Although software applications for seed calculation have been studied in other scenarios (such as cloud-hosted artificial intelligence solutions) [18], this approach is not feasible in Industry 4.0 cryptosystems. Complex computational operations are needed to produce high-randomness and high-entropy seeds, and hardware nodes cannot perform such heavy algorithms. Therefore, Industry 4.0 cryptosystems are based on the so-called “real true random generators” [19], where a physical or electrical unpredictable high-entropy signal is used to produce the seed.

Besides, traditional solutions based on digital circuits have been proven weak against intelligent attacks [9] and Artificial Intelligence learning models [20]. Non-ideal (or non-theoretical) digital True Random Number Generators are always predictable in some sense [21], and advanced algorithms for pattern discovery may capture and reproduce the secret key by exploiting this predictability. Most previously reported digital solutions based on FPGA implementations [22] or digital printed circuits [23] show this vulnerability. To mitigate this problem, analog entropy sources must be injected into random number generators [19]. Four analog sources can be employed: noise, phase jitter, analog chaos, and non-conventional [24].

In the most basic approach, seeds are generated by a regular sampling process of an oscillating natural signal with unpredictable behavior [25]. This scheme is lightweight and fast but requires signals with very high entropy, as seeds are a direct representation of those signals. For example, random telegraph noise generated by CMOS (Complementary Metal Oxide Semiconductor) transistors has been explored as a possible alternative [26]. However, this signal is very temperature-dependent, and cryptosystem operations may be affected by this fact. Some other authors have proposed sampling schemes with a random period for seed generation, so signals with a more stable and lower entropy can be employed, such as natural fluctuations in light signals [27]. However, a clear improvement in seed randomness or entropy has not yet been reported using this approach. Finally, some works describe complex signal processing mechanisms, where features of high entropy signals (such as physiological signals) are extracted, encoded, and concatenated to create complex random seeds [28]. This scheme shows great behavior, but such algorithms are always implemented in computationally powerful servers, and there is no evidence that they can meet the requirements of resource-constrained Industry 4.0 devices. Other approaches, such as quantum random seeds [29], have been reported to be very secure but only feasible in cloud or web services where large amounts of computational resources are available.

However, several authors propose artificial electric signals to produce high-entropy seeds, instead of unpredictable natural signals. The most common approach is based on FPGA (Field Programmable Gate Arrays), used to implement flip-flop ring oscillators [30] or standard sequential circuits but with unpredictable evolution [31]. Similar circuits have also been explored using discrete components [32] or printed circuits [33], although their performance is slightly worse. The main problem with this family of solutions is that they have a limited catalog of possible seeds to be generated, thus reducing the lifetime of nodes.

FPGA-based generators generally employ jitter in logic gates to create a high-entropy signal. However, other random signals are natively present in electronic circuits; mostly, electric noise. Solutions in which noise is amplified and, later, transformed into a bipolar signal using an analog comparator may also be found [34]. However, they show some stability problems.

Finally, a small number of authors have explored how to produce random high-entropy seeds from other kinds of unpredictable signals, such as Internet traffic [35]. However, the required infrastructure to support these algorithms cannot be easily integrated into Industry 4.0 cryptosystems.

As hardware-enabled cryptosystems based on high-entropy analog chaotic signals show the best balance between performance and resource consumption, interest in their study has grown greatly.

Although chaotic solutions are not always considered “true random number generators” (as high entropy dynamics are required), some authors have studied and classified proposals that employ chaos for cryptographic applications [24].

Two approaches may be found. In the first one, numerical methods are employed to iterate discrete chaotic maps, such as the logistic map [36] or the tent map [37]. Hyperchaotic maps with higher entropy have also been studied [38]. However, these mechanisms are highly dependent on numerical errors and hardware data formats. Then, similar schemes, but supported by hardware circuits, have been reported. Circuits representing the Piece-Wise Affine Markov (PWAM) mapping using capacitors [39] or the CMOS discrete-time oscillator where transistors are used as fast electronic switches [40] can be found. However, these circuits have some temporal limitations, so they cannot generate seeds as fast as desired. The limits to the maximum throughput in cryptosystems must be considered (a maximum bitrate of 23 MB/s has been described).

Besides based on numerical methods, some chaotic maps created by neural networks have been used in cryptography [41]. However, those models are too computationally heavy to be maintained on Industry 4.0 nodes.

The second approach is based on continuous chaotic signals, that is, chaotic circuits. During recent years, research on chaotic systems has explored both lines: the simplest dynamics [42] and the highly complex behavior of hyperchaotic systems [43]. Minimum dynamics and circuits have been the most popular ones historically, probably because they are much more stable and easier to manufacture. Schemes based on Chua’s circuit [44], jerk systems [45], and Boolean chaotic oscillators [46] have been proposed. As a main advantage, these schemes are easier to synchronize [44], so remote elements can generate coherent shared symmetric keys in a very efficient manner. However, some authors have shown that these circuits are vulnerable to different attacks and capable of capturing the secret circuit configuration [43].

As a response, in the last years, high-order dynamics have become much more popular. One of the circuits most studied recently is those based on memristors. From traditional circuits with discrete elements that produce only four signals [47], to schemes in which five different unpredictable signals are generated [48] and solutions that mix three different dynamics to enrich the complexity and entropy of signals [49]. But, in most memristive circuits, only chaotic structures are generated [50,51] (only one Lyapunov exponent is positive). Nevertheless, signals, although hyperchaotic, do not show entropy and randomness as high as expected. Mainly because all these circuits are built to be very stable.

Some new dynamics created from mathematical models have been proposed. Some of these new dynamics are inspired by real physical systems, such as temperature fluctuations in circuits [52], but most new dynamics have no clear physical meaning or direct relation to electronic components [43]. Hyperjerck circuits [53], five-dimensional dynamics [54], and unknown structures designed through artificial intelligence mechanisms [55] have been described. The main disadvantage of these circuits is their complexity [56]. In general, they include tens of elements, from multipliers to integrators, amplifiers, etc. Therefore, they are difficult to integrate into Industry 4.0 nodes.

Some FPGA implementations of chaotic cryptosystems have also been reported [57] but, due to the predictable behavior of digital structures (designed to be binary and remove any uncertainty), they typically generate pseudo-random generators [58,59], contrary to the real true random generators required in Industry 4.0 scenarios.

In this paper, we propose a circuit implementation for a new hyperchaotic dynamics, designed to show a great entropy but a compact circuit realization. The proposed four-dimensional circuit only requires nine active components, reducing space and power consumption. But the maximum Lyapunov exponent can reach a value as high as 12.37.

The new hyperchaotic dynamics was initially reported as a solution to magnify the rate of expansion, reduce the rate of contraction, and then improve the security properties of the Lorenz dynamics. The new dynamics include four coupled differential equations with four bifurcation parameters αi (1). (1)x1˙=α4(x2−x1)+2x4x2˙=5x1+α3x2−4x1x3x3˙=x1x2−3x3x4˙=−α2x4−α1(x1−x2)

To calculate the equilibrium points of this dynamics, let xi˙=0∀i∈[1,4] and solve the resulting equation system (2). Three different equilibrium points Ei=(ei1,ei2,ei3,ei4) are, then, found (3). (2){α4(x2−x1)+2x4=05x1+α3x2−4x1x3=0x1x2−3x3=0−α2x4−α1(x1−x2)=0 (3)E0=(0,0,0,0)E1=(34(5+α3)34(5+α3)5+α340)E2=(−34(5+α3)−34(5+α3)5+α340)

Points E1 and E2 are symmetric about the plane {x3=5+c4,x4=0} and have identical behavior, as they have pretty similar Jacobian matrices J(Ei) (4), the same characteristic equation P1,2(λ) and then, eigenvalues (5). The resulting characteristic equations P0(λ) and P1,2(λ) have no analytic solution and must be solved using numerical methods. Other stability criteria, such as Routh-Hurwitz, do not allow for obtaining an analytic expression either. In this context, an exhaustive study of all possible stability situations for the fixed points, to identify the regular, chaotic, and hyperchaotic regions in the dynamics is impractical because four different bifurcation parameters αi should be handled. For some parameters (two, at least), a fixed value must be specified before any further analysis. (4)J(Ei)=(−α4α4025−4ei3c−4ei10ei2ei1−30−α1α10−α2)J(E0)=(−α4α4025α30000−30−α1α10−α2)J(E1)=(−α4α402−α3α3−23(5+α3)034(5+α3)34(5+α3)−30−α1α10−α2)J(E2)=(−α4α402−α3α323(5+α3)0−34(5+α3)−34(5+α3)−30−α1α10−α2) (5)P0(λ)=λ4+(α4−α3+α2+3)λ3+(−α3α2+α4α2−α4α3−2α4−3α3+3α2+2α1)λ2+(−2α4α2−2α1α3−3α2α3−3α4α3−15α4−4α1−α3α4α2)λ+(−30α1−3α4α2α3−15α4α2−6α1α3)P1,2(λ)=λ4+(α4−α3+α2+3)λ3+(−α3α2+3α4+15+α4α2+2α1+3α2)λ2+(3α4α2+15α2+6α4α3+6α1+30α4)λ+(60α1+12α1α3+6α4α2α3+30α4α2)

To obtain a valid value for some of the bifurcation parameters αi we are applying the Liouville theorem (6) to analyze how the trajectory’s volume in the phase space evolves. In stable (chaotic or not) dynamical systems the change rate for the unitary volume occupied by the trajectory must be negative (7). As seen, parameter α3 contributes to the expansion of this volume, while parameters α2 and α4 contribute to the contraction. We propose to compensate for the offset in the change rate by fixing α3=3. Additionally, we propose α2=2, after a heuristic investigation process. In conclusion, to ensure that the dynamic is stable, some restrictions must be observed by parameters α1 and α4 (8). (6)1VdV(t)dt=(∂x1˙∂x1+∂x2˙∂x2+∂x3˙∂x3+∂x4˙∂x4)=trace{J(Ei)}=−α4−3−α2+α3 (7)−α4−3−α2+α3<0 (8){α1∈Rα2=2α3=3α4>−2

Fig. 1 shows a bidimensional numerical analysis of the eigenvalues associated with the fixed point E0. As seen, E0 presents a positive real eigenvalue (λ1) for all possible combinations of bifurcation parameters. Then, E0 is unstable (at least in one direction). The remaining three eigenvalues (λ1,λ2, and λ4) are complex with a negative real part. Approximately, in the region R0 (9) imaginary parts of all these three eigenvalues turn zero. Therefore, the trajectories do not oscillate around the fixed point before exiting the unstable direction. Outside this region, the imaginary parts are positive, and the trajectories oscillate around a fixed point E0 before leaving through an unstable direction. (9)R0={(α1,α4)⋮α1<2α4}

The eigenvalues associated with fixed points E1 and E2 are two pairs of complex conjugate numbers (10), as seen in Fig. 2. Trajectories, then, are always oscillating around these two fixed points. One of these pairs (λ3 and λ4) presents an always-negative real part, while the other two eigenvalues (λ1 and λ2) vary their behavior. Approximately, real parts of these eigenvalues are positive in the region R1 (10), so for those combinations of bifurcation parameters in that region trajectories are unstable (they do not converge to the fixed point) and oscillating. Chaos, hyperchaos, and other regular structures are possible in this situation. Outside region R1 all eigenvalues have negative real parts and trajectories converge to the fixed points from all directions. (10)R1={(α1,α4)⋮α1<−50α4+100}

However, although regions, where trajectories have an oscillating behavior, an unstable evolution, or a convergent flow, have been identified, it is not fully clear if the oscillating trajectories are regular, chaotic, or hyperchaotic. To identify the chaotic regions, the Lyapunov exponents must be analyzed.

In chaotic regions, at least one Lyapunov exponent must be positive, while in hyperchaotic regions, the number of positive exponents increases to two. Then, it is enough to study the two maximum Lyapunov exponents. For this study, we employ an algorithm based on the QR decomposition [60]. Fig. 3 (left) shows a bidimensional study of the maximum Lyapunov exponent Lmax. As seen, this exponent is positive in almost the entire studied area, except for a small square region R2 where it is negative (11). Then, chaotic trajectories are generated for any set of bifurcation parameters outside R2. (11)R2={(α1,α4)⋮α1>70∧α4>6}

In general, the maximum Lyapunov exponent varies in the range [0,2] as most chaotic dynamics do. However, in the region R3 much higher values are observed (12a), as seen in Fig. 3 (right). In this region, values greater than six for the Lyapunov exponent are common, and even values greater than ten are discovered for some bifurcation parameters such as {α1=77.46,α4=0.767}. The maximum value Lmax=12.37 is seen for {α1=63.43,α4=0.16}. (12a)R3={(α1,α4)⋮α1<60∧α4<4}

Fig. 4 shows a bi-dimensional analysis of the second maximum Lyapunov exponent L2. As seen, this exponent is positive in, approximately, half of the studied region. Positive values for this exponent are lower and range in the interval [0,0.5]. Hyperchaos is detected in the region R4, (12b) where the maximum value L2=0.31 is located at the point {α1=73.37,α4=9.2}. As the maximum values for the maximum Lyapunov exponent Lmax and the second maximum Lyapunov exponent L2 are not coincident, the parameter configuration with the higher entropy must be located using heuristic techniques. (12b)R4={(α1,α4)⋮α1<−10α4+100}

Considering all the previous results and some additional heuristic numeric research, in this paper we highlight three different configurations (see Table 1). The first configuration was selected as an example of the regular structures generated by this new Lorenz-based dynamic. Fig. 5 shows the resulting trajectory in the space phase. The structure is a standard limit cycle with a low entropy (as expected from a trajectory where all Lyapunov exponents are non-positive). Hereinafter, we refer to the definition of “entropy” in Kolmogorov-Sinai [61]. This definition is directly related (and is an extension of any physical system) to the Shannon-McMillan-Breiman Theorem and Shannon’s idea of entropy [62], widely applied in engineering and information theory. Then, it is a proper signal definition for a cryptosystem. In this paper, we obtain the Kolmogorov-Sinai entropy through the Gram-Smitch orthogonalization of the Jacobian matrix.

The remaining two configurations share the same value for the α1 parameter, so it is interesting to analyze how the behavior of the dynamics changes as α4 evolves (given the value for α1). Fig. 6 shows the bifurcation diagram of the proposed dynamic, given the following set of parameters {α1=20,α2=2,α3=3}. As seen, chaotic or hyperchaotic trajectories are generated throughout the entire range. Regular structures only appear for values α4>10.38 before disappearing and collapsing in a sink at α4=10.72 (Hopf bifurcation). Besides, there is a very narrow crisis region around α4=4.28, although it is irrelevant to this study.

For most values of the α4 parameter, structures are just chaotic, and have a behavior similar to the Lorenz attractor, as described in Table 1 (second configuration) and can be seen in Fig. 7. Chaotic signals in this region show an entropy like the observed in other chaotic dynamics such as Lorenz or Chua’s circuit (although it is slightly higher). The Lyapunov spectrum, on the other hand, only has one positive exponent; which is coherent with a chaotic trajectory. However, in Industry 4.0 cryptosystems, high-entropy signals are required, and the complexity of traditional chaotic trajectories is not enough. Hyperchaotic flows are needed.

To identify hyperchaotic structures, it is interesting to analyze the entire Lyapunov spectrum in the same range of α4 parameter studied with the bifurcation diagram. Fig. 8 shows the evolution of the Lyapunov spectrum, given the following set of parameters {α1=20,α2=2,α3=3}. As seen, results are coherent with initial analyses and Figs. 3 and 4, but regions may be identified much more precisely. Specifically, hyperchaotic trajectories (those for which the second maximum Lyapunov exponent is positive) are located mainly in the range α4∈[1.02,2.34], although some other narrow hyperchaotic regions (for example around α4=6.24) can be seen. In this paper, as presented in Table 1, we consider the trajectory generated for the parameter set {α1=20,α2=2,α3=3,α4=2}. This hyperchaotic trajectory still has similarities with complex Lorenz’s attractors (see Fig. 9), although it is much denser than the chaotic structures discussed above.

The proposed hyperchaotic signals are not associated with the maximum value of the maximum Lyapunov exponent (see Figs. 3 and 4), but they have the highest entropy (5.13) that we could identify, after exhaustively analyzing the proposed dynamics. This is, in fact, the most important characteristic of Industry 4.0 cryptosystems and key generation applications. The Kaplan-York dimension is, on the other hand, significantly higher than in chaotic structures and is just 10% lower than the possible maximum (four dimensions). Compared to the traditional Lorenz dynamics, in its most complex configuration, the entropy of the proposed dynamics is five times greater (as the maximum Lyapunov exponent and Kolmogorov-Sinai entropy is 0.912 for the Lorenz circuit [63]). Additionally, other stable four-dimensional implementations of Lorenz-based systems cannot increase their maximum Lyapunov exponent above 0.5 [50]. The proposed dynamics in this paper can be implemented in a stable circuit and increases this entropy by up to ten times. Furthermore, compared to other recently reported hyperchaotic circuits (non-Lorenz-based, such as memristive circuits) [64], the Kaplan-York dimension of the proposed circuit is 20% higher. Table 2 shows these comparisons.

In conclusion, the corresponding hyperchaotic signals are complex, unpredictable, and show a high entropy. They have sufficient characteristics to be employed in Industry 4.0 cryptosystems.

The simplest and most direct way to design a circuit implementation (usually named “canonical implementation”) for a dynamic without clear physical meaning (such as the one we are analyzing), is to apply a variable change (13) to the original dynamics (so that the time variable is no longer non-dimensional) and put the equations in their integral form (14); where τ is the current derivation variable and T is a constant with time units. Then, it is possible to implement every resulting differential equation using operational amplifiers (OA), integrators, analog multipliers, etc. (13)τ=T⋅t (14)x1=1T∫α4(x2−x1)+2x4dtx2=1T∫5x1+α3x2−4x1x3dtx3=1T∫x1x2−3x3dtx4=1T∫−α2x4−α1(x1−x2)dt

Fig. 10 shows the proposed canonical implementation for the new hyperchaotic Lorenz-based dynamics. Kirchhoff’s laws describe the circuit behavior using only the voltage at four key points (15). Now, if some special values are selected for resistors and capacitors in the circuit (16), and voltage signals are identified as variables in the dynamic, the proposed circuit is formally equivalent to hyperchaotic dynamics. As seen, control parameters α1 and α4 may be varied using just different values for resistors. Thanks to voltage-controlled resistors, for example, Industry 4.0 nodes could operate with the circuit and generate different trajectories for key generation at various moments.

However, this implementation has an important disadvantage. Any practical realization (using real hardware or simulation tools) requires resistors with very different values: from a few ohms to several megaohms. As the voltage values are similar at all points in the circuit (see Fig. 10), this huge difference among resistors causes currents to be highly variable. Eventually, this causes convergence problems, the circuit becomes unstable, and operational amplifiers tend to get saturated.

This circuit implementation, thus, is not useful for Industry 4.0 cryptosystems, where robust solutions able to operate for hours in a stable range are needed. Therefore, in this section, we propose a compact but robust implementation for the newly described hyperchaotic dynamic, where problems of the canonical implementation are solved.

We propose a new variable change, introducing a new parameter k (17). This parameter is applied to every variable in the dynamic through a different function, so we create a difference in the dynamic range of signals, enough to reduce the dispersion in the resistors and current values. This approach has been proven successful in previous works [65]. We observed currents are higher in the sub-circuit generating x3 signal. Approximately, they show quadratic values, compared to currents at other points of the circuit. Besides, we observe the polarity of the x4 signal is inverted. Then, the proposed variable change corrects those differences (18). (15)v1=1R8C1∫R7(R4(R2+R1)R2(R3+R4)v2−R1R2v1R5+v4R6)dt v2=1R19C2∫R10R18(R15+R17)R15(R18+R16)(v1R11+v2R9)−R17R13R15R14v1v3dt v3=1R24C3∫R22R20v1v2−R23(R22+R20)R20(R21+R23)v3dt v4=1R31C4∫R29R30(R32(R26+R27)R27(R32+R25)v2−R26R27v1)−R33(R29+R30)R30(R33+R28)v4dt R1=R4 R2=R3=R1α4 R5=R7=2R6 R13=4R14 R10=α3R9=5R11 (16)R15=R16=R17=R18 R20=R22 R23R21+R23=32 R29=R30 R26=R32=α1R27=α1R25 R23=α22−α2R28 R31C4=R19C2=R24C3=R8C1=T vi≡xii∈[1,4] (17){u1=x1k,u2=x2k,u3=x3k,u4=−x4k} (18)u1=1T∫α4(u2−u1)−2u4dtu2=1T∫5u1+α3u2−ku1u3dtu3=1T∫u1u2−3u3dtu4=1T∫−α2u4+α1(u1−u2)dt

On the other hand, our objective is to improve the circuit robustness as much as possible, reducing noise and convergence problems. To do that, the number of elements in the circuit must be reduced. So instead of generating all sub-signals independently for every chaotic variable, we reuse all possible sub-signals. As seen in Fig. 11, the sub-circuit generating x2 signal is the largest, with five active elements (two times the number of active circuit elements required to generate other signals). Then, the second equation is rewritten to reduce the number of circuit elements and enable the re-usage of sub-signals (19). (19)u2=1T∫k(5k−u3)u1+α3u2dt

Fig. 11 shows the proposed circuit. This robust circuit implementation only includes nine active components, while the canonical implementation requires fourteen. Besides, standard analog multipliers are being replaced by the AD633 component; a very usual commercial multiplier but with a more complex transference function (see Fig. 12). Analyzing the circuit as before (20) and applying some special restrictions among resistors and capacitors (21), it is possible to confirm that the new circuit represents hyperchaotic dynamics. The value for k parameter must be determined experimentally (see Section 5). (20)u1=R2R5C1∫1R1u2−R4(R1+R2)R1R2(R3+R4)u4−R3(R1+R2)R1R2(R3+R4)u1dtu2=1C2∫1R7u2+1R6(1k−u3)u1dtu3=1C3∫1R10u1u2−1R11u3dtu4=R12R15C4∫1R11u1−R14(R11+R12)R11R12(R13+R14)u2−R13(R11+R12)R11R12(R13+R14)u4dt (21)R2=α4R1R3R1=R2R4R4(R1+R2)=2R1R2(R3+R4)R7C2=Tα3R6C2=TkR5C1=R10C3=TR10=3R11R12=α1R11R11R14=R12R13R13(R11+R12)=α2R11R12(R13+R14)