Spectral sensitivity near exceptional points as a resource for hardware encryption

The spectral sensitivity near exceptional points (EPs) has been recently explored as an avenue for building sensors with enhanced sensitivity. However, to date, it is not clear whether this class of sensors does indeed outperform traditional sensors in terms of signal-to-noise ratio. In this work, we investigate the spectral sensitivity associated with EPs under a different lens and propose to utilize it as a resource for hardware security. In particular, we introduce a physically unclonable function (PUF) based on analogue electronic circuits that benefit from the drastic eigenvalues bifurcation near a divergent exceptional point to enhance the stochastic entropy caused by inherent parameter fluctuations in electronic components. This in turn results in a perfect entropy source for the generation of encryption keys encoded in analog electrical signals. This lightweight and robust analog-PUF structure may lead to a variety of unforeseen securities and anti-counterfeiting applications in radio-frequency fingerprinting and wireless communications.

Then, the temporal response is properly scaled and normalized in the range between 0 and 1 and is uniformly discretized into 64 data points (R1 -R64) in the time domain, as sketched in Fig. 1(b) in the main text. Each data point Ri is digitized into 4-bit binary codes ranging from 0000 to 1111, depending on its floating number value. A 4-bit binary code 0000 will be assigned if the normalized data point is smaller than 0.0625. Subsequently, a 256-bit CRP sequence can be generated based on the temporal response, originally stored in an analog form.
Here to have a statistical analysis of the bitmap, we choose 100 from 10000 simulated PUF keys as a sufficient sample size to repeat the same procedures; a CRP map (i.e., bitmap) can be obtained, as shown in the inset of Fig. 1(a) in the main text and Figs. S2 and S3. We note that the encryption quality provided by the proposed PUF can be further improved by expanding the size of the bitmap. For example, the discretization process can be ameliorated by choosing a high sampling rate to gain more individual data points. In addition, extending the binarycoded length to 8 bits or 16 bits can also increase the size of the bitmap.  Throughout this study, the time-harmonic notation exp( ) i − is adopted. When doing the circuit analysis, we assume the RF pulse generator is removed.
Therefore, the effective Hamiltonian of the system can be written as eff , HH  ) and symmetric with respect to the PT , namely where J is the 33  anti-diagonal identity matrix, I is the 33  identity matrix, and K conducts the operation of complex conjugation. These operations in conjunction leave the system unaltered. The system has six eigenvalues or eigenfrequencies, which can be derived from the secular equation as (in units of 0 We note that in the exact PT-symmetric phase, one may observe an oscillatory motion consisting of the superposition of three harmonics [ Fig. S1]. In the broken phase, due to the positive imaginary part of complex eigenfrequencies, the eigenmodes grow exponentially in time, and, thus, the system exhibits unstable, underdamped behavior. Beyond the point of critical damping, the eigenfrequencies are purely imaginary, and the eigenmodes are either exponentially growing or decaying in the temporal responses. Such a phase is an overdamped mode, with exponential responses arising in the temporal dynamics of charges and displacement currents.

Supplementary Note 3: Design of Negative Impedance Converter
In our experimental validation, a negative impedance converter (NIC) with outstanding performances that can sustain the realization of EP and DEP was designed and fabricated. The circuit structure is shown in the inset of Fig. S5(b), which comprises a unity-gain stable, highprecision, and high-frequency operational amplifier (OPAMP; OPA817, Texas Instruments Inc.) integrated with proper lumped elements. Figure S5(a) shows the photograph of the front and back sides of the RLC − tank (reader) that consists of the NIC and the corresponding capacitor and inductor (coil). Figure S5(b) plots the measured negative impedance of the NIC, from which we could see the effective negative resistance is ~100  with a relatively small parasitic capacitance of within 100 MHz.

Supplementary Note 4: National Institute of Standards and Technology (NIST) randomness tests
An excellent PUF can be regarded as a true random number generator (TRNG) capable of generating perfectly random number sequences. The randomness of bit sequences is of great importance in cryptographic applications. The NIST randomness tests are developed to determine whether a binary sequence generated by a source is genuinely random. The tests include 15 contents in total, which use probability functions (e.g., complete and incomplete gamma functions, error functions, etc.) to examine the binary sequences in different aspects.
For example, the frequency test, also referred to as the monobit test, mainly focuses on the proportion of zeros and ones of the bitstring. It is easy for a sequence to pass this test since it only requires that distributions of zeros and ones are the same. However, the same sequence may not be random if its first half bits are all zeros or ones or if the occurrences of zeros and ones are periodical. To avoid this scenario, other NIST tests are required to be passed. For these tests, P-value is the probability that a perfect random number generator would have produced a sequence less random than the test sequence, given the kind of nonrandomness assessed by

Supplementary Note 5: The influence of pulse-to-pulse variations and time-modulated pulse signals on DEP PUF
The small pulse-to-pulse variations do indeed exist during the measurements, which, however, will not significantly degrade the performances of the proposed DEP PUF. Although the intra-HDs in our paper tend to investigate the temperature stability of the DEP PUF, at the same time, the pulse signals as input challenges to the DEP PUF instances will also have slight fluctuations for these intra-HD measurements, which should also be considered as different operating conditions. Therefore, the experimentally measured intra-HDs are the products of the combined effects of different temperatures and different pulse signals of the DEP PUF, by which the results have rendered a high consistency among different operating conditions. Thus, tiny pulse-to-pulse variations will not significantly influence the DEP PUF.
In addition, we also study the unique transient responses that the DEP PUF may provide concerning the different shapes of pulse signals. As illustrated in Fig. S6, the transient responses will be altered entirely once the pulse or step signal has been added with specific time modulations. By doing so, we may obtain a large number of CRPs by a single PUF instance, and therefore, the DEP PUF may be regarded as a strong PUF.