Parity-time symmetric hardware security system and method

Abstract

This disclosure provides techniques for implementing a Physically Unclonable Function (PUF) in an electronic circuit. One such method includes pairing a transmitter circuit and a receiver circuit via inductive coupling, sending a challenge signal from the transmitter circuit to the receiver circuit to stimulate the receiver circuit, measuring a unique transient response that depends on a natural frequency of the combined system of the transmitter circuit and the receiver circuit, where the natural frequency is a function of a value of a physical property of the receiver circuit, and converting the measurement of the unique transient response into a bit string, where the bit string includes a Physically Unclonable Function (PUF)-based encryption key.

Description

[Technical Field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS This application claims priority to a co-pending U.S. provisional application entitled "Parity-Time Symmetric Hardware Security Systems and Methods," filed on November 11, 2022, and having application serial number 63 / 424,725, and a co-pending U.S. provisional application entitled "Electromagnetically Unclonable Functions Generated by Non-Hermitian Absorber-Emitter," filed on September 8, 2023, and having application serial number 63 / 537,263, each of which is incorporated by reference herein in its entirety.

[0002] STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT This invention was made with government support under awards NSF ECCS-1914420 and NSF2229659 awarded by the National Science Foundation (NSF) and award FA9550-21-1-0202 awarded by the Air Force Office of Scientific Research (AFOSR). The government has certain rights in this invention.

[0003] The present disclosure generally relates to techniques for implementing a Physical Unclonable Function (PUF) in an electronic circuit. [Background technology]

[0004] The development of cryptographic techniques to ensure data security and privacy is an urgent need in modern society. Today, many cryptographic techniques have been proposed for identification, authentication, and anti-counterfeiting. Among various cryptographic techniques, the physically unclonable function (PUF) is widely regarded as one of the most promising hardware security primitives, which can be implemented by applying an input challenge C n , the output response R of the physical system for constructing the encryption key. n (a so-called challenge-response pair or CRP).

[0005] Due to the inoperable variability arising from the inherent process variations in manufacturing, CRPs are device-specific and extremely difficult to clone, allowing them to be used as unique, hard-to-clone encryption keys. Perhaps silicon-based PUFs, which exploit the inherent variability in the complementary metal-oxide semiconductor (CMOS) manufacturing process, are a very common PUF scheme. To date, various silicon-based PUFs have been proposed, including static random access memory (SRAM) PUFs, ring oscillator PUFs, phase-change memory (PCM) PUFs, and memristor PUFs. While these digital circuit-based PUFs benefit from the high throughput and reliability of CMOS technology, they have recently been reported to be vulnerable to approximation and modeling attacks due to their limited device-to-device and intra-die variability. Summary of the Invention [Means for solving the problem]

[0006]

[0003] Embodiments of the present disclosure provide techniques for implementing a Physical Unclonable Function (PUF) in an electronic circuit. One such system includes a challenge generator circuit, a transmitter circuit coupled to the challenge generator circuit, and a receiver circuit configured to receive and be stimulated by the transmitted challenge signal. The combination of the transmitter circuit and the receiver circuit includes a parity-time symmetric structure that operates at an exception point, a divergence exception point, or a coherent perfect absorber laser (CPAL) point of the parity-time symmetric structure. Further, the transmitter circuit is configured to transmit a challenge signal generated by the challenge generator circuit to the receiver circuit, and after transmitting the challenge signal, the transmitter circuit and the receiver circuit are configured to generate a unique transient response that depends on the eigenfrequency and harmonic responses of the parity-time symmetric structure or that depends on the eigenvalues ​​of the scattering matrix of the parity-time symmetric structure, and the transmitter circuit or the receiver circuit is configured to measure the unique transient response and convert values ​​of the measured unique transient response into a bit string, or to measure a spectral (frequency domain) response near the CPAL points and convert values ​​of the measured unique spectral response of the output coefficients into a bit string, and / or the bit string includes a physically unclonable function (PUF)-based encryption key.

[0007] Embodiments of the present disclosure also present methods for implementing a Physical Unclonable Function (PUF) in an electronic circuit. One such method includes pairing a transmitter circuit and a receiver circuit via inductive coupling, sending a challenge signal from the transmitter circuit to the receiver circuit to stimulate the receiver circuit, measuring a unique transient response that depends on a natural frequency of the combined system of the transmitter circuit and the receiver circuit, where the natural frequency is a function of a value of a physical property of the receiver circuit, and / or converting the measurement of the unique transient response into a bit string, where the bit string includes a Physical Unclonable Function (PUF)-based encryption key.

[0008] In one or more aspects of such a method or system, the receiver circuit and the transmitter circuit are complementary metal oxide semiconductor integrated circuits, III-V semiconductor integrated circuits, II-VI semiconductor integrated circuits, and / or combinations of these technologies via heterogeneous integration; the receiver circuit comprises an RLC oscillator having a positive resistance; the transmitter circuit comprises an RLC oscillator and one or more LC oscillators functioning as repeaters; the RLC circuit, the RLC oscillator, and the one or more LC oscillators are based on complementary metal oxide semiconductor integrated circuits; the unique transient response comprises a voltage value measured across a capacitor of the RLC oscillator of the transmitter circuit or the RLC oscillator receiver circuit; the RLC oscillator and the RLC oscillator are configured to be inductively coupled via an on-chip transformer, or capacitively coupled via an on-chip capacitor or a negative capacitance converter, or both inductively and capacitively coupled using the above components; after verifying the bit string, the receiver circuit is configured to transmit content stored in the second memory unit to the transmitter circuit; and / or the challenge generator circuit comprises a pulse generator.

[0009] In one or more aspects, such a method or system may include verifying the bit string by comparing the bit string to a stored valid identifier, and / or transmitting information stored in a memory of the receiver circuit to a transmitter circuit upon verifying the bit string.

[0010] In one or more aspects, such a system may further comprise a first digital memory unit accessible by a transmitter circuit, the transmitter circuit configured to verify a bit string against a valid bit string stored in the first digital memory unit, and / or a second digital memory unit accessible by a receiver circuit, the receiver circuit configured to verify the bit string against a valid bit string stored in the second digital memory unit.

[0011] An embodiment of the present disclosure also presents a system comprising: a challenge generator circuit; and a Physically Unclonable Function (PUF) device configured to receive and be stimulated by a challenge signal transmitted by the challenge generator circuit. Thus, the PUF device comprises a parity-time symmetric structure operating at an exception point, a divergence exception point, and / or a coherent perfect absorber laser (CPAL) point of the parity-time symmetric structure, wherein after transmitting the challenge signal, the PUF device generates a unique transient response that depends on an eigenfrequency and an output harmonic of the parity-time symmetric structure or a unique spectral response that depends on an eigenvalue of a scattering matrix of the parity-time symmetric structure, wherein the PUF device is configured to measure the unique transient response and / or the spectral response and convert the value of the measured transient response into a bit string, and / or the bit string includes a Physically Unclonable Function (PUF)-based encryption key.

[0012] In one or more aspects for such a system, the PUF device comprises a pair of active and passive electromagnetic metasurfaces separated by a dielectric spacer, and the PUF device can also be realized by an equivalent lumped element circuit comprising a negative resistance converter (NRC) and a shunt resistor separated by a transmission line or a T / π transformer, the NRC being implemented using an interconnected coupled pair (XCP) circuit or a current feedback operational amplifier, and the challenge generator circuit generates two coherent waves having a complex amplitude ratio, and / or two pairs of incident coherent waves with different complex value amplitude ratios generate different unique responses.

[0013] Other systems, methods, features, and advantages of the present disclosure will be, or will become, apparent to one with skill in the art upon examination of the following figures and detailed description. It is intended that all such additional systems, methods, features, and advantages be included within this description, be within the scope of the present disclosure, and be protected by the accompanying claims.

[0014] Many aspects of the present disclosure can be better understood with reference to the following drawings. The components in the drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating the principles of the present disclosure. Moreover, in the drawings, like reference numerals designate corresponding parts throughout the several views. [Brief explanation of the drawings]

[0015] [Figure 1(a)] 1 illustrates physically unclonable function (PUF) enabled secure radio frequency (RF) authentication and communication according to the present disclosure. [Figure 1(b)] 10 illustrates exemplary plots of a radio frequency (RF) challenge, time response, and bit sequence processing according to various embodiments of the present disclosure. [Figure 2(a)] 1 illustrates a transmitter-receiver architecture implementing a radio configuration with divergence advantage point (DEP) according to the present disclosure, along with the corresponding pseudospectrum. [Figure 2(b)] 1 illustrates a transmitter-receiver architecture implementing a standard Excellence Point (EP) radio configuration, along with the corresponding pseudospectrum, in accordance with the present disclosure. [Figure 2(c)] 1 illustrates a transmitter-receiver architecture implementing a singularity-free radio configuration according to the present disclosure, along with the corresponding pseudospectrum. [Figure 3(a)] 10 illustrates the distribution of the real part of the natural frequency of a third-order PT telemetry system according to various embodiments of the present disclosure. [Figure 3(b)] 10 illustrates the distribution of the imaginary part of the natural frequency of a third-order PT telemetry system according to various embodiments of the present disclosure. [Figure 4(a)] 1 illustrates a bitmap of 256-bit PUF responses from 100 PUF instances for an exemplary DEP-based RF PUF system, according to various embodiments of the present disclosure. [Figure 4(b)] 1 illustrates an entropy (Ex, Ey) analysis of 256-bit PUF responses from 100 PUF instances for an exemplary DEP-based RF PUF system, according to various embodiments of the present disclosure. [Figure 4(c)]For the exemplary DEP-based RF PUF system used in Figures 4(a) and 4(b), we show plots of inter-Hamming distance (Inter-HD) histograms and intra-Hamming distance (Intra-HD) histograms obtained from 100 PUF instances. [Figure 4(d)] We plot the inter-HD histograms obtained from three different RF-PUF systems in Fig. 2(a)-Fig. 2(c). [Figure 4(e)] We show the inter-HD histograms of PUFs based on (1) a third-order PT system operating near the DEP, (2) a third-order PT system operating away from the DEP, and (3) a standard PT system operating near the exceptional point (EP), and (4) a conventional telemetry system using an NFC coil antenna. [Figure 5(a)] 10 illustrates an entropy (Ex, Ey) analysis of 256-bit PUF responses from 100 readers with different RLC circuits querying the same receiving device, according to various embodiments of the present disclosure. [Figure 5(b)] We show plots of inter-Hamming distance (Inter-HD) and intra-Hamming distance (Intra-HD) histograms obtained from the experimental setup in Fig. 5(a). [Figure 5(c)] We plot the inter-HD histograms obtained from the three different RF-PUF systems in Figures 2(a)-2(c) for the experimental setup used in Figures 5(a) and 5(b). [Figure 6] 1 illustrates a block diagram of an exemplary DEP-based RF-PUF encryption system for use in radio frequency identification (RFID) and wireless access control, according to various embodiments of the present disclosure. [Figure 7] 1 illustrates a block diagram of an exemplary DEP-based RF-PUF cryptographic system used in near field communication (NFC) and authentication processes, according to various embodiments of the present disclosure. [Figure 8(a)]FIG. 1 shows a schematic diagram of an exemplary coherent perfect absorber laser (CPAL) PUF system implemented using active and passive metasurfaces forming a PT-symmetric system in the optical domain, in accordance with various embodiments of the present disclosure. [Figure 8(b)] Fig. 8(a) shows the transmission line network model of the CPAL PUF system. [Figure 8(c)] 1 illustrates a sequence of operations used in a cryptographic pseudorandom number generation process of an exemplary CPAL PUF system, according to various embodiments of the present disclosure. [Figure 9(a)] 1 provides a comparison of randomness between (a) an exemplary CPAL-enabled PUF and (b) a Fabry-Perot Interferometer (FPI)-enabled PUF. [Figure 9(b)] 1 provides a comparison of randomness between (a) an exemplary CPAL-enabled PUF and (b) a Fabry-Perot Interferometer (FPI)-enabled PUF. [Figure 10(a)] Shown are a circuit diagram (top) and a photograph (bottom) of an exemplary CPAL PUF device realized using printed circuit board technology in the radio frequency range. [Figure 10(b)] We show bitmaps measured across 25 CPAL PUF instances under two challenges (phase offsets φ=0, π / 2). [Figure 10(c)] The entropies Ex and Ey measured for the bit string and key string of the CPAL PUF device in Fig. 10(a) are shown. [Figure 10(d)] We present pairwise maps comparing the inter-HD between two arbitrary PUF devices, demonstrating that the produced PUF keys are largely uncorrelated. [Figure 10(e)] Measured inter- and intra-HD of an exemplary CPAL PUF are shown by applying laser (top) and CPA (bottom) challenges, along with their Gaussian fitting results. [Figure 11(a)] Passive FPI (top) and active FPI (bottom) of an FPI-enabled PUF device in the radio frequency domain and their on-board realization are shown. [Figure 11(b)] We show the bitmaps measured across 25 passive FPI-based PUF instances in Fig. 11(a). [Figure 11(c)] Fig. 11(a) shows the measured entropy content of the passive and active FPI PUF instances. [Figure 11(d)] We show the pairwise maps of 25 passive FPI PUF instances in Fig. 11(a). [Figure 11(e)] Fig. 11(a) shows the inter-HD of active-FPI and passive-FPI based PUF instances. [Figure 12(a)] 1 shows a schematic diagram of the Fourier regression (FR) model. [Figure 12(b)] We report the results of an FR modeling attack using the FR model in Figure 12(a). [Figure 12(c)] We report the results of an FR modeling attack using the FR model in Figure 12(a). [Figure 12(d)] We report the results of an FR modeling attack using the FR model in Figure 12(a). [Figure 13(a)] 1 shows a schematic diagram of a generative adversarial network (GAN) structure with a generator and a discriminator. [Figure 13(b)] Figure 1 shows the probability mass function (PMF) of (b) prediction accuracy, (c) correlation coefficient (CC), and (d) Hamming distance (HD) between predicted and simulated CRPs by GANs using an exemplary CPAL PUF system of the present disclosure. [Figure 13(c)] Figure 1 shows the probability mass function (PMF) of (b) prediction accuracy, (c) correlation coefficient (CC), and (d) Hamming distance (HD) between predicted and simulated CRPs by GANs using an exemplary CPAL PUF system of the present disclosure. [Figure 13(d)] Figure 1 shows the probability mass function (PMF) of (b) prediction accuracy, (c) correlation coefficient (CC), and (d) Hamming distance (HD) between predicted and simulated CRPs by GANs using an exemplary CPAL PUF system of the present disclosure. DETAILED DESCRIPTION OF THE INVENTION

[0016] According to various embodiments of the present disclosure, extreme sensitivity to perturbations near exceptional points (EPs) presents a solution to hardware security and authentication. In particular, naturally occurring fabrication errors can be used to construct EP-based electronic circuits to implement physically unclonable functions (PUFs) with excellent statistical properties in terms of the randomness of generated keys and the uniqueness among different keys.

[0017] Traditional security schemes rely on encrypted keys stored inside non-volatile memory chips. These can, in principle, be attacked, posing serious security challenges in nearly every aspect of modern life, including safety, product, food, and drug authentication, radio frequency identification (RFID) authorization, and encrypted communications. In this context, Internet of Things (IoT) systems, in which various data—such as location, financial, and health data—are constantly collected by sensors and different electronic and tracking devices via near-field communication (NFC) interfaces (e.g., built into mobile devices), are particularly vulnerable to such problems. To make matters worse, recent advances in artificial intelligence have made it possible to decrypt some current software-based cryptographic algorithms using machine learning / deep learning-assisted attacks.

[0018] Against this background, PUFs are one of the most promising and cost-effective hardware security primitives for key generation and authentication in cyberspace. Generally, PUFs utilize unique physical variations that naturally occur during the device manufacturing process, and a cryptographic key is generated by mapping a given input (i.e., a "challenge") to an output (i.e., a "response") to form a challenge-response pair (CRP) (e.g., an electrical, mechanical, or optical signal in the time / frequency domain). Therefore, a physical unclonable function (PUF) is a class of hardware-specific security primitives based on a secret key extracted from an integrated circuit, which can protect critical information from cyberattacks and reverse engineering. Typically, PUFs can be classified into two main classes: strong PUFs, which can generate a large number of CRPs, and weak PUFs, which have only a limited number of CRPs. To date, the majority of PUFs are primarily based on digital electronics, namely, complementary metal-oxide semiconductor (CMOS) integrated circuit (IC) technology, including arbiter PUFs, static random access memory (SRAM) PUFs, memristor PUFs, and ring oscillator PUFs.

[0019] While CMOS digital products can have good robustness due to their high-precision micro / nano-fabrication, their application to PUFs is typically affected by their relatively low entropy and power consumption. As a result, CMOS-based PUFs are still potentially vulnerable to machine learning attacks based on predictive regression models and generative adversarial neural networks. Other emerging PUFs with improved randomness, such as quantum electronic PUFs, optical PUFs, and photonic PUFs, as well as PUFs based on randomly distributed nanostructured features, still require high implementation costs and system complexity.

[0020] Over the past decade, the physical properties of exceptional points (EPs) have attracted considerable attention, primarily due to their unusual effects and potential applications in optics, photonics, and electronics. EPs form when two or more eigenstates (eigenvalues ​​and corresponding eigenvectors) of a non-Hermitian Hamiltonian structure become identical. The onset of this unique degeneracy marks the collapse of the eigenspace dimensionality, thereby enhancing the structure's sensitivity to perturbations. This consideration recently led to the proposal to build sensing devices operating at EPs. Subsequent experimental studies have confirmed that EP-based sensors exhibit enhanced responsivity.

[0021] However, careful theoretical analysis and experimental results have raised doubts about the performance of these devices in terms of sensitivity and resolution, which are related to the signal-to-noise ratio. In particular, the presence of EPs leads to enhanced responsivity while amplifying noise by the exact same factor. On the other hand, more recent experimental results on EP-based mechanical accelerators indicate that there exists a regime where the signal enhancement outweighs the noise enhancement, thus demonstrating the benefits of using EPs for sensing applications.

[0022] Besides the noise issue, another problem with known EP-based sensors is that such sensors rely on the implementation of isolated outliers. This presents practical challenges as these systems are highly susceptible to fabrication errors and noisy environment(s), which degrade their performance. For example, two known experiments employed active tuning parameters after fabrication to fine-tune the system to the EP in order to address the sensitivity to nearby perturbations of the EP.

[0023] According to the principles herein, typical variations in the values ​​of standard electrical components (resistors, capacitors, and inductors) can be "amplified" when used to build electrical circuits that operate with special types of EPs, including polar singularities known as divergent EPs or DEPs. Exemplary EPs and DEPs are presented herein. This unique singularity in non-Hermitian physical systems can be experimentally demonstrated, as described herein. The presented results pave the way for building a new generation of hardware-based cryptographic architectures that surpass previous PUF schemes.

[0024] This disclosure considers the extreme eigenvalue sensitivity of systems with EP from different perspectives and demonstrates their usefulness for security applications. In particular, in certain embodiments of this disclosure, EP and DEP-based circuits can be excellent candidates for generating robust, high-quality radio frequency (RF) PUFs, and such circuits can be generalized to realize secure wireless authentication (e.g., RFID and wireless access control) and NFC systems. Furthermore, according to various embodiments, coherent perfect absorber laser (CPAL) devices implemented using parity-time symmetric circuits can be used to generate unique encryption keys.

[0025] 1(a) depicts a general architecture of a PUF-enabled RF wireless identification and communication system 100 having a transmitter (Tx) or reader circuit device 102 that transmits a challenge signal (e.g., an RF pulse signal) (generated by a challenge generator circuit (not shown)) to a receiver (Rx) or tag circuit device 104 that is stimulated by the challenge signal and generates a response in either the Rx or Tx circuit. In various embodiments, the response is converted into an encryption key 108 via a converter circuit (e.g., an analog-to-digital (A / D) converter circuit) 106.

[0026] The security key of this exemplary system is encoded in the unavoidable, non-repeatable manufacturing errors in the values ​​of the electrical components (resistors, capacitors, and / or inductors) used to construct the receiver circuit 104. These manufacturing errors equip each individual circuit with a unique fingerprint that can serve as a PUF-based encryption key 108, which can be looked up as follows: When a transmitter / reader and receiver / tag are paired for PUF encryption as a secure radio frequency identification system, the reader (Tx) stimulates the Rx by initiating an RF pulse known as a "challenge." The time response of the Rx depends strongly on the eigenmodes of the Rx in the combined system, and these eigenmodes are a function of the exact values ​​of the Rx's electrical components.

[0027] Thus, according to various embodiments, different tags exhibit unique time responses given by the instantaneous voltage measured across a capacitor in the reader, and such time responses are referred to as "responses" for short. The time responses are then digitized (e.g., via analog-to-digital conversion) to generate a 256-bit (or greater) string identifier (ID) for a given challenge. Accordingly, Figure 1(b) shows exemplary plots of radio frequency (RF) challenges, time responses, and bit string processing, according to various embodiments of the present disclosure.

[0028] According to various embodiments, once this digitized ID passes validation by a specific IoT database via the Tx (reader), the access request of the Rx (tag) is authenticated. Meanwhile, in various embodiments, when this reader-tag scheme is used for secure wireless communication, the RF signal transmitted by the Tx introduces a unique voltage response drop across the capacitor of the Rx in the time domain. According to various embodiments, only when such time response is digitized and verified by the Rx with a predefined verification is the encrypted data and / or information stored in the memory of the Rx allowed to be sent back to the Tx. Thus, secure wireless communication is achieved, effectively avoiding privacy disclosure.

[0029] An exemplary Tx-Rx architecture capable of implementing DEP in accordance with the principles herein is shown in the left panel of Figure 2(a). Here, the exemplary transmitter circuit comprises an active transmitter (-RLC oscillator) and one or more neutral-point intermediate circuits (LC oscillators) functioning as repeaters. Accordingly, the exemplary receiver circuit comprises a passive receiver (RLC oscillator). According to various embodiments, the RLC oscillator and -RLC oscillator are configured to be inductively coupled via an on-chip transformer, or capacitively coupled via an on-chip capacitor or negative capacitance converter, or both inductively and capacitively coupled via the aforementioned electronic components.

[0030] To evaluate the performance of these DEP-based PUFs, we can compare their performance in terms of security metrics with other circuits implementing EP and non-EP systems, both shown in the left panels of Figures 2(b) and 2(c). The corresponding pseudospectrums are plotted in the right panels of the figures. These plots thus confirm the extreme eigenvalue sensitivity associated with DEP compared to traditional EP, which exceeds the sensitivity of systems without any EP at all.

[0031] The extreme sensitivity of DEP circuits can be better understood by carefully examining the natural frequency of the DEP circuit, which is expressed herein in units of natural frequency ω = 1 / LC.

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[0032] In the exemplary equation, L and M are the self-inductance and mutual inductance of the two coil antennas. From equation (1), the two branch real natural frequencies ω ± is γ EP= It is easy to verify that the point κ=1 / 2 is degenerate. Further analysis confirms that the corresponding eigenstates are also identical, i.e., this point is indeed an EP. Furthermore, the point κ=1 / 2 represents a polar singularity where the eigenfrequencies diverge. The system can be considered to be in an intermediate regime where the singularity enhances the eigenfrequency splitting but does not cause divergence; therefore, the system can be unambiguously defined within the context of a linear circuit, where this DEP splits the system into exact and broken PT-symmetric phases.

[0033] Exemplary systems can be designed to operate accurately with DEP in accordance with the principles herein. Due to the strong bifurcation around this point, any small deviation in the circuit's component values ​​can result in substantial drift in the natural frequency and consequent response to external excitation. This provides a basis for system designs that utilize DEP systems to produce high-quality PUFs by leveraging the process variations that naturally occur in electronic components.

[0034] To elucidate the effect of random physical variations on the intrinsic spectrum of a system, the following scenario may be considered: It is well known that fabrication errors can result in typical variations in the values ​​of electrical components within an approximate range of about 0.001 to 0.05, and such values ​​are close to the percentage errors (0.1%, 5%) found in realistic electrical components. Accordingly, an exemplary combination of DEP circuitry (see Figure 2(a)) can be considered, where the resistor and capacitor (defined by variables R and C) at the receiver end are:

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[0035] Figures 3(a) and 3(b) show plotted distributions of the real and imaginary parts of the eigenfrequency as a function of the desired γ (when the uncertainty in the electronic components R, L, and C is 0%) for a coupling coefficient κ = 0.7. Figure 3(a) illustrates that the real part of the eigenfrequency is randomly distributed around the DEP, showing dark regions inferring high uncertainty and low detection probability. In contrast, when the system operates away from its DEP, the real part of the eigenfrequency has a narrow distribution around the mean value, as indicated by the lighter colors (i.e., high probability) in Figure 3(a). Furthermore, the imaginary part of the eigenfrequency shown in Figure 3(b) has a high probability of being zero at the correct PT phase. This statistical analysis shows that even a typical 4% standard deviation in resistance and capacitance values ​​can result in a highly random eigenspectrum in the vicinity of the DEP, thereby providing an ideal entropy source for PUF and true random number generator applications.

[0036] After establishing the extreme spectral sensitivity of a DEP-based electronic system operating at or near an EP, we can evaluate its performance when the EP is used as an RF-PUF for identification, as illustrated in Figure 1(a). To do so, we can consider two standard metrics: entropy and Hamming distance. Entropy quantifies the randomness of the generated bits for a single device and for different devices, while Hamming distance quantifies how different each device is from other devices.

[0037] In a practical implementation shown in Figure 2(a), a transmitter / reader circuit can be used to query a fully passive ID tag (i.e., an RLC oscillator). The transmitter / reader circuit consists of an active RLC oscillator connected to a neutral LC circuit and a challenge generator circuit (such as a pulse generator). Together with the receiver circuit, the entire circuit forms a third-order PT symmetrical electronic system.

[0038] When the system is turned on, the voltage across the transmitter / reader capacitor can be used to extract the security key and enable wireless access control. One of the minimum requirements for a PUF is the randomness of its keys. Ideally, the bitmap extracted from the transient voltage response should have an unbiased distribution of “0” and “1” states. A highly random two-dimensional bitmap distribution (as shown in Figure 1(a)) has a high entropy pair (E x , E y ) is characterized by E x =-[p x log2p x +(1-p x )log2(1-p x )] E y =-[p y log2p y +(1-p y )log2(1-p y )] (2) In the formula, p x , py are the probabilities of getting the digit "1" along the x-axis and y-axis, respectively. In the case of an ideal random source, the probability of getting a "1" (p x ,p y ) and "0" (1-p x ,1-p y ) are expected to be 50% for both distributions, and the maximum entropy E x , E y= 1, i.e., both are 1. The encryption key 108 in FIG. 1(a) illustrates a bitmap (white means 1, black means 0) generated using the circuit in FIG. 2(a), and FIG. 4(a) plots the entropy function along both axes, E x and E y is close to 1, thus providing high randomness.

[0039] In particular, FIGS. 4(a) and 4(b) show (a) bitmaps and (b) entropy (E) of 256-bit PUF responses from 100 PUF instances for an exemplary DEP-based RF PUF system according to various embodiments of the present disclosure. x , E y ) analysis. Here, the average entropy is found to be 0.93, 0.06, and 0.91, 0.05 along the x-axis and y-axis. Meanwhile, Figure 4(c) plots the inter-device Hamming distance (HD) (defined as the count of different bits between two CRPs under the same challenge) for 100 RF-PUF instances generated in the vicinity of a DEP. Here, the original HD is normalized by the length of the bit string. The inter-HD is measured at 15°C, and the intra-HD is measured at 0, 5, 15, 20, and 25°C.

[0040] The inter-HD can be well fitted by a Gaussian distribution centered around μ = 0.5016 and with an approximate standard deviation σ = 0.0393. This indicates that DEP-based RF-PUF devices indeed exhibit unique responses. It is worth noting that device uniqueness can also be regarded as the degree of correlation between the one-dimensional digitized keys of two different PUFs. The one-dimensional keys from any two different PUF units should be uncorrelated, with the normalized inter-HD equal to 0.5, if possible. A long or short normalized inter-HD between two CRPs degrades the encryption quality, thereby making it possible to decrypt an unknown CRP from another known CRP.

[0041] Another important metric for characterizing the performance of a PUF device is its reliability, defined as its ability to generate the same key after the same challenge is repeated. In other words, the response associated with the same challenge should not change over time, even when environmental conditions (e.g., the temperature of electronic components) change. A reliable RF-PUF system should be sufficiently resistant to temperature changes.

[0042] The effect of temperature on the exemplary devices and systems herein and various architectures associated therewith can be determined, for example, by assuming that each lumped element in FIG. 2(a) has a temperature dependence with a realistic temperature coefficient of resistance or reactance, e.g., R=R ref Assume that [1+α(T-Tref)], where α=1×10 -6Generally, chip-based RF elements can have constant impedance and low noise over a wide temperature range (e.g., 0 to 25°C), and unlike PUFs based on nanomaterials and nanophotonic devices, protective packaging for electronic components can prevent PUF devices from being oxidized and contaminated by physical, chemical, and biological sources. Therefore, for example, RF-PUFs based on printed circuit board or on-chip integrated circuit technology can have very high robustness and reliability.

[0043] To verify the reproducibility of the key, we can compare the intra-HD obtained from five temperature conditions: 0, 5, 15, 20, and 25°C. The results reported in Figure 4(b) show that the intra-HD evaluated at different temperatures is centered at μ = 0.0185 and has a standard deviation σ = 0.0137, which is close to the ideal case where the mean value is zero. Ideally, the mean inter-HD should be 0.5, which can occur when the average is half the bit length, while the mean intra-HD should be close to 0, and their formula is:

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[0044] The sharp contrast in the color map in Figure 4(c) shows a clear difference (i.e., small variation around a mean of 0.5) between the intra-HD of a particular PUF instance (i.e., 0) and the inter-HD between two different PUF instances. Thus, the results obtained in accordance with the principles herein verify that it is possible to build a lightweight, robust solution for ensuring wireless authentication and access control for devices and systems built in accordance with the principles of the present disclosure. Next, the coding capacity can be evaluated. The coding capacity can be defined as the potential number of codes that can be generated by a PUF instance.

[0045] The coding capacity is c n where c=2 (i.e., "0" and "1" in binary) and n is given by n=μ(1-μ) / σ 2 where μ is the average probability and σ is the standard deviation. Based on the results in Figure 4(b), n = 0.5016(1 - 0.5016) / 0.0393 2 ≒ 162, and c n =2 162 ≒5.8×10 48 is.

[0046] Next, the performance of the DEP-based PUF device can be compared with a device that implements standard EP (i.e., does not have polar singularities), as well as a device that does not rely on EP at all. Figure 4(e) then plots the inter-HD histograms obtained from the three different RF-PUF systems considered here. For the third-order PT-symmetric electronic system, the magnetic coupling strength κ = 0.7 = 0.99κ DEP , κ1=0.97κ DEP , and κ 3= 0.85κ DEP Three different cases corresponding to

[0047] On the other hand, a standard (second-order) PT symmetric electronic system used κ = 0.7, ensuring operation near the EP. A conventional telemetry system can be formed by an RLC resonator coupled to a coil antenna with a self-inductance L. In all cases, the input impedance of the RLC oscillator is assumed to follow a Gaussian statistical distribution, which is also used to generate the simulation results in Figure 3(a). Figure 4(d) shows that the conventional telemetry configuration acts as a low-entropy source with a biased distribution of 1s and 0s, downshifting the average inter-HD to 0 and thus reducing the uniqueness of the key. While the EP-responsive bifurcation effect in a standard PT electronic system can increase entropy, its performance still lags far behind that of a third-order PT electronic system operating near the DEP.

[0048] As can be seen from Figure 4(e), the HD of the DEP PUF is highly concentrated around 0.5. Conversely, the HD distribution of the EP-PUF is wider than that of the DEP PUF. Therefore, the PUF performance of the EP PUF is not as good as that of the DEP PUF. This wider distribution of the HD of the EP PUF also results in a smaller coding capacity because it has a larger standard deviation. For the same third-order PT telemetry system, the κ / κ ratio decreases as the system operates farther away from the DEP. DEPAs the ratio decreases, the entropy decreases. The conclusion of this comparison is that implementing a PUF device using DEP-based electronic circuits can significantly increase the entropy and uniqueness of the system, exceeding those of standard devices.

[0049] The exemplary PUF devices described herein can be utilized not only for wireless identification applications, but also for securing near field communication (NFC) or low-power wireless sensors, as illustrated in Figure 1(a). When an NFC transmitter / reader circuit device is paired with a receiving tag or sensor via inductive coupling, a challenge signal (e.g., a pulse signal) transmitted from the reader circuit can generate a unique transient response on the capacitor of each receiving circuit device, and this transient response can be used as a PUF-based encryption key.

[0050] The PUF-based key detected by the receiving device must be verified before transmitting the information stored in the receiving device's digital memory, thus preventing third-party involvement and avoiding the leakage of confidential information. To demonstrate the usefulness of such a device, we evaluated its merits using numerical experiments similar to those associated with Figures 5(a)-5(d). In particular, we used 100 readers with different RLC circuits to query the same receiving device. Here, the lumped elements of the 100 readers are assumed to follow the same Gaussian distribution as previously used.

[0051] The entropy plot in Fig. 5(a) clearly shows a near-ideal scenario that guarantees high-quality randomness. The inter-HD shown in Fig. 5(b) is centered at μ = 0.4975 and σ = 0.0380, which represents excellent uniqueness and security properties that can also be validated by the pairwise evaluation in Fig. 5(c). Furthermore, the robustness analysis shown in Fig. 5(b), quantified by intra-HD at different temperatures, is also promising, with μ = 0.0179 and σ = 0.0128.

[0052] In various embodiments, an exemplary security system of the present disclosure may utilize an RF PUF encryption system. In one such embodiment, the system may include an NFC reader programmed to selectively pair with at least one of a tag or a sensor via inductive coupling and having circuitry programmed to generate a pulse signal output from the reader circuit that includes a unique transient response to a capacitor in the receiving device. The unique transient response may serve as a PUF-based encryption key. The PUF-based encryption key may include verifiable data that prevents a response to mismatched data, such that information stored in the receiving device's digital memory is not transmitted when a mismatch is received. Thus, verification prevents third-party involvement and avoids the disclosure of sensitive information.

[0053] Accordingly, FIG. 6 illustrates a block diagram of an exemplary DEP-based RF-PUF encryption system for use in radio frequency (RF) identification (RFID) and wireless access control. Operation of such a system may include pairing an identification tag (receiver circuit) 604 with a wireless reader (transmitter circuit) 602 such that DEP conditions are met. Operation may further include initiating a challenge signal (e.g., an RF pulse) by the reader 602 to query the ID tag 604. In response, the reader 602 requests or obtains a transient RF response after transmitting the challenge signal and discretizes and digitizes the RF response to obtain a binary PUF encryption key. In various embodiments, a database or memory unit of the reader 602 identifies and validates or revokes the PUF encryption key by comparing the key to valid identifiers stored in a database (e.g., a CRP database) or memory unit. Based on whether the PUF encryption key is validated, access is either denied or granted by the reader 602. For example, access may be associated with, among other things, as a non-limiting example, unlocking an electronic lock to a room or compartment, or, as another non-limiting example, providing access to a computer resource.

[0054] FIG. 7 further illustrates a block diagram of an exemplary DEP-based RF-PUF encryption system used in an exemplary near-field communication (NFC) and authentication process. Operation of such a system may include pairing a transmitter circuit 702 and a receiver circuit 704 such that DEP conditions are met. The operation may further include initiating a challenge signal (e.g., pulsed excitation) by the transmitter 702 to the receiver circuit 704. In response, the receiver circuit 704 requests or obtains a transient RF response after receiving the challenge signal and discretizes and digitizes the RF response to obtain a binary PUF encryption key. In various embodiments, a database or memory unit in the receiver circuit 704 identifies, validates, or revokes the PUF encryption key by comparing the key to a valid identifier stored in a database (e.g., a CRP database) or memory unit. Based on whether the PUF encryption key is validated, the data transmission request is either denied or permitted by the receiver circuit 704. For example, access may be associated with a user of the transmitter circuit 702 requesting access to a computer file or directory that is granted upon successful verification of the PUF encryption key, among other things, as one possible non-limiting example.

[0055] To demonstrate the effectiveness of the exemplary DEP-based RF-PUF encryption system of the present disclosure, the National Institute of Standards and Technology (NIST) randomness test suite was applied to evaluate and validate the performance of the exemplary DEP-based RF-PUF encryption system of the present disclosure. Results show that all nine NIST tests were successfully passed for both authentication and communication applications. Such results again demonstrate that the DEP-based RF-PUF system can be a true random number generator that provides superior randomness and uniqueness for securing short-range wireless access and communications.

[0056] The principles described herein allow us to exploit the extreme sensitivity of PT-symmetric electronic systems near EPs to build new types of physical systems, devices, and schemes (architectures). Such PUF-based lightweight cryptosystems may enable secure authentication and message exchange between systems / devices / schemes. In particular, the unprecedentedly large divergence of eigenvalues ​​that occurs near divergence exceptional points in high-order (i.e., third-order or higher) PT electronic circuits can enhance the randomness, uniqueness, and coding capacity of PUFs generated by the inevitable physical differences between devices due to uncontrolled variations in electronic component values. Results also demonstrate that this new PUF paradigm can function as a perfect entropy source or cryptographic random number generator for encryption and authentication in large-scale wireless communication and identification applications.

[0057] According to embodiments of the present disclosure, we demonstrate that the extreme sensitivity of PT-symmetric electronic systems near the EP can be used in accordance with the principles herein to construct a new type of physically unclonable function (PUF) scheme, and that such a PUF-based lightweight cryptosystem can enable secure authentication and message exchange between devices. In particular, the unprecedentedly large divergence of eigenvalues ​​observed near high-order (i.e., third-order or greater) divergence exceptional points of PT electronic circuits can enhance the randomness, uniqueness, and coding capacity of PUFs generated by the inevitable physical differences between devices due to uncontrolled variations in electronic component values. The results also demonstrate that this new PUF paradigm can function as a perfect entropy source or true random number generator for encryption and authentication in large-scale wireless communication and identification applications. The results herein open up entirely new research avenues that exploit the implications of non-Hermitian physical properties in suitable circuits for new classes of applications, such as hardware security, particularly in electronics platforms.

[0058] The disclosed technology can be utilized to develop cybersecurity devices and systems for hardware security that can evade even machine-learning-assisted cyberattacks. In various embodiments, such devices and systems herein can plug into existing devices or, if desired, function as standalone modules. The spectral sensitivity associated with exceptional points (EPs) can be used to build optical and electronic sensors with enhanced sensitivity. The spectral sensitivity associated with EPs can be used as a resource for hardware security. In particular, the present disclosure introduces a physically unclonable function (PUF) by exploiting the rare spectral singularity of divergent exceptional points (DEPs) present in high-order parity-time (PT) symmetric electronic systems. The extreme eigenvalue divergence near the DEPs can significantly enhance the stochastic entropy caused by the inherent parameter variations of electronic components, resulting in a perfect entropy source for generating cryptographic keys encoded as analog electrical signals (e.g., radio waves). Results demonstrate that the DEP-enhanced entropy source can enable PUFs with truly random and unique inter-device variability while achieving excellent robustness evidenced by small intra-device variability. This lightweight and robust PUF structure can lead to a variety of unexpected security and anti-counterfeiting applications in radio frequency fingerprinting, anti-counterfeiting, wireless communications, etc.

[0059] Security issues are paramount in Internet of Things (IoT) and radio frequency identification (RFID) applications. As mentioned above, physical unclonable functions (PUFs) may be one of the most promising hardware security technologies, leveraging inherent randomness introduced during manufacturing to provide physical entities with unique "fingerprints" or trust anchors. A PUF mechanism inspired by parity-time (PT) symmetry in the field of quantum physics is described herein. An exemplary electronic analogy of a PT-symmetric quantum system (e.g., a PCB or IC chipset) with a non-Hermitian effective Hamiltonian and an exceptional point is described. As discussed, an exceptional point is a singularity in the system's eigenspectrum, which results in rare eigenvalue bifurcation effects. When a system operates around this singularity, manufacturing errors (i.e., device-to-device variations) can make the output response of a PT-symmetric circuit highly unpredictable and unrepeatable.

[0060] Furthermore, when wireless telemetry systems (e.g., RFID or NFC) are implemented in PT-symmetric electronic structures, the highly uncertain output response can be utilized for cryptographic and security applications (i.e., a true random number generator). For example, when a group of RFID or NFC tags (Rx) are fabricated and wirelessly queried by a PT reader (Tx), the output RF responses of those tags can be unique, as manufacturing variations in electronic components (e.g., resistors, capacitors, transistors, etc.) can result in very different electrical output signals being detected by the reader. This opens new possibilities for next-generation wireless access control and secure wireless communications with outstanding cryptographic performance metrics. Other devices and systems are also contemplated in accordance with the principles herein. The devices / systems / schemes herein provide promising cryptographic systems configured to defend against machine-learning-assisted attacks to secure wireless communications.

[0061] While traditional software security relies on pseudorandom number generators, which are known to be vulnerable to machine learning-assisted attacks, the exemplary PT PUF as a new hardware security system / device / method is a simple, low-cost true random number generator that is robust against adversarial attacks. Most importantly, this hardware security technology is compatible with existing wireless and telemetry systems such as Near Field Secure Communication (NFC) and Radio Frequency Identification (RFID).

[0062] Advantageously, a security system constructed in accordance with the principles herein, in an exemplary embodiment, can include two or more RF-PUFs having a first PUF one-dimensional key and a second PUF one-dimensional key. The first and second one-dimensional keys are uncorrelated, and the normalized inter-HD is equal to about 0.5 for the group of PUFs. Furthermore, a security system constructed in accordance with the principles herein, in an exemplary embodiment, can include an RF-PUF further defined by a divergent exceptional point (DEP)-based RF-PUF. Furthermore, a security system constructed in accordance with the principles herein, in an exemplary embodiment, can include an NFC reader having circuitry programmed to selectively pair with at least one of a tag or a sensor via inductive coupling and generate a pulse signal output from the reader circuitry including a unique transient response to a capacitor of the receiving device, the unique transient response serving as a PUF-based encryption key. A security system constructed in accordance with the principles herein may also, in an exemplary embodiment, include a PUF-based cryptographic key containing verifiable data that prevents responses to non-matches and prevents transmission of information stored in the system's digital memory when a non-match is received, thus preventing the involvement of third entities and avoiding the disclosure of sensitive information. Thus, a security system constructed in accordance with the principles herein may prevent generative adversarial network-based machine learning-assisted cyber attacks.

[0063] As previously mentioned, according to various embodiments, DEP-based circuits may be excellent candidates for creating PUF devices within coherent perfect absorber laser (CPAL) devices to generate unique cryptographic keys. In accordance with the principles herein, new lightweight, low-cost PUFs can be developed that are significantly less predictable and, most importantly, more resilient to machine learning-assisted attacks, and that are easily implemented as hardware security primitives.

[0064] Since the experimental validation of acoustic, optical, opto-mechanical, and photonic analogs of parity-time (PT)-symmetric non-Hermitian Hamiltonians across the spectrum, the field of non-Hermitian physics continues to blossom, leading to many promising applications such as unidirectional invisibility, non-reciprocal unidirectional optical devices, coherent perfect absorber lasers (CPALs), high-performance telemetry for sensing, and wireless power transfer.

[0065] PT symmetry has revolutionized the design paradigm of wave propagation and scattering by extending wave control into non-Hermitian domains. However, it has recently been reported that at or near singular points of PT systems, where the completeness and continuity of the Hamiltonian eigenbasis are broken, e.g., exceptional points (EPs) or CPAL points, the distribution of complex eigenvalues / eigenstates can be highly disordered and noisy. This can cause significant sample-to-sample variations that affect the reproducibility and scalability of non-Hermitian physical systems with exceptional points. In another situation, the high entropy near exceptional points or exceptional / CPAL points can be exploited to generate high-quality PUF-based cryptographic keys. At exceptional / CPAL points, the output of the system can be switched from laser mode to its time-reversed mode, i.e., coherent perfect-body absorption (CPA) mode, by adjusting the complex-valued amplitude ratio of the incident waves.

[0066] In particular, due to the self-dual singularity at the exception / CPAL point, both the CPA mode and the laser mode are narrowband effects. More interestingly, both modes are subject to variability in the material properties of the gain and loss elements and their coupling velocities. In this regard, even small but unavoidable process variations in the fabrication of PT-symmetric structures can produce very different output responses from device to device. While such characteristics may be considered an enemy for sensing purposes, they may find useful applications in generating high-quality PUF keys. In particular, the rapid emergence of manufacturing technologies in the microelectronics industry minimizes the inherent device variability on electronic or photonic microchips, thereby posing a challenge to improving the randomness and uniqueness of digital-circuit-based PUFs. In this regard, exemplary PUF keys generated from frequency-domain electromagnetic responses may significantly enhance the entropy inherent to manufacturing defects, thereby enabling high cryptographic randomness and uniqueness, given the extreme sensitivity at the self-dual singularity-CPAL point.

[0067] This paper describes a novel electromagnetic PUF paradigm that exploits the probabilistic and variability characteristics between CPAL devices to generate unique cryptographic keys. For example, its optical realization is illustrated in Figure 8(a), which shows a schematic diagram of an exemplary CPAL PUF system implemented using active and passive metasurfaces in the optical domain. Here, manufacturing variations can cause random variations between unit cells (or meta-atoms), resulting in device-to-device variations. Accordingly, Figure 8(b) shows a two-port transmission line network (TLN) model of the generalized PT-symmetric CPAL device of Figure 8(a), which consists of spatially distributed, balanced gain and loss, and the shunt / surface conductivities—G (gain) and G (loss)—of the model, separated by a transmission line segment with an electrical distance x = kd, where k is the propagation constant and d is the physical length. This generalized PUF paradigm can be easily translated to any physical system spanning the broad electromagnetic spectrum, including optics, photonics, radio frequency (RF), and microwave electronics. In the infrared and optical regions, as shown in Fig. 8(a), the equivalent TLN model in Fig. 8(b) can be implemented, for example, using a pair of active and passive electromagnetic metasurfaces separated by a thin dielectric spacer.

[0068] In the RF and microwave domains, the structure can be implemented using integrated circuit or printed circuit board (PCB) technology, where the negative resistance converter (NRC) and shunt resistor are separated by a transmission line or a section of a T / π transformer. In the PUF key retrieval process, the nonlinear analog output response of a CPAL-based PUF instance can be appropriately discretized and digitized into a bit-string-based authentication code with excellent cloning resistance, as shown in Figure 8(c). Here, two incident waves (α = ψ i - / ψ i + Different challenges achieved using different complex amplitude ratios of (C n ) to the PUF device to generate a unique response (R n) The response is then discretized and digitized as a binary bit string to form a digital cryptographic map for PUF applications. The output response in Figure 8(c) is a simulation result.

[0069] The PUF key retrieved from the output response of a CPAL-based PUF instance can be reconstructed by adjusting the complex amplitude ratio of the incident wave. Such a property may not be possible in most CMOS-based digital PUFs, where each instance corresponds to one or more bits in the response. Therefore, increasing the number of CRPs comes at the expense of increased size, device area, and design complexity.

[0070] The scattering parameters of the two-port TLN model in Figure 8(b) can be calculated using the transfer matrix method. The CPAL point can be obtained by setting x = π / 2,

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[0071] Naturally occurring process variations in device dimensions, defects, and material profiles can cause variations in shunt conductivity (see Figure 8(b)). Here, we simply assume δG = δG' and the electrical distance (assumed here to be δx), thus resulting in different output responses for CPAL device combinations. As an example, when a CPAL device is initially operated in CPA mode (i.e., challenged by φ = π / 2), the output response as a function of the normalized conductivity perturbation ν = δG / Y0 can be written as follows:

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[0072] In an ideal scenario where (ν,δx)=(0,0), a null output response is obtained. Considering that manufacturing imperfections are small, i.e., δx,ν<<1, applying a Taylor series expansion to equation (4) gives

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[0073] Such a result shows that the output is sensitive to ν and 2 Furthermore, the sensitivity of the output response is 1 / (δx) 2 The sensitivity is enhanced by a factor of . Due to small changes in (ν,δx), it is possible to envision that the device can hop from a low scattering (absorption) mode to a high scattering (emission) mode, resulting in a significantly different scattering response and net emitted energy flux. Note that a similarly sensitive output response can be obtained even if the system is initially operated in laser mode (i.e., challenged by φ=π / 2). Figure 9(a) shows the sensitivity as a function of δx and ν, plotted using equation (4).

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[0074] By comparing the simulation results in Figures 9(a) and 9(b), we can see that CPAL PUF instances exhibit greater variability in their output responses and therefore potentially higher entropy than those of their passive / active FPI instances due to the self-dual singularity of the CPAL point. Such diverse distributions of output responses caused by device-to-device variations can ensure high randomness and uniqueness for PUF applications. This disclosure also considers the sensitivity of output responses to v and δx for the same PT system operating at the EP. Theoretical results show that devices operating near the CPAL point can exhibit greater variability in output responses and therefore higher randomness than devices operating near the EP.

[0075] In this disclosure, an exemplary electromagnetic PUF paradigm is experimentally demonstrated in the RF domain. However, it should be noted that the disclosed concepts can be similarly implemented in the fields of optics, photonics, opto-mechanics, and possibly acoustics, as long as the equivalent TLN model in FIG. 8(b) is valid. At low frequencies, the CPAL-based PUF instance shown in FIG. 8(b) can be implemented using lumped-element circuitry on a PCB or monolithically integrated chipset, as shown in FIG. 10(a), where the transmission line segments can be converted into compact integrated "T" networks and the shunts G and G can be realized using NRCs and resistors, respectively. FIG. 10(a) also shows a photograph of a prototype of a CPAL PUF instance on a substrate. In this prototype, the NRC is based on a single current-feedback operational amplifier. To effectively evaluate the PUF performance, 25 PUF instances were built and tested. Due to manufacturing process tolerances, the NRC was found to be insufficient at a CPAL frequency of 16.7 MHz.

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[0076] Another important performance metric, PUF uniqueness, can be evaluated by the inter-device Hamming distance (or inter-HD) between the (digitized) response bit strings of all PUF instances under the same challenge. The ideal inter-HD should be 0.5. This means that, on average, half of the response bits are not repeated, thus ensuring the best quality of encryption from a statistical point of view. Figure 10(d) shows a pairwise comparison of inter-HD among 25 PUF keys. We can see that the inter-HD in the off-diagonal region fluctuates only slightly around the average value of 0.46. This result suggests that all PUF keys have high uniqueness, i.e., each key is unique, completely uncorrelated, and unpredictable from history. It is worth noting that PUFs can be classified into strong and weak categories according to the number of CRPs. A PUF is considered strong (weak) if the number of CRPs increases or decreases exponentially (linearly or polynomially) with its size. Weak PUFs with a limited number of CRPs are often used for cryptographic key generation in identification applications, while strong PUFs that can generate a large number of CRPs are utilized for authentication and secure communication applications. The output response of a CPAL device is a function of the input challenge (i.e., JPEG2025537263000022.jpg6144 ), it may be possible to create a large CRP space by adjusting both the amplitude and / or phase offset between the two incident waves. This disclosure illustrates the above idea by applying two challenges to a CPAL-based PUF instance: laser (φ=0) and CPA (φ=π / 2), where the phase shift is varied.

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[0077] For comparison, we also constructed 25 PUF instances based on conventional passive and active FPI structures, as illustrated in Figure 11(a). The lumped elements used to construct these FPIs and input challenges are the same as those used in CPAL devices. The measured output responses of the passive and active FPI-based PUF instances are nearly identical, implying insufficient randomness and uniqueness. Figure 11(b) plots the bitmaps obtained from 25 passive FPI-based CPAs (applied with challenges φ=0 and φ=π), clearly showing a decrease in uniformity compared to the results in Figure 10(b). The entropy content of the active (open circles) and passive (filled circles) FPI-based PUFs is plotted in Figure 11(c). Under the same manufacturing tolerances, the passive FPI-based PUF key (E x = 0.54 ± 0.41, E y =0.86±0.04) and active FPI-based PUF key (E x =0.35±0.35, E yThe entropy of both the FPI CPA instances (=0.15±0.10) is much lower than that of the CPAL-based PUF key. Figure 11(d) reports the pairwise map of inter-HD for the passive FPI CPA-based PUF. From Figure 11(d), we can see that most FPI CPA instances are highly correlated, i.e., the extracted PUF key may be vulnerable to attacks. Figure 11(e) is similar to Figure 10(e), but obtained using active and passive FPI-based PUF instances. From these histograms, we can see that the average inter-HD value for the passive (active) FPI-based PUF is only 0.31 (0.12). Although the passive (active) FPI-based PUF instance is also initially locked in CPA (laser) mode, the resulting uniqueness or randomness is much worse than that of the CPAL-based PUF instance. The results in Figure 11(e) are in stark contrast to Figure 10(e), obtained using the CPAL-based PUF. Therefore, the existence of self-dual CPAL singularities in PT non-Hermitian systems indeed plays a key role in amplifying the output response deviations caused by device-to-device variations, thus providing a unique and hard-to-clone encryption key.

[0078] From a practical perspective, reliability refers to the essential consistency of CRP when subjected to environmental changes (e.g., ambient temperature). In PUF applications, reliability can be described by intra-device HD (intra-HD), which is defined as the bit error rate (BER) between responses generated by the same PUF instance under different operating conditions for a given challenge. To evaluate the reliability of the fabricated CPAL PUF circuit, each instance was measured at 11 different temperatures (-20°C to 80°C at 10°C intervals). The measured intra-HD histogram of the disclosed PUF is shown in the top panel of Figure 10(e). This histogram is normally distributed with a mean of 0.05 and a standard deviation of 0.05. Such values ​​are low enough to ensure good robustness against environmental changes. In addition to temperature stability, "dynamic" noise, such as phase / flicker noise and thermal noise introduced by charge diffusion, can also generate time variations in the system's response. We also investigated the time dependence of reliability by measuring the output response of six CPAL PUF instances every 30 seconds (for a total of 3 minutes) and calculating the Intra-HD of the generated CRP. The Intra-HD histogram associated with temporal stability is plotted in the bottom panel of Figure 10(e), which shows a near-zero mean, implying that CPAL PUFs may be robust to "dynamic" noise. In fact, previous studies on CPAL systems have shown that thermal noise, as the dominant noise source, contributes only slightly to the signal-to-noise ratio (SNR). Finally, we note that because the encryption key of a CPAL PUF exists in analog form (before digitization), its bit length can be arbitrarily long depending on how the discretization and digitization are performed. Such interesting properties enable the disclosed PUF to outperform conventional digital PUFs, such as multi-valued logic PUFs, monostable PUFs, and voltage divider PUFs, whose bit length cannot be increased without introducing a significant increase in the number of cells / bits. Furthermore, by adjusting the various input challenges, and therefore the complex-valued amplitude ratio of the two input waves, a potentially large number of CRPs can be generated.As a result, taking all these advantages into consideration, a single CPAL PUF instance can form a much larger CRP space than the aforementioned digital PUFs. Next, we consider the resilience of an exemplary CPAL PUF device against machine learning-assisted attacks. Machine / deep learning has emerged in recent years and can provide powerful support for pattern recognition, signal processing, and reverse engineering in electromagnetics. In particular, it has been reported to be a powerful tool for password guessing and decryption. Several recent studies have pointed out that some PUFs with relatively low randomness and uniqueness may also be vulnerable to attacks based on machine learning models, such as Fourier regression (FR) models and generative adversarial networks (GANs). FR- and GAN-based modeling attacks have also been performed against CPAL-based PUFs. FR-based modeling attacks do not require a large training database and are therefore widely used to attack PUFs. Figure 12(a) shows a. 0i , a ni , and b ni (n=1,2,…,N)) are the Fourier coefficients determined by least squares fitting (order of regression), and x i is a random input in the range (0, 1) i) is shown. Here, 8-, 16-, and 32-order regressions were performed, and only the 16-order results are presented because the 16-order results provide optimal performance. The FR model extracts randomness features from the training dataset (estimator CRP) to predict PUF responses. In the attack defense experiment, 50 256-bit measured CRPs obtained from 25 devices with two challenges were divided into a training dataset (40 estimator CRPs) and a test dataset (10 CRPs). After completing the training, the FR model was used to generate 10 CRPs to be compared with the test dataset. The performance of the PUF can be understood from the prediction accuracy (ACC), i.e., the number of correctly predicted bits as a percentage of the total number of predictions. Here, the correlation coefficient (CC), defined as the linear correlation level between the predicted CRP and the measured CRP, two random sequences, and the HD, was used to evaluate the resilience of the CPAL PUF against attacks. In an ideal scenario, the ACC, CC, and HD are expected to be sufficiently close to 50%, 0, and 0.5, respectively. Figures 12(b) to 12(d) report the results of FR modeling attacks. The average values ​​of ACC, CC, and HD are 54%, 0.03, and 0.46, respectively. Compared with the prediction accuracy of conventional silicon-based PUFs (typically over 90%), these results demonstrate superior robustness against FR-based modeling attacks.

[0079] As illustrated in Figure 13(a), a GAN-based modeling attack, which includes two deep neural networks (a generator and a discriminator), is another powerful password guessing tool. The discriminator is a binary classifier used to distinguish whether an input CRP belongs to a training dataset or is generated by a generator. Meanwhile, the generator attempts to generate fake CRPs similar to the training data to fool the discriminator. In many applications, such as image synthesis, a well-trained GAN can generate new data instances similar to the training data. As shown in Figure 13(a), the GAN structure adopted here includes a generator network with four layers and a discriminator network with three layers. Here, we adopted a 4 / 3 linear layer configuration because too many hidden layers can lead to convergence failure and reduce the model's prediction accuracy. Because the GAN structure requires a large amount of training data, we simulated 1,000 PUF instances, extracted from experimental results to account for manufacturing tolerances, and applied 10 input challenges (10 phase differences between two input signals) to generate 10,000 CRPs. 2,000 CRPs were used for testing, and the rest were reserved for training. The GAN was trained for 3,000 epochs. Without loss of generality, we replaced all "0"s in the CRPs to train the GAN with "-1". The distributions of ACC, CC, and HD between the GAN-predicted CRPs and the simulated CRPs are shown in Figures 13(b)–13(d), respectively. Here, we employ a normal probability mass function (PMF). We clearly see that the ACC is narrowly distributed and centered at 50%, which means that the CPAL-based PUF is resilient to GAN-based modeling attacks. The maximum (minimum) prediction accuracy is 73% (35%), corresponding to a probability of success of only 0.0001 (0.0003). Such results suggest that GANs cannot generate CRPs similar to those extracted from PUF instances. The average CC and HD are 0.07 and 0.46, further confirming that the exemplary PUF is resilient to machine learning-assisted attacks.

[0080] Briefly, this disclosure presents a robust, high-quality PUF primitive based on the CPAL effect, enabled by a PT-symmetric non-Hermitian electromagnetic structure. It is theoretically demonstrated that the self-dual singularity can make the output response of a CPAL-based PUF instance highly sensitive to inevitable device-to-device variations. Furthermore, experimental studies conducted on CPAL-based PUF keys implemented using RF circuits show that performance metrics, including randomness, uniqueness, correlation level, coding capacity, and thermal / temporal stability, can outperform PUF instances based on conventional passive and active FPIs. Furthermore, CPAL-based PUFs are highly robust against state-of-the-art machine learning-assisted attacks, such as FR and GAN modeling attacks. The disclosed PUF technique may pave a promising path toward next-generation identification, authentication, encryption, and security systems that can be widely applied in modern society. The disclosed hardware security paradigm can also be extended to other wave systems, such as photonics, acoustics, elasticity, and opto-mechanics.

[0081] Therefore, this disclosure proposes a type of PUF that emerges in the electromagnetic domain due to a self-dual emitter-absorber singularity uniquely present in a parity-time (PT) symmetric structure. At this self-dual singularity, reconfigurable emission and absorption properties with orders of magnitude difference in scattering power can sensitively respond to admittance or phase perturbations, for example, caused by manufacturing defects. Therefore, entropy arising from inevitable manufacturing variations can be amplified, resulting in superior PUF security metrics in terms of randomness and uniqueness. It has been shown that this electromagnetic PUF can be robust against machine-learning-assisted attacks based on Fourier regression and generative adversarial networks. Furthermore, the disclosed PUF concept is wavelength-scalable and transformative in radio frequency, terahertz, infrared, and optical systems, paving the way for promising applications in cryptography, including using the probabilistic and variability characteristics between CPAL devices to generate unique encryption keys.

[0082] It should be emphasized that the above-described embodiments of the present disclosure are merely possible examples of implementations, set forth merely for a clear understanding of the principles of the present disclosure. Many variations and modifications can be made to the above-described embodiment(s) without substantially departing from the principles of the present disclosure. All such modifications and variations are intended to be included herein within the scope of the present disclosure and protected by the following claims.

Claims

1. 1. A system comprising: a challenge generator circuit; a transmitter circuit coupled to the challenge generator circuit; a receiver circuit configured to receive the transmitted challenge signal and be stimulated by said transmitted challenge signal; the combination of the transmitter circuit and the receiver circuit comprises a parity time symmetric structure, the parity time symmetric structure operating at its exceptional point, divergence exceptional point, or coherent perfect absorber laser (CPAL) point; the transmitter circuit is configured to transmit one or more challenge signals generated by the challenge generator circuit to the receiver circuit; after transmitting the challenge signal(s), the transmitter circuitry and the receiver circuitry are configured to generate a unique transient response that depends on the eigenfrequency and harmonic response of the parity time-symmetric structure or on the eigenvalues ​​of a scattering matrix of the parity time-symmetric structure; the transmitter circuit or the receiver circuit is configured to measure the unique transient response and convert the measured unique transient response values ​​into a bit string, or to measure a spectral (frequency domain) response near the CPAL point and convert the measured unique spectral response values ​​of output coefficients into a bit string; The system, wherein the bit string comprises a Physical Unclonable Function (PUF)-based encryption key.

2. 10. The system of claim 1, wherein the receiver circuit and the transmitter circuit are complementary metal oxide semiconductor integrated circuits, III-V semiconductor integrated circuits, II-VI semiconductor integrated circuits, and / or a combination of two or more complementary metal oxide semiconductor integrated circuits, III-V semiconductor integrated circuits, or II-VI semiconductor integrated circuits.

3. 3. The system of claim 2, wherein the receiver circuit comprises an RLC oscillator having a positive resistance, and the transmitter circuit comprises a negative RLC oscillator and one or more LC oscillators.

4. 4. The system of claim 3, wherein the RLC oscillator and the −RLC oscillator are configured to be inductively coupled via an on-chip transformer, or configured to be capacitively coupled via an on-chip capacitor or negative capacitance converter, and / or are reconfigured to be both inductively and capacitively coupled via the aforementioned electronic components.

5. 4. The system of claim 3, further comprising a first digital memory unit accessible by the transmitter circuit, the transmitter circuit configured to verify the bit string against a valid bit string stored in the first digital memory unit.

6. 4. The system of claim 3, wherein the unique transient response comprises a voltage value measured across a capacitor of the RLC oscillator of the transmitter circuit or a receiver circuit of the RLC oscillator.

7. 7. The system of claim 6, further comprising a second digital memory unit accessible by the receiver circuit, the receiver circuit configured to verify the bit string against a valid bit string stored in the second digital memory unit.

8. 8. The system of claim 7, wherein after verifying the bit string, the receiver circuitry is configured to transmit the content stored in the second memory unit to the transmitter circuitry.

9. The system of claim 2 , wherein the challenge generator circuit comprises a pulse generator.

10. 1. A system comprising: a challenge generator circuit; a physical unclonable function (PUF) device configured to receive the challenge signal transmitted by the challenge generator circuit and to be stimulated by the challenge signal; the PUF device comprises a parity time symmetric structure, the parity time symmetric structure operating at its exceptional point, divergence exceptional point, and / or coherent perfect absorber laser (CPAL) point; after transmitting the challenge signal, the PUF device generates a unique transient response that depends on the eigenfrequencies and power harmonics of the parity time symmetric structure, or a unique spectral response that depends on the eigenvalues ​​of a scattering matrix of the parity time symmetric structure; the PUF device is configured to measure the unique transient and / or spectral response and convert the measured transient response values ​​into a bit string; The system, wherein the bit string comprises a Physical Unclonable Function (PUF)-based encryption key.

11. 11. The system of claim 10, wherein the CPAL-based PUF device comprises a pair of active and passive electromagnetic metasurfaces separated by a dielectric spacer.

12. 12. The system of claim 11, wherein the PUF device is realized by an equivalent lumped element circuit comprising a negative resistance converter (NRC) and a shunt resistor separated by a transmission line or a T / π transformer.

13. The system of claim 12 , wherein the NRC is implemented using cross-coupled pair (XCP) circuits.

14. The system of claim 12 , wherein the NRC comprises a current feedback operational amplifier.

15. The system of claim 11 , wherein the challenge generator circuit generates two coherent waves having a complex amplitude ratio.

16. The system of claim 12 , wherein two pairs of incident coherent waves with different complex-valued amplitude ratios produce different unique responses.

17. 1. A method comprising: pairing a transmitter circuit with a receiver circuit via inductive coupling; transmitting a challenge signal from the transmitter circuit to the receiver circuit to stimulate the receiver circuit; measuring a unique transient response dependent on a natural frequency of the combined system of the transmitter circuit and the receiver circuit, the natural mode being a function of a value of a physical property of the receiver circuit; and converting the unique transient response measurement into a bit string, the bit string comprising a Physical Unclonable Function (PUF) based encryption key.

18. validating the bit string by comparing it with a stored valid identifier; 18. The method of claim 17, further comprising: upon verifying the bit sequence, transmitting information stored in a memory of the receiver circuit to the transmitter circuit.

19. 20. The method of claim 18, wherein the receiver circuit comprises an RLC oscillator having a positive resistance, the transmitter circuit comprises an RLC oscillator and one or more LC oscillators functioning as repeaters, and wherein the RLC circuit, the RLC oscillator, and the one or more LC oscillators are based on a complementary metal oxide semiconductor integrated circuit, a III-V semiconductor integrated circuit, a II-VI semiconductor integrated circuit, and / or a combination of two or more complementary metal oxide semiconductor integrated circuits, III-V semiconductor integrated circuits, or II-VI semiconductor integrated circuits.

20. 20. The method of claim 19, wherein the unique transient response comprises a voltage value measured across a capacitor of the RLC oscillator or the −RLC oscillator.

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