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This letter proposes using intermittent chaotic clocks, generated from chaotic maps, to drive cryptographic chips running the Advanced Encryption Standard as a countermeasure against Correlation Power Analysis attacks. Five different chaotic maps -- namely: the Logistic map, the Bernoulli shift map, the Henon map, the Tent map, and the Ikeda map -- are used in this work to generate chaotic clocks. The performance of these chaotic clocks is evaluated in terms of timing overhead and the resilience of the driven chip against Correlation Power Analysis attacks. All proposed chaotic clocking schemes successfully protect the driven chip against attacks, with the clocks produced by the optimized Ikeda, Henon, and Logistic maps achieving the lowest timing overhead. These optimized maps, due to their intermittent chaotic behavior, exhibit lower timing overhead compared to previous work. Notably, the chaotic clock generated by the optimized Ikeda map approaches the theoretical limit of timing overhead, i.e., half the execution time of a reference periodic clock.

In this paper we propose a generalized distribution function of relaxation times (DFRT) considering the Davidson-Cole model as an elementary process instead of the standard Debye model. The distribution function is retrieved from the inverse of the generalized Stieltjes transform expressed in terms of iterated Laplace transforms. We derive computable analytical expressions of the generalized DFRT for some of the most known normalized impedance (or admittance) models including the constant phase element, the Davidson-Cole, Havriliak-Negami and the Kohlrausch-Williams-Watts models.

The interpretation of electrochemical impedance spectroscopy data by fitting it to equivalent circuit models has been a standard method of analysis in electrochemistry. However, the inversion of the data from the frequency domain to a distribution function of relaxation times (DFRT) has gained considerable attention for impedance data analysis, as it can reveal more detailed information about the underlying electrochemical processes without requiring a priori knowledge. The focus of this paper is to provide a general procedure for obtaining analytically the DFRT from an impedance model, assuming an elemental Debye relaxation model as the kernel. The procedure consists of first representing the impedance function in terms of the Fox $H$-function, which possesses many useful properties particularly that its Laplace transform is again an $H$-function. From there the DFRT is obtained by two successive iterations of inverse Laplace transforms. In the passage, one can easily obtain an expression for the response function to a step excitation. The procedure is tested and verified on some known impedance models.

This paper is devoted to the study of the $M$-Wright function ($M_{\alpha}(t)$) which is the inverse Laplace transform of the single-parameter Mittag-Leffler (ML) function ($E_{\alpha}(-s)$). Because $E_{\alpha}(-s)$ can be viewed as the fractional-order generalization of the exponential function for $0<\alpha<1$, to which it reduces for $\alpha=1$, i.e. $E_{1}(-s)=\exp(-s)$, its inverse Laplace transform, being $M_{\alpha}(t)$, can be viewed as a generalized fractional-order Dirac delta function. At the limiting case of $\alpha = 1$ the $M$-Wright function reduces to $M_{1}(t)=\delta(t-1)$. We investigate numerically the behavior of this fractional-order delta function as well as it integral, the fractional-order unit-step function. Subsequently, we validate our results with experimental data for the charging of a supercapacitive device., Comment: 6 pages, 5 figures

The capacitance of capacitive energy storage devices can not be directly measured, but can be estimated from the input and output signals expressed in the time or frequency domains. Here the time-domain voltage-charge relationship in non-ideal electrochemical capacitors is treated as an ill-conditioned convolution integral equation where the unknown capacitance kernel function is to be found. This comes from assuming \emph{a priori} that in the frequency domain the charge is equal to the product of capacitance by voltage. The computation of a stable solution to this problem particularly when dealing with experimental data is highly sensitive to noise as it may lead to an oscillating output even in the presence of small errors in the measurements. In this work, the problem is treated using Tikhonov's regularization method, where a degree of damping is added to each singular value decomposition (SVD) component of the solution, thus effectively filtering out the components corresponding to the small singular values., Comment: 7 pages, 5 figures

In this work, we show that a pinched-loop can be observed in the voltage-current plane when a series R-L circuit is current excited. Specifically, the resistance (R) in this circuit is variable and is voltage-controlled by the voltage developed across the inductor due to the exciting current. In this context, we confirm our previous results that the pinched-loop is not a characteristic of memrsitors or memrsitive systems and that it can be observed in many other nonlinear systems. Numerical simulations, circuit simulations and experimental results validate the theory., Comment: 4 pages, 7 figures

Supercapacitors are mostly recognized for their high power density capabilities and fast response time when compared to secondary batteries. However, computing their power in response to a given excitation using the standard formul{\ae} of capacitors is misleading and erroneous because supercapacitors are actually non-ideal capacitive devices that cannot be characterized with a single constant capacitance. In this study we show how to estimate accurately the time-domain power and energy of supercapacitors in response to any excitation signal represented in terms of its Fourier series coefficients with the sole knowledge of the frequency-domain impedance parameters of device. The presented theory is first verified and validated with simulations conducted on an equivalent fifth-order RC circuit emulating the behavior of a fractional circuit consisting of a resistance (Rs) in series with a constant phase element (CPE) of fractional impedance ZCPE = 1/C{\alpha}s{\alpha}. Then we do the same for a commercial supercapacitor modeled as an Rs-CPE circuit, and subjected to both a periodic triangular voltage waveform and a random voltage excitation. The results are conclusive and very promising for adopting the proposed procedure to estimate the power and energy performance of supercapacitors in response to real-world charging and discharging signals., Comment: 8 pages, 5 figures

This study provides new sets of experimental results supporting Westerlund's conjecture that Dead Matter Has Memory. Memory effects in the dynamic response of electric double-layer capacitors (EDLCs) that integrate its prior history of stimulation and state have been experimentally observed and reported in a few recent studies. The different excitation signals used to quantify such effects in these studies aimed at charging a device to the same voltage value and the exact same accumulated charge level but in different manners. Having reached the same unique voltage-charge point, it was observed that different yet repeatable discharge patterns occur, proving the existence of memory. The aim of this work is to provide further experimental evidence of the inherent memory effect in EDLCs in response to time-varying stationary input excitations with different statistical properties. In particular, different sets of charging voltage waveforms composed of fixed dc values with superimposed uniformly-distributed random fluctuations of different amplitudes were created and used to charge the same EDLC device to a unique voltage-charge point. The duration of these signals was the same but with different values of variance around the mean value. We observed different time-charge responses depending on the extent of the noise level in these charging waveforms. This is interpreted and discussed in the context of inherent memory using fractional-order voltage-charge equations of non-ideal capacitors., Comment: 5 pages, 4 figues

Electrochemical capacitors are a class of energy devices in which complex mechanisms of accumulation and dissipation of electric energy take place when connected to a charging or discharging power system. Reliably modeling their frequency-domain and time-domain behaviors is crucial for their proper design and integration in engineering applications, knowing that electrochemical capacitors in general exhibit anomalous tendency that cannot be adequately captured with traditional integer-order-based models. In this study we first review some of the widely used fractional-oder models for the description of impedance and relaxation functions of dissipative resistive-capacitive system, namely the Cole-Cole, Davidson-Cole, and Havriliak-Negami models. We then propose and derive new q-deformed models based on modified evolution equations for the charge or voltage when the device is discharged into a parallel resistive load. We verify our results on anomalous spectral impedance response and time-domain relaxation data for voltage and charge obtained from a commercial supercapacitor., Comment: 9 pages, 3 figures

In this work, we aim to show that there are generally four possible mapping functions that can be used to map the time-domain or frequency-domain representations of an applied voltage input to the resulting time-domain or frequency-domain electrical charge output; i.e. when the capacitive device is voltage-charged. Alternatively, there are four more possible combinations when the device is current-charged. The dual relationship between each pair of functions for the case of voltage or charge input are provided in terms of single or double Fourier transforms. All eight system functions coincide with each other if and only if a constant time- and frequency-independent capacitance is considered.

In this study, we show that the discharge voltage pattern of a fractional-order supercapacitor from the same initial steady-state voltage into a constant resistor is dependent on the past charging voltage profile. The charging voltage was designed to follow a power-law function, i.e. $v_c(t)=V_{cc} \left( {t}/{t_{ss}}\right)^p \;(0