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

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.

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.

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 ⁢ ( t ) in the i - t ⁢ h 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 [ 0 , T ] , being T e ⁢ x the duration of the excitation. A square window signal W ⁢ ( t ) represents this restriction Eq. (2).

c i ⁢ ( t ) ∈ C ∞(1) (2)

The material will generate a response r i ⁢ ( t ) 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 ⁢ ( ⋅ ) , 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.

r i = f i ⁢ ( c i ⋅ W )(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 ⁢ ( t ) containing the amplitude information Eq. (5), and the instantaneous carrier frequency c i f ⁢ ( t ) containing the frequency information Eq. (6).

c i ⁢ ( t ) = ℛ ⁢ e ⁢ { c i a ⁢ ( t ) ⋅ e j ⋅ c i f ⁢ ( t ) ⋅ t } = c i p ⁢ h ⁢ ( t ) ⁢ cos ⁡ ( ω i c ⁢ ( t ) ⁢ t ) - c i Q ⁢ ( t ) ⋅ sin ⁡ ( ω i c ⁢ ( t ) ⋅ t )(4) c i a ⁢ ( t ) = c i p ⁢ h ⁢ ( t ) + j ⁢ c i Q ⁢ ( t )(5) c i f ⁢ ( t ) = ω i c ⁢ ( t )(6)

OPEN IN VIEWER

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

c i 𝑐𝑜𝑠 = c i ⁢ ( t ) ⋅ cos ⁡ ( ω s ⋅ t ) = 1 2 ⁢ c i p ⁢ h ⁢ ( t ) + 1 2 ⁢ [ c i p ⁢ h ⁢ ( t ) ⋅ cos ⁡ ( [ ω i c + ω s ] ⋅ t ) - c i Q ⁢ ( t ) ⋅ sin ⁡ ( [ ω i c + ω s ] ⋅ t ) ] c i 𝑠𝑖𝑛 = c i ⁢ ( t ) ⋅ sin ⁡ ( ω s ⋅ t ) = - 1 2 ⁢ c i Q ⁢ ( t ) + 1 2 ⁢ [ c i p ⁢ h ⁢ ( t ) ⋅ cos ⁡ ( [ ω i c + ω s ] ⋅ t ) - c i Q ⁢ ( t ) ⋅ sin ⁡ ( [ ω i c + ω s ] ⋅ t ) ](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 ⁢ ( s ) in the Laplace domain Eq. (8) is easy to represent using the Butterworth polynomial B K ⁢ ( a ) Eq. (3.1), where s is the complex frequency in the Laplace domain.

H ⁢ ( s ) = G B K ⁢ ( s B )(8) B K ⁢ ( a ) = ∑ m = 0 K a m ⋅ s m a 0 = 1 a m = ∏ μ = 1 m cos ⁡ ( [ μ - 1 ] ⋅ π 2 ⁢ K ) sin ⁡ ( μ ⋅ π 2 ⁢ K )(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).

c i p ⁢ h ⁢ ( t ) = c i 𝑐𝑜𝑠 ⁢ ( t ) ∗ h K ⁢ ( t ) = ∫ - ∞ ∞ c i 𝑐𝑜𝑠 ⁢ ( τ ) ⋅ h K ⁢ ( t - τ ) ⁢ 𝑑 τ c i Q ⁢ ( t ) = c i 𝑠𝑖𝑛 ⁢ ( t ) ∗ h K ⁢ ( t ) = ∫ - ∞ ∞ c i 𝑠𝑖𝑛 ⁢ ( τ ) ⋅ h K ⁢ ( t - τ ) ⁢ 𝑑 τ ⁢ h K ⁢ ( t ) = ℒ - 1 ⁢ { H ⁢ ( s ) }(10)

On the other hand, the frequency signal c i f ⁢ ( t ) can be calculated through the Fourier transform. The instant carrier frequency ω is the one for which the power spectrum C i ⁢ ( ω , t ) of the original challenge is maximum Eq. (11). This spectrum is directly obtained from the Fourier transform Eq. (12).

c i f ⁢ ( t ) = ω i c ⁢ ( t ) = 𝑎𝑟𝑔 ⁢ max ω { C i ⁢ ( ω , t ) } (11) C i ⁢ ( ω , t ) = | 1 2 ⁢ π ⁢ ∫ - ∞ ∞ c i ⁢ ( t ) ⋅ e - j ⁢ ω ⁢ t ⁢ 𝑑 t | 2(12)

The same process may be applied to response signal r i ⁢ ( t ) and function f i ⁢ ( t ) Eq. (3.1).

r i a ⁢ ( t ) = f i a ⁢ ( c i a ) r i f ⁢ ( t ) = f i f ⁢ ( c i f )(13)

With this approach, function f i a ⁢ ( ⋅ ) is complex, while f i f ⁢ ( ⋅ ) 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 ⁢ ( ⋅ ) and f i f ⁢ ( ⋅ ) 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 ⁢ ( ⋅ ) . 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 γ 1 the interior border of the anulus Eq. (15), and γ 2 the exterior border Eq. (16), function f i a ⁢ ( ⋅ ) 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).

A z 0 = { z ⋮ R 2 ⩽ | z - z 0 | ⩽ R 1 }(14) γ 1 = { z ⋮ | z - z 0 | = R 1 }(15) γ 2 = { z ⋮ | z - z 0 | = R 2 }(16)

f i a ⁢ ( z ) = ∑ m = 0 ∞ a i m ⋅ ( z - z 0 ) m + ∑ m = 1 ∞ l i m ( z - z 0 ) m a i m = 1 2 ⁢ π ⁢ j ⋅ ∫ γ 2 f i a ⁢ ( τ ) ( τ - z 0 ) m + 1 ⁢ 𝑑 τ b i m = 1 2 ⁢ π ⁢ j ⋅ ∫ γ 1 f i a ⁢ ( τ ) ( τ - z 0 ) - m + 1 ⁢ 𝑑 τ(17)

f i a ⁢ ( z ) = ∑ m = 0 ∞ ξ i m ⋅ z m + ∑ m = 1 ∞ η i m ⋅ z - m(18)

Thus, unknown function f i a ⁢ ( ⋅ ) may be described as a polynomial with unknown coefficients ξ i m and η 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 ⁢ ( ⋅ ) as a polynomial around point c 0 , in a very similar way to the process described above Eq. (19). Equally, unknown coefficients β i m will be initially estimated through the Kane Yee formula and later improved through the proposed correction process.

f i f ⁢ ( c ) = ∑ m = 0 ∞ f i f ⁢ ( m ) ⁢ ( c 0 ) m ! ⁢ ( c - c 0 ) m = ∑ m = 0 ∞ β i m ⋅ 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) 𝒟 is, in all situations, a known function. It is also known the transformation 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.

𝒟 ⁢ ( b i c ) = c i ⁢ ( t )(20) ℰ ⁢ ( r i ⁢ ( t ) ) = b i r(21)

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 Γ i ⁢ ( ⋅ ) , where S N is the set of natural numbers that may be represented with N bits Eq. (3.2). Besides, we can understand function Γ ⁢ ( ⋅ ) as a vector function where each component Γ n ⁢ ( ⋅ ) obtains the n -th bit y n in the response b i r from the challenge b i c Eq. (3.2).

Γ i : S N → S M Γ i ⁢ ( b i c ) = b i r(22) Γ ⁢ ( b i c ) = ( Γ i 1 ⁢ ( b i c ) , Γ i 2 ⁢ ( b i c ) , … , Γ i n ⁢ ( b i c ) , … , Γ i M ⁢ ( b i c ) ) = ( y 1 , y 2 , … , y n , … , y M ) = b i r(23)

To define our mathematical model, we are developing the function Γ i ⁢ ( ⋅ ) as a series or as a binary polynomial [30]. Properly, Γ i ⁢ ( ⋅ ) is not a Boolean function, as its image set is not ℤ 2 = { 0 , 1 } but S N . However, each component Γ i n ⁢ ( ⋅ ) is a Boolean function. So hereafter we are working with function Γ i n ⁢ ( ⋅ ) .

We consider the definition of positive cofactor Γ x j n , i and negative cofactor Γ 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 Γ i n ⁢ ( ⋅ ) may be expressed as a combination of these cofactors and the exclusive binary addition (XOR), ⊕ Eq. (25).

Γ x j n , i ⁢ ( b i c ) = Γ i n ⁢ ( x 1 , … , x j - 1 , 1 , x j + 1 , … , x N )(24) Γ x j ¯ n , i ⁢ ( b i c ) = Γ i n ⁢ ( x 1 , … , x j - 1 , 0 , x j + 1 , … , x N ) ∂ ⁡ Γ i n ∂ ⁡ x j ⁢ ( b i c ) = Γ x j n , i ⁢ ( b i c ) ⊕ Γ x j ¯ n , i ⁢ ( b i c )(25)

Equally, any derivation Γ i n ⁢ ( r ) 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 Γ i n ⁢ ( ⋅ ) 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 ρ i n , m is obtained for each individual PUF item Eq. (3.2).

Γ i n ⁢ ( r ) ⁢ ( b i c ) = ∂ r ⁡ Γ i n ∂ ⁡ x 1 r 1 ⋅ … ⋅ ∂ ⁡ x j r j ⋅ … ⋅ ∂ ⁡ x N r N ⁢ ( b i c ) = ∂ ⁡ Γ i n ∂ ⁡ x 1 r 1 ⁢ ( ∂ r ⁡ Γ i n ∂ ⁡ x 2 r 2 ⋅ … ⋅ ∂ ⁡ x j r j ⋅ … ⋅ ∂ ⁡ x N r N ) ⁢ ( b i c ) = … = ∂ ⁡ Γ i n ∂ ⁡ x 1 r 1 ⁢ … ⁢ ( ∂ ⁡ Γ i n ∂ ⁡ x j r j ⁢ … ⁢ ( ∂ ⁡ Γ i n ∂ ⁡ x N r N ) ) ⁢ ( b i c )(26)

Γ n ⁢ ( b i c ) = Γ x j ¯ n ⁢ ( b i c ) ⊕ x j ⋅ ∂ ⁡ Γ n ∂ ⁡ x j ⁢ ( b i c )(27)

Γ i n ⁢ ( b i c ) = 𝑚𝑎𝑥 ⁢ { S N } ⊕ r = 0 [ Γ i n ⁢ ( r ) ⁢ ( d i n ) ⋅ ∏ j = 1 N ( x j ⁢ ⊕ w j ) r j ](28)

Γ i n ⁢ ( b i c ) = ⊕ r ∈ S 2 ⁢ N r j ≠ r j + N ∀ j ∈ [ 1 , N ] [ ρ i n , r ⁢ ∏ j = 1 N ( x j ) r j ⋅ ( x j ¯ ) r j + N ] ρ i n , r ∈ ℤ 2 = { 0 , 1 }(29)

In this case we must remember that coefficients ρ 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 𝒟 and ℰ .

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 ⁢ ( v ) defined in different parts by polynomials P i m ⁢ ( v ) (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 [ c 0 , c Q ] . Where Q is the number of known challenges, and coefficients γ m 1 , γ m 2 , γ m 3 and γ m 4 are unknown.

(30)

P i m ⁢ ( v ) = γ m 1 ⋅ v 3 + γ m 2 ⋅ v 2 + γ m 3 ⋅ v + γ m 4(31)

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

P i m ⁢ ( v ) = α i m 6 ⁢ h i m ⁢ ( c m + 1 - v ) 3 + α i m + 1 6 ⁢ h i m ⁢ ( v - c m ) 3 + ( r m + 1 h i m + α i m + 1 ⁢ h i m 6 ) ⋅ ( v - c m ) + ( r m h i m - α i m ⁢ h i m 6 ) ⋅ ( c m + 1 - v ) ⁢ 𝑏𝑒𝑖𝑛𝑔 ⁢ h i m = c m + 1 - c m ⁢ m ∈ [ 0 , Q ](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 α 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 𝒟 and ℰ .

h i m - 1 ⁢ α i m - 1 + 2 ⁢ ( h i m + h i m + 1 ) ⁢ α i m + h i m ⁢ α i m + 1 = 6 h i m - 1 ⁢ ( r m + 1 - r m ) - 6 h i m - 1 ⁢ ( r m - r m - 1 ) m ∈ [ 1 , Q - 1 ](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.

When the spline P i ⁢ ( v ) is fully defined, it is simple to generate a new prediction r i for challenge c i just applying the function Eq. (34).

r 𝑝𝑟𝑒𝑑 = P i ⁢ ( c 𝑝𝑟𝑒𝑑 )(34)

At this point, when a new prediction must be made, three different responses are obtained: the one from the analog model r i 𝑎𝑛 , the one from the Boolean model r i 𝑏𝑜𝑙 , and the one from the numerical model r i 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 𝑔𝑙𝑜 , all three partial responses must be combined. In this work we are using an arithmetic weighted average Eq. (35).

r i 𝑔𝑙𝑜 = λ i 𝑎𝑛 ⋅ r i 𝑎𝑛 + λ i 𝑏𝑜𝑙 ⋅ r i 𝑏𝑜𝑙 + λ i 𝑛𝑢𝑚 ⋅ r i 𝑛𝑢𝑚(35)

Weights λ i 𝑎𝑛 , λ i 𝑏𝑜𝑙 and λ i 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 𝑛𝑜𝑖𝑠𝑒 , inter-challenge distance P 𝑖𝑛𝑡𝑒𝑟 and PUF stability P 𝑠𝑡𝑏 .

PUF noise P 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 c ⁢ o i over the total number of predictions t ⁢ o i .

P 𝑛𝑜𝑖𝑠𝑒 = c ⁢ o i t ⁢ o i(36)

The inter-challenge distance P 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 ⁢ ( ⋅ , ⋅ ) . 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).

P 𝑖𝑛𝑡𝑒𝑟 =(37) 2 - N + 1 L ⁢ ( L - 1 ) ⁢ ∑ j = 1 L - 1 ∑ i = j + 1 L [ 2 N d H ⁢ ( C i , C j ) ⋅ d H ⁢ ( r i , r j ) M ]

In general, the probability of two independent responses r i , r j to be different may be calculated through the Hamming distance d H ⁢ ( r i , r j ) ; 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 N d H ⁢ ( C i , C j ) 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 [ 0 , 1 ] .

Finally, the PUF stability P 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 ⁢ ( ⋅ , ⋅ ) . The remaining constants are just included in the expression to ensure the probability takes values in the interval [ 0 , 1 ] .

P 𝑠𝑡𝑏 = 1 - 1 L ⁢ ∑ j = 1 L d H ⁢ ( r j , r i ) M(38)

Now, in order to calculate each weight λ i 𝑎𝑛 , λ i 𝑏𝑜𝑙 and λ i 𝑛𝑢𝑚 from probabilities P 𝑛𝑜𝑖𝑠𝑒 , P 𝑖𝑛𝑡𝑒𝑟 and P 𝑠𝑡𝑏 a multivariable mapping function is required.

This function must be monotonically non-decreasing in the interval [ 0 , 1 ] . 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).

λ i 𝑎𝑛 + λ i 𝑏𝑜𝑙 + λ i 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 τ 𝑛𝑜𝑖𝑠𝑒 , τ 𝑖𝑛𝑡𝑒𝑟 and τ 𝑠𝑡𝑏 representing the increasing speed of mapping function with respect to probabilities P 𝑛𝑜𝑖𝑠𝑒 , P 𝑖𝑛𝑡𝑒𝑟 and P 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.

σ i s =(40) exp ⁡ { ( P 𝑛𝑜𝑖𝑠𝑒 s - 1 ) τ 𝑛𝑜𝑖𝑠𝑒 s + ( P 𝑖𝑛𝑡𝑒𝑟 s - 1 ) τ 𝑖𝑛𝑡𝑒𝑟 s + ( P 𝑠𝑡𝑏 s - 1 ) τ 𝑠𝑡𝑏 s } λ i s = σ i s ∑ s σ i s(41) s = { an, bol, num }(42)

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 ( c j , r j ) 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 ξ i m , η i m and β i m in the analog model; parameters ρ i n , r in the Boolean model, and parameters α 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 ( c i , r i ) must be obtained. Using these pairs, parameters α i m may be obtained using Algorithm 1. Parameters ξ i m and η i m may be obtained by replacing (18) all Q challenge-response pairs ( c i , r i ) 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 β i m Eq. (19) and parameters ρ 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 ξ i m , η i m and β i m in the analog model and parameters ρ 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 ( c i , r i ) , we are looking for the best set of parameters ξ i m , η i m and β i m , so when introduced in the model, the predicted responses r i 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 Γ ⁢ ( ⋅ ) and coefficients ρ i n , r Eq. (44).

{ { ξ i m } , { η i m } , { β i m } } = 𝑎𝑟𝑔𝑚𝑖𝑛 ⁢ { ∑ i = 1 Q ( r i a ⁢ n - r i ) 2 Q }(43) { ρ i n , r } = 𝑎𝑟𝑔𝑚𝑖𝑛 ⁢ { ∑ i = 1 Q ( r i 𝑏𝑜𝑙 - r i ) 2 Q }(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 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.

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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 g ⁢ l ⁢ o . 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 σ 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 ( c j , r j ) 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 ( c j , r j ) 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.

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.

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.

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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.

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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.

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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.

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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.

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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.

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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).

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].

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.