobility conditions.

Finally, results about resource consumption in microcontrollers are based on a system-on-chip device (ESP32) where the communication firmware is embedded and provided by the manufacturer. Other devices, where the full communication stack must be stored in the user programming space, may have very hard limits to the user variables and consumed memory. Further and specific analyses are needed to study the compatibility of the proposed predictor-corrector algorithm with those devices.

6. Conclusions

In this paper, we propose a predictor-corrector model to enable lightweight management of secure massive machine-type communication in Industry 4.0 scenarios. Models focus on Physical Unclonable Functions included in most Industry 4.0 sensing nodes. The lightweight models are based on complex functions managed as Laurent series, cubic spline interpolations, and Boolean functions also developed as series. Unknown parameters in these models are predicted and eventually corrected to calculate the output value for each given key. The initial parameters are based on the Kane Yee formula.

In general, we can conclude that the proposed predictor-corrector model enables secure mMTC communication in Industry 4.0 scenarios. Results show that it is lightweight and scalable. Memory usage in gateways reduces by up to 60%, while processing delay may decrease by one million times compared to the traditional approach (where all PUF challenge-response pairs are stored in databases). Besides, sensitivity, specificity, precision, and accuracy are above 80% in general scenarios, reaching value around 90% (or higher) under certain circumstances.

As a global conclusion, algebraic frameworks are a valid approach to model the behavior of Physical Unclonable Functions. They achieve, or even improve, the accuracy of other approaches such as SVM models, but with a higher flexibility. Computational resource consumption (mainly memory) is much more reduced than in traditional key-value databases where exhaustive challenge-response pair catalogues are maintained. As a result, scalability is highly improved.

Future works will analyze the impact of other models in the performance of the predictor-corrector scheme (additional to the analog, Boolean and interpolation models described in this paper). For example, nature-inspired algorithms [51] can be used to replicate very accurately the behavior of many physical phenomena. Additionally, SVM-based models (although current approaches have a poor performance) will be improved using customized kernels [52] and incorporated to the predictor-corrector scheme, in order to analyze their impact on the global precision and accuracy.

On the other hand, the proposed model will be deployed in real Industry 4.0 scenarios to analyze its performance under non-ideal conditions in future works.

Footnotes

Acknowledgments

The publication is produced within the framework of Ramón Alcarria and Borja Bordel research projects on the occasion of their stay at Argonne National Laboratory (José Castillejo’s 2021 grant). This work is supported by Comunidad de Madrid within the framework of the Multiannual Agreement with Universidad Politécnica de Madrid to encourage research by young doctors (PRINCE project).

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Predictor-corrector models for lightweight massive machine-type communications in Industry 4.0

Abstract

Keywords

1. Introduction

2.1 Massive machine-type communications management

2.2 Physical Unclonable Functions modeling

Table 1 State of the art in PUF modeling

Table 2 Configuration parameters

4.2 Performance analysis: Methodology

5. Results and discussions

6. Conclusions

Footnotes

Acknowledgments

References

Table 1
State of the art in PUF modeling

Table 2
Configuration parameters