CROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims priority to co-pending U.S. provisional application entitled, “Parity-Time Symmetric Hardware Security Systems and Methods,” having application number 63/424,725, filed November 11 , 2022, and co-pending U.S. provisional application entitled, “Electromagnetically Unclonable Functions Generated by Non-Hermitian Absorber-Emitter,” having application number 63/537,263, filed September s, 2023, each of which is entirely incorporated herein by reference.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0002] This invention was made with government support under awards NSF ECCS- 1914420 and NSF 2229659 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 the invention.

TECHNICAL FIELD

[0003] The present disclosure is generally related to techniques for implementing physically unclonable functions (PUFs) in electronic circuits.

BACKGROUND

[0004] Development of cryptographic techniques to ensure data security and privacy is critical and an urgent need in modem society. Nowadays, many cryptographic techniques have been proposed for identification, authentication, and anti-counterfeiting. Among various cryptographic techniques, physically unclonable function (PUF) has been widely regarded as one of the most promising hardware security primitives, which maps input challenges C _n to output responses R _n (so-called challenge-response pairs or CRPs) of a physical system for constructing cryptographic keys.

[0005] Due to unmanipulable variations arising from process variations inherent to manufacturing, CRPs are device-specific and prohibitively challenging to replicate, such that they can be used as unique and unclonable encryption keys. Perhaps, silicon-based PUFs, which exploit intrinsic variations in the complementary metal-oxide-semiconductor (CMOS) manufacturing process, are by far the most common PUF schemes. Up to date, a variety of silicon-based PUFs has been proposed, including static random-access memory (SRAM) PUFs, ring oscillator PUFs, phase change memory (PCM) PUFs, and memristor PUFs. Although these digital circuit-based PUFs benefit from the high throughput and reliability of CMOS technologies, they have been recently reported to be vulnerable to approximation and modeling attacks due to the limited inter-device and intra-die variations

SUMMARY

[0006] Embodiments of the present disclosure provide techniques for implementing physically unclonable functions (PUFs) in electronic circuits. One such system comprises 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. A combination of the transmitter circuit and the receiver circuit, which comprises a parity-time-symmetric structure that operates at an exceptional point, a divergent exceptional point, or a coherent perfect absorber-laser (CPAL) point of the parity time-symmetric structure. Additionally, the transmitter circuit is configured to transmit a challenge signal generated by the challenge generator circuit to the receiver circuit; after transmitting the challenge signal, the transmitter circuit and the receiver circuit are configured to generate a unique transient response that is dependent on eigenfrequencies and harmonic responses of the parity-time-symmetric structure or on the eigenvalues of the 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 a value of the measured unique transient response to a bit string, or is configured to measure the spectral (frequency-domain) response near the CPAL point and convert a value of the measured unique spectral response of output coefficient to a bit string; and/or wherein the bit string comprises a physically unclonable function (PUF)-based cryptographic key.

[0007] Embodiments of the present disclosure also present a method for implementing physically unclonable functions (PUFs) in electronic circuits. One such method comprises pairing a transmitter circuit and a receiver circuit through inductive coupling; transmitting a challenge signal from the transmitter circuit to the receiver circuit to stimulate the receiver circuit; measuring a unique transient response that depends on eigenfrequencies of a combined system of the transmitter circuit and the receiver circuit, wherein the eigenmodes are a function of values of physical properties of the receiver circuit; and/or converting a measured value of the unique transient response to a bit string, wherein the bit string comprises a physically unclonable function (PUF)-based cryptographic key.

[0008] In one or more aspects for such a method or system, the receiver circuit and the transmitter circuit are complementary metal-oxide semiconductor integrated circuits, lll-V semiconductor integrated circuits, ll-VI semiconductor integrated circuits, and/or a combination of these techniques via the heterogeneous integration; the receiver circuit comprises an RLC oscillator having a positive resistance and the transmitter circuit comprises a -RLC oscillator and one or more LC oscillators that act 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 coupled inductively via an on-chip transformer, or are configured to be coupled capacitively via an on-chip capacitor or a negative capacitance converter, or are configured to be coupled both inductively and capacitively using the aforementioned 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 method or system may involve verifying the bit string by comparing the bit string against a stored valid identifier; and/or upon verifying the bit string, transmitting information stored in a memory of the receiver circuit to the transmitter circuit. [0010] In one or more aspects, such a system may further comprise a first digital memory unit that is accessible by the transmitter circuit, wherein the transmitter circuit is configured to verify the bit string against a valid bit string that is stored in the first digital memory unit; and/or a second digital memory unit that is accessible by the receiver circuit, wherein the receiver circuit is configured to verify the bit string against a valid bit string that is stored in the second digital memory unit.

[0011] Embodiments of the present disclosure also present a system comprising a challenge generator circuit; and a physically unclonable function (PUF) device that is configured to receive and be stimulated by a challenge signal transmitted by the challenge generator circuit. As such, the PUF device comprises a parity-time-symmetric structure that operates at an exceptional point, divergent exceptional point, and/or coherent perfect absorber-laser (CPAL) point of the parity time-symmetric structure; after transmitting the challenge signal, the PUF device generates a unique transient response that is dependent on eigenfrequencies and output harmonics of the parity-time-symmetric structure or a unique spectral response that is dependent on the eigenvalue of the scattering matrix of the parity-time-symmetric structure; the PUF device is configured to measure the unique transient and/or spectral response and convert a value of the measured transient response to a bit string; and/or the bit string comprises a physically unclonable function (PUF)-based cryptographic 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; the PUF device can also be realized with an equivalent lumped-element circuit that comprises a negative resistance converter (NRC) and a shunt resistor separated by a transmission line or a T/7T-transformer; the NRC is implemented using a cross-coupled pair (XCP) circuit or a current-feedback operational amplifier; the challenge generator circuit generates two coherent waves with a complex amplitude ratio; and/or two pairs of incoming coherent waves having different complex-valued amplitude ratios generate different unique responses.

[0013] Other systems, methods, features, and advantages of the present disclosure will be or become apparent to one with skill in the art upon examination of the following drawings 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.

BRIEF DESCRIPTION OF THE DRAWINGS

[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.

[0015] FIG. 1 (a) illustrates physically unclonable function (PUF)-enabled secure radio-frequency (RF) authentication and communication in accordance with the present disclosure.

[0016] FIG. 1 (b) shows exemplary plots of radio frequency (RF) challenge, temporal responses, and bit string processing in accordance with various embodiments of the present disclosure.

[0017] FIG. 2(a)-2(c) shows a transmitter-receiver architecture that implements a wireless setup having a divergent excellent point (DEP), a standard excellent point (EP), and no singular points, respectively, along with corresponding pseudospectra, in accordance with the present disclosure.

[0018] FIGS. 3(a)-3(b) show distributions of (a) real and (b) imaginary parts of eigenfrequencies for a third-order PT telemetry system in accordance with various embodiments of the present disclosure.

[0019] FIGS. 4(a)-4(b) show a (a) bitmap and (b) entropy E _x , E _y ) analysis of a 256- bit PUF response from 100 PUF instances for an exemplary DEP-based RF PUF system in accordance with various embodiments of the present disclosure. [0020] FIG. 4(c) shows a plot of Inter-Hamming distance (inter-HD) and intra- Hamming distance (intra-HD) histograms obtained from 100 PUF instances for the exemplary DEP-based RF PUF system used in FIGS. 4(a)-4(b).

[0021] FIG. 4(d) depicts the inter-HD histograms obtained from three different RF- PUF systems of FIGS. 2(a)-2(c).

[0022] FIG. 4(e) shows inter-HD histograms of PUFs based on (1 ) a third-order PT system operating near DEP, (2) a third-order PT system operating away from DEP , and (3) a standard PT system operating near the exceptional point (EP); and (4) a conventional telemetry system using an NFC coil antenna.

[0023] FIG. 5(a) shows an entropy (E _x ,E _y ) analysis of a 256-bit PUF response from 100 readers with discrepant RLC circuits interrogating the same receiving device in accordance with various embodiments of the present disclosure.

[0024] FIG. 5(b) shows a plot of Inter-Hamming distance (inter-HD) and intra- Hamming distance (intra-HD) histograms obtained from the experimental setup of FIG. 5(a).

[0025] FIG. 5(c) depicts the inter-HD histograms obtained from three different RF- PUF systems of FIGS. 2(a)-2(c) the experimental setup used in FIGS. 5(a)-5(b).

[0026] FIG. 6 shows a block diagram for an exemplary DEP-based RF-PUF cryptographic system used in radio-frequency identification (RFID) and wireless access control in accordance with various embodiments of the present disclosure.

[0027] FIG. 7 shows a block diagram for an exemplary DEP-based RF-PUF cryptographic system used in near-field wireless communication (NFC) and authorization process in accordance with various embodiments of the present disclosure. [0028] FIG. 8(a) shows a schematic of an exemplary coherent perfect absorber-laser (CPAL) PUF system implemented using the active and passive metasurfaces, which form the PT-symmetric system, in the optical region in accordance with various embodiments of the present disclosure.

[0029] FIG. 8(b) shows a transmission-line network model of the CPAL PUF system of FIG. 8(a).

[0030] FIG. 8(c) shows a sequence of operations used in the cryptographic random number generation process of an exemplary CPAL PUF system in accordance with various embodiments of the present disclosure.

[0031] FIG. 9(a)-9(b) provide a comparison of randomness between (a) an exemplary CPAL-enabled PUF and (b) a Fabry-Perot Interferometer (FPI)-enabled PUF.

[0032] FIG. 10(a) shows a circuit diagram (top) and a photograph (bottom) of an exemplary CPAL PUF device realized using printed circuit board technology in the radiofrequency range.

[0033] FIG. 10(b) shows a bitmap measured over 25 CPAL PUF instances under two challenges (phase offset <p = 0, /2).

[0034] FIG. 10(c) shows measured entropies E _x and E _y for the bit-strings and keystrings of the CPAL PUF device of FIG. 10(a).

[0035] FIG. 10(d) shows a pairwise map comparing the inter-HDs between two arbitrary PUF devices, showing that the fabricated PUF keys are almost uncorrelated.

[0036] FIG. 10(e) shows measured inter-HDs and intra-HDs of an exemplary CPAL PUF by applying lasing (top) and CPA (bottom) challenges and their Gaussian-fitting results. [0037] FIG. 11 (a) shows a passive (top) and active (bottom) of FPI-enabled PUF devices and their onboard realizations in the radio-frequency region.

[0038] FIG. 11 (b) shows a bitmap measured over 25 passive FPI-based PUF instances of FIG. 11 (a).

[0039] FIG. 11 (c) shows measured entropy contents of passive FPI and active FPI PUF instances of FIG. 11 (a).

[0040] FIG. 11 (d) shows a pairwise map of the 25 passive FPI PUF instances of FIG. 11 (a).

[0041] FIG. 11 (e) shows inter-HDs of the active and passive FPI-based PUF instances of FIG. 11 (a).

[0042] FIG. 12(a) shows a schematic of a Fourier regression (FR) model.

[0043] FIGS. 12(b)-12(d) report the results of a FR modeling attack utilizing the FR model of FIG. 12(a).

[0044] FIG. 13(a) shows a schematic of a generative adversarial network (GAN) structure having a generator and a discriminator.

[0045] FIGS. 13(b)-13(d) shows probability mass functions (PMFs) of (b) prediction accuracies, (c) correlation coefficients (CCs), and (d) Hamming distances (HDs) between the predicted CRPs by GAN and the simulated CRPs by an exemplary CPAL PUF system of the present disclosure.

DETAILED DESCRIPTION

[0046] In accordance with various embodiments of the present disclosure, an extreme sensitivity to perturbation near exceptional points (EPs) presents a solution to hardware security and authentication. In particular, naturally occurring fabrication errors can be used to build EP-based electronic circuits, implementing physically unclonable functions (PUFs) with excellent statistical characteristics in terms of the randomness of the generated keys and the uniqueness between different keys.

[0047] Traditional security schemes rely on encrypted keys stored inside nonvolatile memory chips. These can in principle be attacked, which poses a serious security challenge in almost every aspect of modem life, including safety, authentication of goods, foods and drugs, radio-frequency identification (RFID) authorizations, and encrypted communications. In this context, internet-of-things (loTs) systems in which various data, such as location, financial, and health, are constantly collected by sensors and different electronic and tracking devices through near-field communication (NFC) interface (built in, for example, mobile devices), are particularly vulnerable to such a problem. To make things worse, the recent progress in artificial intelligence has made it possible to decrypt some current software-based cryptographic algorithms by using machine/deep learning- assisted attacks.

[0048] Against this backdrop, PUFs are among the most promising and cost-effective hardware security primitives for key generations and authentications in the cyberspace. In general, PUFs exploit unique physical variations that occur naturally during the device manufacturing process, and the encrypted key is generated by mapping a given input (i.e. , “challenge”) to an output (i.e., “response”), forming a challenge-response pair (CRP) (e.g., electrical signals in time/frequency domain, mechanical or optical signals). Accordingly, physically unclonable functions (PUFs) are a class of hardware-specific security primitives based on secret keys extracted from integrated circuits, which can protect important information against cyber-attacks and reverse engineering. Typically, PUFs can be categorized into two major classes: the strong PUFs capable of generating a large number of CRPs, and the weak PUFs possessing only a limited number of CRPs. To date, the majority of PUFs are primarily based on digital electronics, i.e., complementary metal- oxide semiconductor (CMOS) integrated circuit (IC) technologies, including arbiter PUFs, static random-access memory (SRAM) PUFs, memristor PUFs, and ring oscillator PUFs.

[0049] Although CMOS digital products can have good robustness through micro- /nano-manufacturing with high precision, their applications in PUFs, on the flip side, are usually affected by the relatively low entropy and power consumption. As a result, CMOSbased 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 and photonic PUFs, and those based on features of randomly distributed nanostructures, are still subject the implementation cost and system complexity.

[0050] Over the past decade, the physics of exceptional points (EP) has attracted considerable attention due to their exotic effects and potential applications, mainly in optics, photonics, and electronics. An EP is formed when two or more eigenstates (eigenvalues and corresponding eigenvectors) of a non-Hermitian Hamiltonian structure coalesce and become identical. The onset of this peculiar degeneracy signals the collapse of the eigenspace dimensionality, which in turn enhances the structure’s sensitivity to perturbations. This observation has inspired the recent proposal of building sensing devices operating at EPs. Subsequent experimental studies have confirmed that EP-based sensors enjoy enhanced responsivity.

[0051] However, careful theoretical analysis and experimental results have raised doubts about the performance of these devices in terms of sensitivity and resolvability associated with the signal-to-noise ratio. In particular, while the presence of EP leads to enhancement in the responsivity, at the same time it also amplifies the noise by exactly the same factor. On the other hand, very recent experimental results on EP-based mechanical accelerators suggest that there exists a regime where the signal enhancement outweighs that of the noise, thus showing the merit of attempting to use EPs for sensing applications.

[0052] Beyond the noise issues, another problem with known EP-based sensors is that they rely on an implementation of isolated exceptional points. This poses a practical challenge because those systems then become very susceptible to fabrication error and noisy environment(s), which degrade their performance. For instance, in two known experiments, active tuning parameters were employed after the fabrication in order to fine tune the system to the EP to address the sensitivity to perturbation near EPs.

[0053] In accordance with the principles herein, typical variations in the values of standard electric components (resistors, capacitors and inductors) can be “amplified” when used to build an electric circuit operating at a special type of EPs that also involve pole singularity, which are known as divergent EPs or DEPs. Exemplary EPs and DEPs are demonstrated herein. This peculiar singularity in non-Hermitian physical systems can be experimentally demonstrated, as set forth herein. The results shown pave the way for building a new generation of hardware-based encryption architectures that outperform previous PUFs schemes.

[0054] In the present disclosure, the extreme eigenvalue sensitivity of systems with EPs from a different point of view are considered, and their utility for security applications is demonstrated. In particular, in certain embodiments of the present disclosure, EP- and DEP-based circuits can be excellent candidates for producing robust, high-quality radiofrequency (RF) PUFs, which can be generalized to realize the secure wireless authentication (e.g., RFID and wireless access control) and NFC systems. Further, in accordance with various embodiments, coherent perfect absorber-laser (CPAL) devices realized with parity-time-symmetric circuits can be used to generate unique encryption keys.

[0055] In accordance with various embodiments, FIG. 1 (a) depicts a generic 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., 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 produces a response in either the Rx or Tx circuit. In various embodiments, the response is converted to a cryptographic key 108 via a converter circuit (e.g., analog to digital (A/D) converter circuitry) 106.

[0056] The security keys in this exemplary system are encoded in the unavoidable, irreproducible fabrication errors in the values of the electric components (resistors, capacitors and/or inductors) that are used to build the receiver circuit 104. These fabrication errors equip each individual circuit with a unique fingerprint that can serve as a PUF-based cryptographic key 108, which can be probed as follows: when the transmitter/reader and receiver/tag are paired for PUF encryption as secure wireless identification system, the reader (Tx) launches an RF pulse, known as a “challenge” to stimulate the Rx; the temporal response of the latter strongly depends on its eigenmodes of the combined system, which in turn are functions of the exact values of its electric components.

[0057] Thus, in accordance with various embodiments, different tags will exhibit unique temporal responses as given by the instantaneous voltages measured across a reader’s capacitor, called for short as the “response.” Next, the temporal response is digitized (e.g., via analog/digital conversion) to generate a 256-bit (or greater) string identifier (ID) for a given challenge. Correspondingly, FIG. 1 (b) shows exemplary plots of radio frequency (RF) challenge, temporal responses, and bit string processing in accordance with various embodiments of the present disclosure.

[0058] In accordance with various embodiments, once this digitized ID passes the validation by specific loT database via the Tx (reader), the access request of the Rx (tag) will be authorized. On the other hand, in various embodiments, when this reader-tag scheme is used for secure wireless communication, the RF signals transmitted by Tx will introduce unique voltage response drop across a Rx’s capacitor in time domain. Only when such a temporal response is digitized and verified by the Rx with pre-defined verification, will the encrypted data and/or information stored in the Rx’s memory be allowed to be transmitted back to the Tx, in accordance with various embodiments. The secure wireless communication is therefore achieved, effectively avoiding the disclosure of privacy. [0059] In accordance with the principles herein, an exemplary Tx-Rx architecture that can implement DEPs is shown in the left panel of FIG. 2(a). Here, an exemplary transmitter circuit comprises an active transmitter (-RLC oscillator) and one or multiple neutral intermediator (LC oscillator) that act as repeaters. Accordingly, an exemplary receiver circuit comprises a passive receiver (RLC oscillator). In accordance with various embodiments, the RLC oscillator and the -RLC oscillator are configured to be coupled inductively via an on-chip transformer or are configured to be coupled capacitively via an on-chip capacitor or a negative capacitance converter, or are configured to be coupled both inductively and capacitively via the aforementioned electronic components.

[0060] To evaluate the performance of these DEP-based PUFs, their performance in terms of security metrics can be compared with other circuits that realize EPs as well as non-EP systems, both shown in the left panel of FIGS. 2(b) and 2(c). On the right panel of the figures, corresponding pseudospectra are plotted. These plots thus confirm the extreme eigenvalue sensitivity associated with DEP as compared with conventional EPs, which in turn exceeds that of systems with no EPs at all.

[0061] The extreme sensitivity of the DEP circuit can be better understood by closely inspecting its eigenfrequencies, which we express here in unit of the natural frequency a> ₀ = 1/LC: where the dimensionless non-Hermitian (expressing gain or loss) parameter and the normalized coupling factor are given by y = R~ ^L/C and K = M/L, respectively.

[0062] In the exemplary formula, L and M are self and mutual inductances of the two coil antennas. From Eq. (1), it is straightforward to check that two bifurcating real eigenfrequencies, w _± become degenerate at y _EP = 1/2. Further analysis confirms that the corresponding eigenstates become also identical, i.e., this point is indeed an ER In addition, the point K = 1/2 represents a pole singularity, at which the eigenfrequencies diverge. The system is considered in an intermediate regime where the singularity enhances the eigenfrequency splitting but without causing any divergency, and hence the system can be well defined within the context of a linear circuit, in which this DEP divides the system into exact and broken PT symmetry phases.

[0063] An exemplary system can be designed to operate exactly at the DEP in accordance with the principles herein. Due to the strong bifurcation around this point, any small deviations in the values of the circuit’s components can lead to a substantial drift in the eigenfrequencies and, consequently, the response to external excitations. This is exactly the basis for system designs that utilize DEP systems for producing a high-quality PUF by leveraging the process variation naturally occurring in electronic components.

[0064] To elucidate the effect of random physical variations on the system’s eigenspectrum, one may consider the following scenario. It is well known that fabrication errors can lead to a typical variation in the values of electric components in the approximate range of about 0.001 to 0.05; such values are close to percentage errors found in realistic electric components (0.1%, 5%). In line with this, an exemplary ensemble of DEP-circuits (see FIG. 2(a)) can be considered, where resistors and capacitors at the receiver end (defined by the variables R and C) follow Gaussian (normal) distribution given b , where . is the mean resistance or capacitance value and G is the standard deviation induced by fabrication errors, which can be c = 0.04 throughout the present disclosure for PUF evaluations (applicable for most chip resistors and chip capacitors).

[0065] FIGS. 3(a) and 3(b) show plot distributions of real and imaginary parts of eigenfrequencies as a function of the desired y (with 0% uncertainty in electronic components R, L, and C) under coupling coefficient K = 0.7. FIG. 3(a) illustrates that real parts of eigenfrequencies are randomly distributed around the DEP, showing high uncertainty and a dark region that infers a low probability of detection. On the contrary, when the system is operated away from its DEP, real parts of eigenfrequencies have a narrow distribution centered at the mean value, as indicated by brighter colors (i.e. , high probabilities) in FIG. 3(a). In addition, imaginary parts of eigenfrequencies shown in FIG. 3(b) has a high probability of being zero in the exact PT phase. This statistical analysis shows that even the typical 4% standard deviation in resistance and capacitance values can result in a highly random eigenspectrum in the vicinity of DEP, thereby providing an ideal entropy source for PUF and true random number generator applications.

[0066] Having established the extreme spectral sensitivity of DEP-based electronic systems operating at or near their EPs, their performance can be assessed when used as RF-PUF for identification purposes, as illustrated in FIG. 1 (a). To do so, two standard metrics can be reviewed, namely entropy and Hamming distance. The first quantifies the randomness of the bits generated for a single device and different devices, while the latter quantifies how each device is distinct from another.

[0067] In the practical implementation shown in FIG. 2(a), a transmitter/reader circuit can be used for interrogating fully-passive ID tags (i.e., RLC oscillator). It is composed of an intermediary neutral LC circuit and an active RLC oscillator connected to a challenge generator circuit (e.g., pulse generator). Together with the receiver circuit, the whole circuit forms the third-order PT-symmetric electronic system.

[0068] When the system is turned on, the voltage across the transmitter/reader’s capacitor can be exploited to extract the security key to enable the wireless access control. One of the minimal requirements of PUFs is the randomness of their keys. Ideally, the bitmap extracted from the transient voltage responses should have an unbiased distribution of “0” and “1” states. A highly random two-dimensional bitmap (such as that shown in FIG. 1 (a)) distribution is characterized by a high entropy pair (E _x , E _y ) defined by: E _x = ~[Px log ₂ p _x + (1 - p log ₂ 1 - p _x )];

E _y = -[p _y log ₂ Py + (1 - Py) log ₂ ( 1 - Py)]; (2) where p _x , p _y are the probabilities of obtaining digit “1”, along the x- and y-axis, respectively. For an ideal random source, the distributions of “1” (p _x ,p _y ) and “0” (1 - p _x , 1 - p _y ) in a bitstring are both expected to be 50%, resulting in the maximum entropy E _x , E _y = 1, i.e. , both are unity. The cryptographic key 108 of FIG. 1 (a) illustrates a bitmap (white means 1 and black means 0) generated by using the circuit in FIG. 2(a) and FIG. 4(a) depicts the entropy functions along both axes, showing that E _x and E _y are close to 1 and thus high randomness is obtained.

[0069] In particular, FIGS. 4(a)-4(b) show the (a) bitmap and (b) entropy E _x , E _y ) analysis of a 256-bit PUF response from 100 PUF instances for an exemplary DEP-based RF PUF system in accordance with various embodiments of the present disclosure. Here, the average entropy is found to be 0.93, 0.06 and 0.91 , 0.05 along the x- and y-axis. On the other hand, FIG. 4(c) plots the inter-device Hamming distance (HD) (defined as the counts of different bits between two CRPs under the same challenge) for 100 RF-PUF instances generated in the vicinity of the DEP; here, the original HD is normalized by the length of bitstring. The inter-HD is measured at 15 Celsius, while the intra-HD is measured at 0, 5, 15, 20 and 25 Celsius.

[0070] The inter-HD can be well fitted by a Gaussian distribution centered at approximately = 0.5016 having the approximate standard deviation G = 0.0393. This indicates that the DEP-based RF-PUF devices do indeed exhibit a unique response. It is worthwhile noting that the device uniqueness is also 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, if possible, should be uncorrelated with a normalized inter-HD equaling to 0.5. A long or short normalized inter-HD between two CRPs would deteriorate the quality of encryption, such that one could decipher an unknown CRP from another known CRP.

[0071] Another important metric characterizing the performance of PUF devices is their reliability, which is defined as the ability to generate identical keys after the same repeated challenges. In other words, the response associated with the same challenge should not change overtime, even though the environmental conditions are changed (e.g., temperatures in electronic components). A reliable RF-PUF system should have sufficient tolerance against the temperature variation.

[0072] The effect of temperature on the exemplary devices and systems herein, and various architectures associated therewith, can be determined. For example, assume that each lumped element in FIG. 2(a) has temperature dependency, with realistic temperature coefficient of resistance or reactance, e.g., R = R _ref [l + a(T - Tref)] where a = 1 x 10 ^-6 . In general, chip-based RF elements can have constant impedance and low noise over a wide temperature range (0 to 25 Celsius, for example), and, unlike PUFs based on nanomaterial and nanophotonic devices, protective packages for electronic components can prevent PUF devices from oxidization and contamination from physical, chemical, and biological sources. Thus, RF-PUFs, based on the printed circuit board or on-chip integrated circuit technologies for example, can be quite robust and reliable.

[0073] To verify the key reproducibility, intra-HD obtained from 5 temperature conditions at 0, 5, 15, 20 and 25 Celsius can be compared. The results reported in FIG. 4(b) show that the intra-HD, evaluated at different temperatures, is centered at . = 0.0185 with the standard deviation G = 0.0137, which are close to the ideal case with zero mean value. Ideally, the mean inter-HD should be 0.5, which can occur when on average half of the bit length, whereas the mean intra-HD should be close to 0, and their expressions can be given by: where N represents the number of PUF devices (N = 100), L is the length of bit-strings digitized from the RF signal (L = 256), K _t is the i-th PUF key, and m is the number of repeated measurement under different environmental conditions, ® denotes the XOR logic operation. Applying Eq. (3), the mean inter-HD and the mean intra-HD are 0.49 and 0.06, respectively. The low mean intra-HD shows a good robustness against temperature variations. To clearly illustrate the lack of correlation between two arbitrary PUF units, a pairwise map of 50 CRPs can be plotted, where the diagonal line indicates the intra-HD for the PUF instance itself and the off-diagonal points represent the inter-HD values compared to other PUF instances. [0074] The sharp contrast of the colormap in FIG. 4(c) shows a distinct difference between the intra-HD of a specific PUF instance (i.e., 0) and the inter-HD between two different PUF instances (i.e., small fluctuation around its average of 0.5). Accordingly, results obtained in accordance with the principles herein verify the possibility of building a lightweight, robust solution to secure wireless authorization and access control for devices and systems constructed in accordance with the principles of the present disclosure. Next, the encoding capacity can be evaluated. Encoding capacity can be defined as the potential number of codes that can be generated by a PUF instance.

[0075] The encoding capacity is given by c ⁿ . Here, c = 2 (i.e., “0” and “1” in a binary digit) and n is the key size given by n = / (l - /T)/o- ² , where is the mean probability and a is the standard deviation. Based on the results of FIG. 4(b), n = 0.5016(1 - 0.5016)/0.0393 ² « 162 and c ⁿ = 2 ¹⁶² « 5.8 x 10 ⁴⁸ .

[0076] Next, the performance of DEP-based PUF devices can be compared with those that implement a standard EP (i.e., without the pole singularity) as well as those that do not rely on EPs at all. Next, FIG. 4(e) depicts the inter-HD histograms obtained from three different RF-PUF systems considered herein. For the third-order PT- symmetric electronic systems, three different cases corresponding to magnetic coupling strengths /q = 0.7 = 0.99K _O£P , K _± = 0.97K _DEP , and K ₃ = 0.85K _O£P were tested.

[0077] On the other hand, for the standard (second-order) PT-symmetric electronic system, j = 0.7 was used, which ensures operation near the EP. The traditional telemetry system can be formed by an RLC resonator coupled to a coil antenna with selfinductance L In all cases, the input impedance of the RLC oscillator is assumed to follows a Gaussian statistical distribution similarly used in producing the simulation results of FIG. 3(a). From FIG. 4(d), it is shown that the traditional telemetry setup acts as a low entropy source with a biased distribution of “1” and “0”, causing the mean inter-HD to downshift to 0 and thus degraded key uniqueness. The EP-enabled bifurcation effect in the standard PT electronic system can increase the entropy, however, its performance still lags far behind that of the third-order PT electronic system operating nearby the DEP.

[0078] As can be seen from FIG. 4(e), HDs of DEP PUF are highly concentrated around 0.5. On the contrary, the distribution of HDs of EP-PUF is wider than that of DEP PUF. Hence, the PUF performance of EP PUF is not as good as the DEP one. This wider distribution of HDs of EP PUF also yields a smaller encoding capacity, since it has a larger standard deviation. Even in the same third-order PT telemetry system, the entropy decreases as the K/K _DEP ratio decreases, since the system operates far away from DEP. The upshot of this comparison is that using DEP-based electronic circuits to implement PUF devices can significantly boost the system’s entropy and uniqueness beyond their values in standard devices.

[0079] In addition to its application in wireless identification, the exemplary PUF device set forth herein can also be exploited to secure the near-field communication (NFC) or low-power wireless sensors, as illustrated in FIG. 1 (a). When an NFC transmitter/reader circuit device is paired with receiving tags or sensors through inductive coupling, a challenge signal (e.g., pulse signal) sent from the reader circuit can generate unique transient response on each receiving circuit device’s capacitor, which can be used as the PUF-based cryptographic key.

[0080] The PUF-based key detected by the receiving device must be verified before sending out the information stored in its digital memory, thus preventing the participation of the third entity and avoiding leakage of confidential information. To demonstrate the utility of such a device, a similar numerical experiments to that associated with FIGS. 5(a)-5(d) was used to evaluate the device merits. In particular, 100 readers with discrepant RLC circuits were used to interrogate the same receiving device. Here, the lumped elements of the 100 readers are assumed to follow the same Gaussian distribution used before.

[0081] The entropy plots of FIG. 5(a) clearly indicate a nearly ideal scenario that guarantees high quality randomness. The inter-HD shown in FIG. 5(b) is centered at . = 0.4975 and G = 0.0380, which represent excellent uniqueness and security properties which can also be validated by the pairwise evaluation in FIG. 5(c). In addition, the robustness analysis shown in FIG. 5(b), quantified by the intra-HD at different temperatures, is also promising with . = 0.0179 and G = 0.0128.

[0082] In various embodiments, an exemplary security system of the present disclosure can utilize an RF PUF cryptographic system. In one such embodiment, the system can comprise: an NFC reader having a circuit programmed to selectively pair with at least one of tags or sensors via inductive coupling, and to generate a pulse signal output from the reader circuit containing a unique transient response for a receiving device’s capacitor. The unique transient response can serve as a PUF-based cryptographic key. The PUF-based cryptographic key can contain verifiable data preventing a response for a data non-match so that no information stored in the receiving device digital memory is sent when a non-match is received. The verification thus prevents the participation of the third entity and avoiding leakage of confidential information. [0083] Accordingly, FIG. 6 shows a block diagram of an exemplary DEP-based RF- PUF cryptographic system used in radio frequency (RF) identification (RFID) and wireless access control. Operations for such as system can include pairing an identification tag (receiver circuit) 604 and a wireless reader (transmitter circuit) 602 such that a DEP condition is satisfied. The operations can further include a launching of a challenge signal (e.g., RF pulse) by the reader 602 to interrogate the ID tag 604. Correspondingly, the reader 602 acquests or acquires a transient RF response after transmitting the challenge signal and discretizes & digitizes the RF response to obtain a binary PUF cryptographic key. In various embodiments, a database or memory unit of the reader 602 identifies and validates or invalidates the PUF cryptographic key by comparing the key against a stored valid identifier in the database (CRP database) or memory unit. Based on whether the PUF cryptographic key is validated, access is either denied or allowed by the reader 602. For example, access may be associated with unlocking an electrical lock to a room or compartment, as a non-limiting example, or may be associated with providing access to a computer resource, as another non-limiting example, among others.

[0084] Additionally, FIG. 7 shows a block diagram of an exemplary DEP-based RF- PUF cryptographic system used in an exemplary near-field wireless communication (NFC) and authorization process. Operations for such as system can include pairing a transmitter circuit 702 and a receiver circuit 704 such that a DEP condition is satisfied. The operations can further include a launching of a challenge signal (e.g., pulse excitation) by the transmitter 702 to the receiver circuit 704. Correspondingly, the receiver circuit 704 acquests or acquires a transient RF response after receiving the challenge signal and discretizes & digitizes the RF response to obtain a binary PUF cryptographic key. In various embodiments, a database or memory unit of the receiver circuit 704 identifies and validates or invalidates the PUF cryptographic key by comparing the key against a stored valid identifier in the database (CRP database) or memory unit. Based on whether the PUF cryptographic key is validated, a data transmission request is either denied or allowed 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 will be granted upon successful validation of the PUF cryptographic key, as one possible nonlimiting example, among others.

[0085] To demonstrate its effectiveness, the National Institute of Standards and Technology (NIST) randomness test suite was applied to evaluate and validate the performance of an exemplary DEP-based RF-PUF cryptographic system of the present disclosure. The results show that all 9 kinds of NIST tests are passed successfully for both authentication and communication applications. Such results again indicate that the DEP-based RF-PUF system can be a true random number generator that provides excellent randomness and uniqueness for securing near-field wireless access and communication.

[0086] In accordance with the principles herein, the extreme sensitivity of PT symmetric electronic systems near EPs can be utilized for building a new type of physical system, devices and schemes (architectures), and these lightweight PUF-based cryptosystems may enable secure authentication and message exchange among the system/devices/schemes. In particular, the unprecedentedly large eigenvalues bifurcation arising near divergent exceptional points in higher-order (i.e., third-order and beyond) PT electronic circuits can enhance the randomness, uniqueness, encoding capacity of PUFs generated by inevitable physical differences between devices due to uncontrolled variations in the values of their electronic components. The results also indicate that this new PUF paradigm can serve as a perfect entropy source or cryptographic random number generator for encryption and authentication in enormous wireless communication and identification applications.

[0087] In accordance with embodiments of the present disclosure, the extreme sensitivity of PT symmetric electronic systems near EPs can be used, in accordance with the principles herein, for building a new type of physical unclonable function (PUF) schemes and have shown that these lightweight PUF based cryptosystems may enable secure authentication and message exchange among the devices. In particular, the unprecedentedly large eigenvalues bifurcation observed near divergent exceptional points in higher-order (i.e., third order and beyond) PT electronic circuits can enhance the randomness, uniqueness, encoding capacity of PUFs generated by inevitable physical differences between devices due to uncontrolled variations in the values of their electronic components. The results also indicate that this new PUF paradigm can serve as a perfect entropy source or true random number generator for encryption and authentication in enormous wireless communication and identification applications. The results herein open a totally new research direction exploiting the implications of non-Hermitian physics, particularly in electronics platforms, for a new class of applications in suitable circuits, such as hardware security and the like.

[0088] Utilizing the disclosed technologies, cybersecurity devices and systems can be developed for hardware security that can avoid even machine learning assisted cyberattacks. In various embodiments, such devices and systems herein can be a plugin to existing devices, and can also work as a standalone module, if desired. The spectral sensitivity associated with exceptional points (EPs) can be used for building 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 virtue of exotic spectral singularity of a divergent exceptional point (DEP) existing in higher-order parity-time (PT) symmetric electronic system. The drastic eigenvalues bifurcation near the DEP can significantly enhance the stochastic entropy caused by inherent parameter fluctuations in electronic components, resulting in a perfect entropy source to generate encryption keys encoded in analog electrical signals (e.g., radio wave). The results reveal that a DEP-boosted entropy source can enable PUFs with truly random and unique inter-device variation, while achieving great robustness proven by a small intra-device variation. This lightweight and robust PUF structure may lead to a variety of unforeseen securities and anticounterfeiting applications in radio-frequency fingerprinting, anti-counterfeiting, wireless communications, and the like.

[0089] Security issues are of paramount importance in the internet-of-things (loT) and radio-frequency identification (RFID) applications. As set forth hereinabove, physically unclonable function (PUF) may be one of the most promising hardware security technologies, which exploits inherent randomness introduced during manufacturing to give a physical entity a unique ‘fingerprint’ or trust anchor. A PUF mechanism inspired by the parity-time (PT) symmetry in the field of quantum physics is set forth herein. An exemplary electronic analogy (e.g., PCB or IC chipsets) of the PT-symmetric quantum system which has a non-Hermitian effective Hamiltonian and exceptional points is set forth. As discussed, an exceptional point is a singular point in the eigenspectrum of the system, which leads to the exotic eigenvalue bifurcation effect. When the system is operated around this singular point, manufacturing errors (i.e., device-to-device fluctuations) could make the output response of PT-symmetric circuit become highly unpredictable and unrepeatable.

[0090] Moreover, if a wireless telemetry system (e.g., RFID or NFC) is implemented into the PT-symmetric electronic structure, the highly uncertain output responses can be exploited for encryption and security applications (i.e., true random number generator). For example, if a group of RFID or NFC tags (Rx) are fabricated and wirelessly interrogated by the PT reader (Tx), their output RF responses can be unique, as manufacturing variations of electronic components (e.g., resistors, capacitors, and transistors etc.) could result in very different electrical output signals detected by the reader. This opens up new possibilities for making the next-generation wireless access control and secure wireless communication, with outstanding cryptography performance metrics. Other devices and systems are contemplated as well in accordance with the principles herein. Devices/systems/schemes herein provide promising cryptosystems configured to defend against machine learning-assisted attacks to secure wireless communications.

[0091] While traditional software security relies on pseudo random number generator which is known to be vulnerable to machine learning-assisted attacks, an exemplary PT PUF, as a new hardware security system/device/method, is a simple, low-cost true random number generator, which is robust against adversarial attacks. Most importantly, this hardware security technique is compatible to the existing wireless and telemetric systems, such near-field secure communication (NFC), radio frequency identification (RFID), etc.

[0092] Advantageously, a security system constructed in accordance with the principles herein can comprise, in an exemplary embodiment, two or more RF-PUF’s having a first PUF one-dimensional key and a second PUF one-dimensional key. The first and second one dimensional keys are uncorrelated, with a normalized inter-HD approximately equal to 0.5 for a group of PUFs. Further, a security system constructed in accordance with the principles herein can comprise, in an exemplary embodiment, the RF-PUF’s being further defined by divergent exceptional point (DEP)-based RF-PUFs. Additionally, a security system constructed in accordance with the principles herein can comprise, in an exemplary embodiment, an NFC reader having a circuit programmed to selectively pair with at least one of tags or sensors via inductive coupling, and to generate a pulse signal output from the reader circuit containing a unique transient response for a receiving device’s capacitor, wherein the unique transient response serves as a PUF- based cryptographic key. A security system constructed in accordance with the principles herein can also comprise, in an exemplary embodiment, the PUF-based cryptographic key containing verifiable data preventing a response for a non-match to prevent sending out the information stored in its digital memory when a non-match is received, thus preventing the participation of the third entity and avoiding leakage of confidential information. As such, a security system constructed in accordance with the principles herein can prevent generative adversarial network-based machine learning assisted cyber-attacks. [0093] As previously mentioned, in accordance with various embodiments, DEP- based circuits can also be excellent candidates for producing PUF devices among coherent perfect absorber-laser (CPAL) devices to generate unique encryption keys. In accordance with the principles herein, emerging lightweight, low-cost PUFs with outstandingly lower predictability and, most importantly, higher resilience against machine learning-assisted attacks, can be developed so as to facilitate their practice as hardware security primitives.

[0094] Ever since the experimental validation of acoustic, optical, optomechanical, and photonic analogues of parity-time (PT)-symmetric non-Hermitian Hamiltonian, across the spectrum, the field of non-Hermitian physics continues to blossom, leading to many promising applications, such as unidirectional invisibility, non-reciprocal one-way optical devices, coherent perfect absorber-laser (CPAL), high-performance telemetry for sensing, and wireless power transmission.

[0095] PT-symmetry has been innovating the design paradigms of wave propagation and scattering by expanding the control of waves into the non-Hermitian realm. However, it has been recently reported that, at or close to singular points of the PT system where the completeness and continuity of the Hamiltonian’s eigenbasis are broken, e.g., exceptional point (EP) or CPAL point, the distribution of complex eigenvalues/eigenstates could be quite turbulent and noisy. This may cause pronounced sample-to-sample fluctuations that affect the reproducibility and scalability of non-Hermitian physical systems having exceptional points. In a different circumstance, high entropy nearby the exceptional point or the exceptional/CPAL point may be leveraged to generate high- quality PUF-based encryption keys. At the exceptional/CPAL point, the output of the system can be switched from the lasing mode to its time-reverse counterpart, coherent perfect absorption (CPA) mode, by adjusting the complex-valued amplitude ratio of incident waves.

[0096] Notably, due to the self-dual singular nature of the exceptional/CPAL point, both CPA and lasing modes are narrowband effects, and, more interestingly, both modes are susceptible to fluctuations in materials properties of gain and loss elements and their coupling rate. In this regard, even small but inevitable process variations in manufacturing PT-symmetric structures may produce very different output responses from device to device. Although such a property may be seen as a foe for sensing purposes, on the flip side, it may find useful applications in generating high-quality PUF keys. Particularly, with the rapid advent of manufacturing technologies in the microelectronics industry, the inherent device variability on an electronic or photonic microchip has been minimized, which poses a challenge to improve the randomness and uniqueness of digital circuitbased PUFs. In this regard, the exemplary PUF keys generated from the frequencydomain electromagnetic responses may remarkably enhance the entropy inherent in fabrication flaws, in light of the extreme sensitivity at the self-dual singularity - CPAL point, thereby enabling high cryptographic randomness and uniqueness.

[0097] Herein, an emerging electromagnetic PUF paradigm is set forth that exploits stochastic and fluctuating properties among CPAL devices to generate unique encryption keys. For instance, its optical realization is illustrated in FIG. 8(a) that shows a schematic of an exemplary CPAL PUF system implemented using the active and passive metasurfaces in the optical region. Here, manufacturing variations can cause random fluctuations among unit cells (or meta-atoms), resulting in device-to-device variations. Correspondingly, FIG. 8(b) shows the two-port transmission-line network (TLN) model for the generalized PT-symmetric CPAL device of FIG. 8(a) that is composed of spatially- distributed and balanced gain and loss, of which the shunt/surface conductances —G

(gain) and G (loss) are 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 readily translated to any physical system spanning a broad electromagnetic spectrum, including optics, photonics, radio-frequency (RF), and microwave electronics. In the infrared and optical regions, the equivalent TLN model in FIG. 8(b) can be implemented using, for example, a pair of active and passive electromagnetic metasurfaces separated by a thin dielectric spacer, as shown in FIG. 8(a).

[0098] In the RF and microwave regions, the structure can be implemented using integrated circuit or printed circuit board (PCB) technology, where a negative resistance converter (NRC) and a shunt resistor are separated by a portion of a transmission line or a T/n-transformer. In the PUF key retrieval process, the nonlinear analog output responses of CPAL-based PUF instances can be appropriately discretized and digitized into the bitstring-based authorization codes with excellent unclonability, as represented in FIG. 8(c). Here, different challenges ( C _n ) achieved with different complex-valued amplitude ratios of two incoming waves (a = ip~ /-i *) are applied to the PUF devices to generate unique responses (7? _n ). The responses are then discretized and digitized into binary bit-strings, forming digital cryptographic maps for PUF applications. The output responses in FIG. 8(c) are simulation results.

[0099] The PUF keys retrieved from the output responses of the CPAL-based PUF instances can be reconfigured by adjusting the complex amplitude ratio of incident waves. Such a property may not be possible for most CMOS-based digital PUFs, in which each instance corresponds to one or multiple bits in the response, and thus, increasing the number of CRPs comes at the cost of increased size, device area and design complexity. [00100] The scattering parameters of the two-port TLN model in FIG. 8(b) can be computed using the transfer matrix method. The CPAL point can be obtained by setting x = n/2 and G = ^2Y ₀ , where Y _o is the port admittance and the characteristic admittance of the finite transmission-line segment. At the CPAL point, the eigenvalues of the system’s scattering matrix approach zero and infinity, corresponding to the CPA mode and the lasing mode, respectively. When characterizing the CPAL properties in terms of energy flux, the output is defined as the ratio of total output power to the total input power: R = reflection/transmission (i.e. , R — > 0) can be obtained, whereas the lasing mode with large scattering coefficients (i.e., R — > °°) can be achieved under the condition: 'i’/ 'i ¹ ’ * i( 2 - 1); throughout the present disclosure, the notation of - 1 \e ^i(t> is adopted. In the optical region, such a complex-valued amplitude ratio of two incident light waves can be achieved with a polarizer and a voltage-controlled liquid crystal phase shifter (LCPS). In the low-frequency range, various analog/digital phase shifters and attenuators can be used to precisely tune the complex amplitude ratio of two input radio signals.

[00101] Naturally occurring process variations in device dimensions, defects, and material profiles may cause fluctuations in the shunt conductances (see FIG. 8(b)). Here, we simply assume 8G = 8G') and electrical distance (here, we assume 8x), thus resulting in different output responses for an ensemble of CPAL devices. As an example, if the CPAL device is initially operated at the CPA mode (i.e., challenged by <p = n/2), the output response as a function of the normalized conductive perturbation v = 8G/Y ₀ can be written as:

[00102] In the ideal scenario with v,8x) = (0, 0), a null output response is obtained. Given that the manufacturing-induced flaws are small, i.e. , 8x, v « 1, applying the Taylor series expansion to Eq. (4) leads to:

[00103] Such a result implies that the output is sensitive to v, with a v ² dependence. Further, the sensitivity of output response is enhanced by a factor of l/(5%) ² . One may envision that due to a small change in (v,5x), a device could hop from the low-scattering (absorptive) mode to the high-scattering (emissive) mode, resulting in drastically different scattering responses and net outgoing energy flux. It should be noted that even when the system is initially operated in the lasing mode (i.e., challenged by <p * n/2), a similar sensitive output response can be obtained. FIG. 9(a) presents the theoretical results for the contour of the output response P^PAL ^{as a} function of 8x and v, plotted using Eq. (4).

The scattered points represent the simulation results. The 1000 randomly generated CPAL-based PUF instances are initially locked in the CPA mode; here, 8x (v) is assumed to follow a Gaussian distribution with mean . = 10 ^-2 (10 ^-3 ) and standard deviation G = 10 ^-2 (10 ^-3 ) due to device-to-device variations. From FIG. 9(a), we find that, surprisingly, even with subtle fluctuations in conductance and electrical distance, the outputs of a set of devices can differ by ~60 dB. For comparison, FIG. 9(b) plots the same contours as FIG. 9(a) but for the Fabry-Perot interferometer (FPI)-based PUF system under the same perturbation levels. The device configuration of FPI is similar to that of the CPAL device but with a pair of surface/shut conductances G = cY ₀ and a separation distance x = n, where passive and active FPIs are obtained with c > 0 and c < 0, respectively. In contrast to the passive FPI that acts as a CPA, the active FPI exhibits only lasing properties. The results shown in FIG. 9(b) indicate that the output responses of FPI-based PUF systems, regardless of their type, are rather insensitive to small-valued 8x and v, with a theoretical expression approximated by:

[00104] By comparing the simulation results in FIGS. 9(a) and 9(b), we find that the CPAL PUF instances exhibit a larger variation in their output responses and, thus, potentially higher entropy than those of their passive/active FPI counterparts, thanks to the self-dual singular nature of the CPAL point. Such diversified distribution of output responses caused by the inter-device variation may ensure high randomness and uniqueness for PUF applications. The present disclosure also studies the sensitivity of output responses to v and 8x for the same PT system operating at the EP. The theoretical results show that devices operating near the CPAL point can exhibit greater variations in output responses and thus higher randomness than those operating near the EP. [00105] In the present disclosure, an exemplary electromagnetic PUF paradigm is experimentally demonstrated in the RF region. However, it should be noted that as long as the equivalent TLN model in FIG. 8(b) is valid, the disclosed concept can be similarly implemented in the fields of optics, photonics, optomechanics, and even acoustics. At low frequencies, the CPAL-based PUF instance shown in FIG. 8(b) can be implemented with the lumped-element circuitry on the PCB or monolithically integrated chipset, as shown in FIG. 10(a), where a transmission-line segment can be transformed into a compact, lumped “T”-network, and the shunt —G and G can be realized with the NRC and resistor, respectively. FIG. 10(a) also shows a photograph of a prototype of an onboard CPAL PUF instance. In this prototype, 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 the manufacturing process tolerance, NRCs have an averaged negative conductance of -G _NRC « -0.0284 + iO.0001 S at the CPAL frequency of 16.7 MHz, and the averaged shunt conductance G = 0.0284 S. The fabricated “T”-networks also have phase variation (6x) of ±3°, alongside other parasitics from the board and package. It is already known that if the system is initially designed to work at the CPAL point, any small perturbation in lumped element values may cause the output response to differ substantially. This makes possible the generation of PUF keys. In the synthesis of PUF keys (FIG. 8(c)), each CPAL device as a PUF instance translates various input challenges C _n (i.e. , different sets of ip~ /-i *) to unique output responses R _n . In the present experiment, the output response was measured over a frequency range of 16.7 ± 0.1 MHz. The output response spectrum was first normalized in the range of (0, 1 ) and then discretized into 64 points. Subsequently, the 64 points were digitized into a 4-bit binary code to form a 256 (4 x 64)-bit CRP as the device-specific unique identifier. For example, points with values smaller than 0.0625 will be given the binary code “0000” (details are schematically shown in FIG. 8(c). It is noted that by changing the complex amplitude ratio 0“/0 ₍ ⁺ = where 0 < <p < n, it may be possible to produce a large number of CRPs from a single PUF instance by adjusting the phase shift , which, in turn, creates new input challenges. FIG. 10(b) plots the extracted 50 x 256 cryptographic bitmap for two challenges with <p = 7T/2 and 0 = 0 (corresponding to the CPA and lasing modes, respectively).

The quality of PUF keys, by and large, is determined by three main metrics, namely randomness, uniqueness, and reliability. Different PUF instances should have random and unique responses when interrogated by the same challenge, while reliability represents the consistency of the response of the same PUF instance at different environmental conditions. We first evaluate the randomness of an exemplary CPAL- based PUF, i.e., the entropy of the cryptographic map associated with the uniformity Ideally, the number of “0”s and “1”s in the cryptographic map should occur with equal probability, which results in E _x , _y = 1. FIG. 10(c) shows the entropies E _x and E _y extracted from the bitmap in FIG. 10(b). We find that E _y (0.97 ± 0.02) exhibits a nearly perfect distribution and E _x (0.90 ± 0.11) deviates only slightly from unity, indicating excellent randomness in the generated PUF keys. The entropy quality is a direct measure but may not be sufficient to describe the randomness of a PUF. The National Institute of Standards and Technology (NIST) randomness tests suite may be used to fully assess the randomness of the CPAL-based PUF. The results show that the CPAL-based PUF can pass all 9 NIST randomness tests (the rest 6 require a bit length greater than 10 ⁶ ) with P- values larger than 0.01 . Therefore, the exemplary PUF can be regarded as a true random number generator.

[00106] The uniqueness of PUF, as another important performance metric, can be assessed by the inter-device Hamming Distance (or inter-HD) between the (digitized) response bitstrings of all PUF instances under the same challenge. An ideal inter-HD should be 0.5, which means that, on average, half the response bits are not repeated, thus ensuring the best quality of encryption from the statistical perspective. FIG. 10(d) shows the pairwise comparisons of inter-HDs among 25 PUF keys. It is seen that inter- HDs in the off-diagonal areas only fluctuate slightly around the mean value of 0.46. The result suggests that all PUF keys have great uniqueness; that is, each key is unique, highly uncorrelated, and unpredictable from history. It is worth mentioning that PUF can be divided into strong and weak categories, which are classified according to the number of CRPs. A PUF is considered strong (weak) if the number of CRPs scales exponentially (linear or polynomial) with its size. Weak PUFs with a limited number of CRPs are often used for cryptographic key generation in identification applications, whereas strong PUFs capable of generating a large number of CRPs are utilized for authentication and secure communication applications. Since the output response of the CPAL device is very sensitive to input challenges (i.e. , it may be possible to create a large CRP space by adjusting both the amplitude and/or phase offset between two incident waves. The present disclosure illustrates this idea by applying two challenges to the CPAL-based PUF instances: lasing (< > = 0) and CPA (< > = n/2 , here, the absolute value of amplitude ratio is fixed to V2 - 1, while the phase shift is varied. FIG. 10(e) plots the histogram of average inter-HDs measured over 25 PUF instances under the two challenges. The CRPs on the upper (lower) panel of FIG. 10(e) are obtained by applying the lasing (CPA) challenge. The inter-HDs of the lasing challenge have a near-ideal distribution = 0.4834 and G = 0.0354, while that of the CPA challenge performs even better (/ = 0.4910 and G = 0.0294). The inter-HD histogram fitted by a Gaussian distribution is centered around the ideal value of 0.5. The binary encoding capacity of a PUF related to information density can be expressed as c ⁿ , where c = 2 and n = / (l - /T)/o- ² . The statistical results in FIG. 10(e) demonstrate a high encoding dimensionality of c ⁿ = 2 ²⁸⁹ , which may enable the development of high-capacity encryption systems. It is noted that the number of CRPs can be further increased by introducing more challenges in terms of phase shifts. We should also point out the average electrical length extracted from all “T”-networks is x « 1771/40, which is slightly away from CPAL point. The PUF metrics may be further improved by using advanced fabrication techniques that minimize the board parasitics, allowing the devices to operate in close proximity to the exceptional/CPAL point.

[00107] For comparison, 25 PUF instances were also built based on the conventional passive and active FPI structures, as illustrated in FIG. 11 (a). The lumped elements used to build these FPIs and input challenges remain the same as those used in CPAL devices. The measured output responses of passive and active FPI-based PUF instances are nearly identical, implying poor randomness and uniqueness. FIG. 11 (b) plots the bitmap obtained from the 25 passive FPI-based CPAs (with the challenges <p = 0 and <p = n applied), clearly showing a deteriorated uniformity compared to the results in FIG. 10(b). The entropy contents of the active (hollow circle) and passive (solid dot) FPI-based PUFs are plotted in FIG. 11 (c). Under the same manufacturing tolerance, the entropies of both the passive FPI-based PUF keys (E _x = 0.54 ± 0.41, E _y = 0.86 ± 0.04) and active FPI- based PUF keys (E _x = 0.35 ± 0.35, E _y = 0.15 ± 0.10) are much lower than those of the CPAL-based PUF keys. FIG. 11 (d) reports the pairwise map of inter-HDs for the passive FPI CPA-based PUF. From FIG. 11 (d), we find that most FPI CPA instances are highly correlated, namely, the extracted PUF keys could be vulnerable to attacks. FIG. 11 (e) is similar to FIG. 10(e) but obtained with active and passive FPI-based PUF instances. From these histograms, the mean inter-HD value for the passive (active) FPI-based PUFs is found to only be 0.31 (0.12). Although the passive (active) FPI-based PUF instances are also initially locked at the CPA (lasing) mode, the resulting uniqueness or randomness is much worse than that of their CPAL counterparts. The results in FIG. 11 (e) are in sharp contrast to FIG. 10(e) obtained with the CPAL-based PUF. Therefore, the presence of the self-dual CPAL singularity in PT non-Hermitian systems, indeed, plays a key role in amplifying the output response deviation caused by the inter-device variation, thus providing unique and unclonable encryption keys.

[00108] From the perspective of practical application, reliability refers to the consistency of CRPs under environmental variations (e.g., ambient temperature) is essential. In PUF applications, reliability can be described by intra-device HD (intra-HD), defined as the bit error rate between responses generated by the same PUF instance at different operating conditions for a given challenge. To assess the reliability of the fabricated CPAL PUF circuits, each instance was measured at eleven different temperatures (from -20 °C to 80 °C with an interval of 10 °C). The measured intra-HD histogram of the disclosed PUF is shown in the upper panel of FIG. 10(e), which is normally distributed with a mean of 0.05 and a standard deviation of 0.05. Such values are sufficiently low to ensure good robustness against environmental variations. In addition to the temperature stability, “dynamic” noises, such as phase/flicker noises and thermal noises introduced by the agitation of electrical charges, may also generate temporal fluctuations in a system’s responses. We also studied the time dependence of reliability by measuring the output responses of 6 CPAL PUF instances every 30 s (for a total of 3 minutes) and calculating the intra-HDs of the generated CRPs. The intra-HD histogram associated with the temporal stability is plotted in the lower panel of FIG. 10(e), which shows a near-zero mean, implying that the CPAL PUF can be robust against “dynamic” noises. In fact, previous studies on the CPAL systems have shown that thermal noises as the dominant noise source contributes negligibly to the signal-to-noise ratio (SNR). Finally, we note that since the encryption keys of the CPAL PUF exist in an analog form (before digitization), the bit length can be arbitrarily long depending on how discretization and digitization are performed. Such intriguing properties may make the disclosed PUF outperform the traditional digital PUFs, such as the multiple-valued logic PUF, monostable PUF, and voltage divider PUF, of which the bit length can only be increased by introducing considerably increasing the number of cells/bits. Additionally, various input challenges and, thus, a potentially large number of CRPs may be created by adjusting the complex-valued amplitude ratio of two input waves. As a result, considering all these merits, a single CPAL PUF instance may create a large CRP space, far exceeding the aforementioned digital PUFs. Next, the resilience of exemplary CPAL PUF devices to machine learning-assisted attacks are considered. Machine/deep learning has emerged in recent years, which may provide strong assistance in pattern recognition, signal processing, and inverse design in electromagnetics. In particular, it is also reported to be a powerful tool for password-guessing attacks and decrypting. Some recent works pointed out that some PUFs with relatively low randomness and uniqueness may also be vulnerable to attacks based on machine learning models, such as the Fourier regression (FR) model and generative adversarial network (GAN). FR- and GAN-based modeling attacks were also performed on the CPAL-based PUF. FR-based modeling attacks do not require a large training database and are, therefore, commonly used in attacking PUFs. FIG. 12(a) shows the model expression F(%j), in which a _oi , a _ni , and b _ni (n = 1, 2,—,N) are Fourier coefficients determined by the least square fitting (the order of regression), and x _t is the random input in the range (0, 1); here, the 8/16/32 -order regressions were performed, and only the results of 16-order are presented because it gives the optimum performance. The FR model extracts the features of randomness from the training data set (estimator CRPs) to predict PUF responses. In the attack-defense experiment, 50 256-bit measured CRPs, which are obtained from 25 devices with two challenges, are divided into the training dataset (40 estimator CRPs) and the 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 PUF could be understood from the prediction accuracy (ACC), namely the number of correctly predicted bits as a percentage of the total number of predictions. Here, the correlation coefficients (CCs), defined as the linear correlation level between two random sequences and HDs between the predicted and measured CRPs, were employed to evaluate the resilience of our 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. FIGS. 12(b)-12(d) report the results of the FR modeling attack. The mean values of ACC, CC, and HD are 54%, 0.03, and 0.46, respectively. Compared with the prediction accuracy in traditional silicon-based PUFs (typically larger than 90%), such results show excellent robustness to the FR-based modeling attack.

[00109] The GAN-based modeling attack comprising two deep neural networks - a generator and a discriminator, as illustrated in FIG. 13(a), is another strong passwordguessing tool. The discriminator is a binary classifier used to distinguish whether an input CRP belongs to the training dataset or is produced by the generator. The generator, on the flip side, strives to generate fake CRPs that resemble the training data to fool the discriminator. In many applications, such as image synthesis, the well-trained GAN can generate new data instances that resemble the training data. The GAN structure adopted herein, as shown in FIG. 13(a), includes a generator network with 4 layers and a discriminator network with 3 layers; here, 4/3 linear layers were adopted because too many hidden layers could lead to convergence failure, which reduces the prediction accuracy of the model. Since the GAN structure requires a large amount of training data, we simulated 1000 PUF instances with the fabrication tolerance extracted from experimental results and applied 10 input challenges (10 phase differences between two input signals) to generate 10000 CRPs, where 2000 CRPs were used for testing, and the rest were kept for training. The GAN was trained with 3000 epochs. Without the loss of generality, all “0”s of the CRPs for training the GAN were replaced with “-1”s. The distributions of ACC, CC, and HD between the GAN-predicted and simulated CRPs are shown in FIGS. 13(b)-13(d), respectively; here, the probability mass function (PMF) of a normal distribution is adopted. It can be clearly seen that the ACC is narrowly distributed and centered at 50%, implying that the CPAL-based PUF is resilient to the GAN-based modeling attacks. The maximum (minimum) prediction accuracy is 73% (35%), corresponding to a success possibility of only 0.0001 (0.0003). Such results suggest that the GAN may fail to generate CRPs resembling the ones extracted from PUF instances. The mean CC and HDs are 0.07 and 0.46, which further confirms an exemplary PUF is resilient to machine learning-assisted attacks.

[00110] In brief, the present disclosure presents a robust, high-quality PUF primitive based on the CPAL effect enabled by PT-symmetric non-Hermitian electromagnetic structures. It has been theoretically demonstrated that the self-dual singularity may make output responses of CPAL-based PUF instances highly sensitive to inevitable device-to- device variations. Besides, experimental studies conducted for the CPAL-based PUF keys implemented using the RF circuits have shown that the performance metrics, including randomness, uniqueness, correlation level, encoding capacity, and thermal/temporal stability, can outperform PUF instances based on traditional passive and active FPIs. Furthermore, the CPAL-based PUF is highly robust against state-of-the- art machine learning-assisted attacks, such as FR and GAN modeling attacks. The disclosed PUF techniques may pave a promising avenue toward next-generation identification, authentication, encryption, and security systems, which may find widespread applications in modem society. The disclosed hardware security paradigm may also be extended to other wave systems, such as photonics, acoustics, elastics, and optomechanics.

[00111] Accordingly, the present disclosure puts forward an emerging type of PUF in the electromagnetic domain by virtue of the self-dual emitter-absorber singularity that uniquely exists in the parity-time (PT)-symmetric structures. At this self-dual singular point, the reconfigurable emissive and absorptive properties with order-of-magnitude differences in scattered power can respond sensitively to admittance or phase perturbations caused by, for example, manufacturing imperfectness. Consequently, the entropy sourced from inevitable manufacturing variations can be amplified, yielding excellent PUF security metrics in terms of randomness and uniqueness. It is shown that this electromagnetic PUF can be robust against machine learning-assisted attacks based on the Fourier regression and generative adversarial network. Moreover, the disclosed PUF concept is wavelength-scalable and transformative in radio-frequency, terahertz, infrared, and optical systems, paving a promising avenue toward applications of cryptography and encryption, including using stochastic and fluctuating properties among CPAL devices to generate unique encryption keys..

[00112] It should be emphasized that the above-described embodiments of the present disclosure are merely possible examples of implementations, merely set forth for a clear understanding of the principles of the disclosure. Many variations and modifications may be made to the above-described embodiment(s) without departing substantially from the principles of the disclosure. All such modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims.