A Mathematical Approach for Generating a Highly Non-Linear Substitution Box Using Quadratic Fractional Transformation

Nowadays,one of the most important difficulties is the protection and privacy of confidential data.To address these problems,numerous organizations rely on the use of cryptographic techniques to secure data from illegal activities and assaults.Modern cryptographic ciphers use the non-linear component of block cipher to ensure the robust encryption process and lawful decoding of plain data during the decryption phase.For the designing of a secure substitution box(S-box),non-linearity(NL)which is an algebraic property of the S-box has great importance.Consequently,the main focus of cryptographers is to achieve the S-box with a high value of non-linearity.In this suggested study,an algebraic approach for the construction of 16×16 S-boxes is provided which is based on the fractional transformation Q(z)=1/α(z)^(m)+β(mod257)and finite field.This technique is only applicable for the even number exponent in the range(2-254)that are not multiples of 4.Firstly,we choose a quadratic fractional transformation,swap each missing element with repeating elements,and acquire the initial S-box.In the second stage,a special permutation of the symmetric group S256 is utilized to construct the final S-box,which has a higher NL score of 112.75 than the Advanced Encryption Standard(AES)S-box and a lower linear probability score of 0.1328.In addition,a tabular and graphical comparison of various algebraic features of the created S-box with many other S-boxes from the literature is provided which verifies that the created S-box has the ability and is good enough to withstand linear and differential attacks.From different analyses,it is ensured that the proposed S-boxes are better than as compared to the existing S-boxes.Further these S-boxes can be utilized in the security of the image data and the text data.

An Improved Robust Model Predictive Control Approach to Systems with Linear Fractional Transformation Perturbations

In this paper, a robust model predictive control approach is proposed for a class of uncertain systems with time-varying, linear fractional transformation perturbations. By adopting a sequence of feedback control laws instead of a single one, the control performance can be improved and the region of attraction can be enlarged compared with the existing model predictive control （MPC） approaches. Moreover, a synthesis approach of MPC is developed to achieve high performance with lower on-line computational burden. The effectiveness of the proposed approach is verified by simulation examples.

From fractional Fourier transformation to quantum mechanical fractional squeezing transformation

By converting the triangular functions in the integration kernel of the fractional Fourier transformation to the hyperbolic function,i.e.,tan α → tanh α,sin α →〉 sinh α,we find the quantum mechanical fractional squeezing transformation（FrST） which satisfies additivity.By virtue of the integration technique within the ordered product of operators（IWOP） we derive the unitary operator responsible for the FrST,which is composite and is made of e^iπa＋a/2 and exp[iα/2（a^2 ＋a^＋2）.The FrST may be implemented in combinations of quadratic nonlinear crystals with different phase mismatches.

Two mutually conjugated tripartite entangled states and their fractional Fourier transformation kernel

We newly construct two mutually-conjugate tripartite entangled state representations, based on which we propose the formulation of three-mode entangled fractional Fourier transformation （EFFT） and derive the transformation kernel. The EFFT＇s additivity property is proved and the eigenmode of EFFT is derived. As an application, we calculate the EFFT of the three-mode squeezed vacuum state.

The Factors for Transformation between the Fractions of Speciation of Trace Metals in Lake Sediments

Objective Despite the fact that the bioavailability of trace metals indicated by their speciation has been an indispensable parameter in the assessment and treatment of the environmental pollution of trace metals, many studies have suggested that the bioavailability of trace metals may change according to the conditions of the environment, and the speciation of trace metals can also transform between some fractions. These transformations are related with these factors such as the compositions, microorganism, time, and other physical-chemical conditions of the system. Our work aims to systematically investigate and probe the factors to affect the transformation aside from analysis at certain time-place. The results of these understanding and investigations can be used for reasonably determining the allocation of financial and technical resources in natural and engineered processes, with bringing about inspirations from the evolution of the speciation of the trace metals on environmental impacts.

Quantum entangled fractional Fourier transform based on the IWOP technique

In our previous papers,the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics,and the calculation produced quantum mechanical operators corresponding to the generation of fractional Fourier transform.The core function of the coordinate-momentum exchange operators in the addition law of fractional Fourier transform was analyzed too.In this paper,the bivariate operator Hermite polynomial theory and the technique of integration within an ordered product of operators(IWOP)are used to establish the entanglement fractional Fourier transform theory to the extent of quantum.A new function generating formula and an operator for generating quantum entangled fractional Fourier transform are obtained using the fractional Fourier transform relationship in a pair of conjugated entangled state representations.

On Nonlinear Conformable Fractional Order Dynamical System via Differential Transform Method

This article studies a nonlinear fractional order Lotka-Volterra prey-predator type dynamical system.For the proposed study,we consider the model under the conformable fractional order derivative(CFOD).We investigate the mentioned dynamical system for the existence and uniqueness of at least one solution.Indeed,Schauder and Banach fixed point theorems are utilized to prove our claim.Further,an algorithm for the approximate analytical solution to the proposed problem has been established.In this regard,the conformable fractional differential transform(CFDT)technique is used to compute the required results in the form of a series.Using Matlab-16,we simulate the series solution to illustrate our results graphically.Finally,a comparison of our solution to that obtained for the Caputo fractional order derivative via the perturbation method is given.

SAR target detection based on the optimal fractional Gabor spectrum feature

In this paper,an algorithm based on a fractional time-frequency spectrum feature is proposed to improve the accuracy of synthetic aperture radar(SAR)target detection.By extending the fractional Gabor transform(FrGT)into two dimensions,the fractional time-frequency spectrum feature of an image can be obtained.In the achievement process,we search for the optimal order and design the optimal window function to accomplish the two-dimensional optimal FrGT.Finally,the energy attenuation gradient(EAG)feature of the optimal time-frequency spectrum is extracted for high-frequency detection.The simulation results show the proposed algorithm has a good performance in SAR target detection and lays the foundation for recognition.

High-sensitive state perception method for inverter-fed machine turn insulation based on FrFT-Mel

Amidst the swift advancement of new power systems and electric vehicles,inverter-fed machines have progressively materialized as a pivotal apparatus for efficient energy conversion.Stator winding turn insulation failure is the root cause of inverter-fed machine breakdown.The online monitoring of turn insulation health can detect potential safety risks promptly,but faces the challenge of weak characteristics of turn insulation degradation.This study proposes an innovative method to evaluate the turn insulation state of inverter-fed machines by utilizing the fractional Fourier transform with a Mel filter(FrFT-Mel).First,the sensitivity of the high-frequency(HF)switching oscillation current to variations in turn insulation was analyzed within the fractional domain.Subsequently,an improved Mel filter is introduced,and its structure and parameters are specifically designed based on the features intrinsic to the common-mode impedance resonance point of the electrical machine.Finally,an evaluation index was proposed for the turn insulation state of inverter-fed machines.Experimental results on a 3kW permanent magnet synchronous machine(PMSM)demonstrate that the proposed FrFT-Mel method significantly enhances the sensitivity of turn insulation state perception by approximately five times,compared to the traditional Fourier transform method.

Security-Enhanced Directional Modulation Based on Two-Dimensional M-WFRFT

Directional modulation(DM)is one of the most promising secure communication techniques.However,when the eavesdropper is co-located with the legitimate receiver,the conventional DM has the disadvantages of weak anti-scanning capability,anti-deciphering capability,and low secrecy rate.In response to these problems,we propose a twodimensional multi-term weighted fractional Fourier transform aided DM scheme,in which the legitimate receiver and the transmitter use different transform terms and transform orders to encrypt and decrypt the confidential information.In order to further lower the probability of being deciphered by an eavesdropper,we use the subblock partition method to convert the one-dimensional modulated signal vector into a twodimensional signal matrix,increasing the confusion of the useful information.Numerical results demonstrate that the proposed DM scheme not only provides stronger anti-deciphering and anti-scanning capabilities but also improves the secrecy rate performance of the system.

Aerial target threat assessment based on gated recurrent unit and self-attention mechanism

Aerial threat assessment is a crucial link in modern air combat, whose result counts a great deal for commanders to make decisions. With the consideration that the existing threat assessment methods have difficulties in dealing with high dimensional time series target data, a threat assessment method based on self-attention mechanism and gated recurrent unit(SAGRU) is proposed. Firstly, a threat feature system including air combat situations and capability features is established. Moreover, a data augmentation process based on fractional Fourier transform(FRFT) is applied to extract more valuable information from time series situation features. Furthermore, aiming to capture key characteristics of battlefield evolution, a bidirectional GRU and SA mechanisms are designed for enhanced features.Subsequently, after the concatenation of the processed air combat situation and capability features, the target threat level will be predicted by fully connected neural layers and the softmax classifier. Finally, in order to validate this model, an air combat dataset generated by a combat simulation system is introduced for model training and testing. The comparison experiments show the proposed model has structural rationality and can perform threat assessment faster and more accurately than the other existing models based on deep learning.

An Efficient Radar Detection Method of Maneuvering Small Targets

Detection of maneuvering small targets has always been an important yet challenging task for radar signal processing.One primary reason is that target variable motions within coherent processing interval generate energy migrations across multiple resolution bins,which severely deteriorate the parameter estimation performance.A coarse-to-fine strategy for the detection of maneuvering small targets is proposed.Integration of small points segmented coherently is performed first,and then an optimal inter-segment integration is utilized to derive the coarse estimation of the chirp rate.Sparse fractional Fourier transform(FrFT)is then employed to refine the coarse estimation at a significantly reduced computational complexity.Simulation results verify the proposed scheme that achieves an efficient and reliable maneuvering target detection with-16dB input signal-to-noise ratio(SNR),while requires no exact a priori knowledge on the motion parameters.

Kerogen Kinetic Distributions and Simulations Provide Insights into Petroleum Transformation Fraction (TF) Profiles of Organic-Rich Shales

Two hundred and fifty single first-order Arrhenius reactions are simulated to generate S2 pyrograms at three heating rates 25,15,and 5°C·min-1.The activation energy(E)and pre-exponential factor(A)of the reactions simulated follow a long-established trend of those variable values displayed by shales and kerogens.The characteristics of the transformation fraction(TF)profiles(product generation window temperatures)of the simulated single reactions are compared to the TF profiles of recorded shale pyrograms generated by multiple reactions with different E-A values lying near the defined E-A trend.Important similarities and differences are observed between the TF profile values of the two datasets.The similarities support the spread of E-A values involved in shale pyrogram best fits.The differences are most likely explained by the complexity of the multiple kerogen first-order and second-order reactions contributing to the recorded shale pyrograms versus the simplicity and crispness of the single first-order reactions simulated.The results also justify the validity of using the previously described“variable E-A pyrogram-fitting method”of multi-heating-rate shale pyrograms enabling optimizers to choose multiple reactions from an unlimited range of E-A values.In contrast,further doubt is cast on the validity of the constant-A pyrogram-fitting method used by the Easy%Ro technique,in that a distribution of reactions with a single A value is unlikely to represent the complex variety of kerogen macerals observed in shale formations.TF profiles generated by the variable E-A pyrogram-fitting method lie close to the established E-A trend and are likely to provide more realistic TF generation window temperatures than TF profiles generated by the constant-A pyrogram-fitting method.

Chirp-Rate Resolution of Fractional Fourier Transform in Multi-component LFM Signal

Distinguishing close chirp-rates of different linear frequency modulation （LFM） signals under concentrated and complicated signal environment was studied. Firstly, detection and parameter estimation of multi-component LFM signal were used by discrete fast fractional Fourier transform （FrFT）. Then the expression of chirp-rate resolution in fractional Fourier domain （FrFD） was deduced from discrete normalize time-frequency distribution, when multi-component LFM signal had only one center frequency. Furthermore, the detail influence of the sampling time, sampling frequency and chirp-rate upon the resolution was analyzed by partial differential equation. Simulation results and analysis indicate that increasing the sampling time can enhance the resolution, but the influence of the sampling frequency can he omitted. What＇s more, in multi-component LFM signal, the chirp-rate resolution of FrFT is no less than a minimal value, and it mainly dependent on the biggest value of chirp-rates, with which it has an approximately positive exponential relationship.

Fractional Envelope Analysis for Rolling Element Bearing Weak Fault Feature Extraction

The bearing weak fault feature extraction is crucial to mechanical fault diagnosis and machine condition monitoring.Envelope analysis based on Hilbert transform has been widely used in bearing fault feature extraction. A generalization of the Hilbert transform, the fractional Hilbert transform is defined in the frequency domain, it is based upon the modification of spatial filter with a fractional parameter, and it can be used to construct a new kind of fractional analytic signal. By performing spectrum analysis on the fractional envelope signal, the fractional envelope spectrum can be obtained. When weak faults occur in a bearing, some of the characteristic frequencies will clearly appear in the fractional envelope spectrum. These characteristic frequencies can be used for bearing weak fault feature extraction.The effectiveness of the proposed method is verified through simulation signal and experiment data.

Order selection in fractional Fourier transform based beamforming

Traditionally,beamforming using fractional Fourier transform（FrFT） involves a trial-and-error based FrFT order selection which is impractical.A new numerical order selection scheme is presented based on fractional power spectra（FrFT moment） of the linear chirp signal.This method can adaptively determine the optimum FrFT order by maximizing the second-order central FrFT moment.This makes the desired chirp signal substantially concentrated whereas the noise is rejected considerably.This improves the mean square error minimization beamformer by reducing effectively the signal-noise cross terms due to the finite data length de-correlation operation.Simulation results show that the new method works well under a wide range of signal to noise ratio and signal to interference ratio.

Image recovery from double amplitudes in fractional Fourier domain

The classical Gerchberg-Saxton algorithm is introduced into the image recovery in fractional Fourier domain after adaptation. When this algorithm is applied directly, its performance is good for smoothed image, but bad for unsmoothed image. Based on the diversity of fractional Fourier transform on its orders, this paper suggests a novel iterative algorithm, which extracts the information of the original image from amplitudes of its fractional Fourier transform at two orders. This new algorithm consists of two independent Gerchberg-Saxton procedures and an averaging operation in each circle. Numerical simulations are carried out to show its validity for both smoothed and unsmoothed images with most pairs of orders in the interval [0, 1].

New analytical exact solutions of time fractional KdV KZK equation by Kudryashov methods

In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov （KdV-KZK） equa- tion are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann-Liouville derivative is used to convert the nonlinear time fractional KdV-KZK equation into the non- linear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV-KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV-KZK equation.

Research Progress of the Sampling Theorem Associated with the Fractional Fourier Transform

Sampling is a bridge between continuous-time and discrete-time signals,which is import-ant to digital signal processing.The fractional Fourier transform(FrFT)that serves as a generaliz-ation of the FT can characterize signals in multiple fractional Fourier domains,and therefore can provide new perspectives for signal sampling and reconstruction.In this paper,we review recent de-velopments of the sampling theorem associated with the FrFT,including signal reconstruction and fractional spectral analysis of uniform sampling,nonuniform samplings due to various factors,and sub-Nyquist sampling,where bandlimited signals in the fractional Fourier domain are mainly taken into consideration.Moreover,we provide several future research topics of the sampling theorem as-sociated with the FrFT.

Digital watermarking for still image based on discrete fractional fourier transform

Presents a digital watermarking technique based on discrete fractional Fourier transform (DFRFT), discusses the transformation of the original image by DFRFT, and the modification of DFRFT coefficients of the original image by the information of watermark, and concludes from experimental results that the proposed technique is robust to lossy compression attack.