In this paper we compute the Fourier spectrum of the Fractal Interpolation Functions FIFs as introduced by Michael Barnsley. We show that there is an analytical way to compute them. In this paper we attempt to solve the inverse problem of FIF by using the spectrum
Differential uniformity is a significant concept in cryptography as it quantifies the degree of security of S-boxes respect to differential attacks. Power functions of the form $F(x)=x^d$ with low differential uniformity have been extensively studied
in the past decades due to their strong resistance to differential attacks and low implementation cost in hardware. In this paper, we give an affirmative answer to a recent conjecture proposed by Budaghyan, Calderini, Carlet, Davidova and Kaleyski about the differential uniformity of $F(x)=x^d$ over $mathbb{F}_{2^{4n}}$, where $n$ is a positive integer and $d=2^{3n}+2^{2n}+2^{n}-1$, and we completely determine its differential spectrum.
The total uncertainty measurement of basic probability assignment (BPA) in evidence theory has always been an open issue. Although many scholars have put forward various measures and requirements of bodies of evidence (BoE), none of them are widely r
ecognized. So in order to express the uncertainty in evidence theory, transforming basic probability assignment (BPA) into probability distribution is a widely used method, but all the previous methods of probability transformation are directly allocating focal elements in evidence theory to their elements without specific transformation process. Based on above, this paper simulates the pignistic probability transformation (PPT) process based on the idea of fractal, making the PPT process and the information volume lost during transformation more intuitive. Then apply this idea to the total uncertainty measure in evidence theory. A new belief entropy called Fractal-based (FB) entropy is proposed, which is the first time to apply fractal idea in belief entropy. After verification, the new entropy is superior to all existing total uncertainty measurements.
In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020] considered an infinite family of quadrinomials over $mathbb{F}_{2^{n}}$ of the form $x^3+a(x^{2^s+1})^{2^k}+bx^{3cdot 2^m}+c(x^{2^{s+m}+2^m})^{2^k}$, where $n=2m$ with $m$
odd. They proved that such kind of quadrinomials can provide new almost perfect nonlinear (APN) functions when $gcd(3,m)=1$, $ k=0 $, and $(s,a,b,c)=(m-2,omega, omega^2,1)$ or $((m-2)^{-1}~{rm mod}~n,omega, omega^2,1)$ in which $omegainmathbb{F}_4setminus mathbb{F}_2$. By taking $a=omega$ and $b=c=omega^2$, we observe that such kind of quadrinomials can be rewritten as $a {rm Tr}^{n}_{m}(bx^3)+a^q{rm Tr}^{n}_{m}(cx^{2^s+1})$, where $q=2^m$ and $ {rm Tr}^n_{m}(x)=x+x^{2^m} $ for $ n=2m$. Inspired by the quadrinomials and our observation, in this paper we study a class of functions with the form $f(x)=a{rm Tr}^{n}_{m}(F(x))+a^q{rm Tr}^{n}_{m}(G(x))$ and determine the APN-ness of this new kind of functions, where $a in mathbb{F}_{2^n} $ such that $ a+a^q eq 0$, and both $F$ and $G$ are quadratic functions over $mathbb{F}_{2^n}$. We first obtain a characterization of the conditions for $f(x)$ such that $f(x) $ is an APN function. With the help of this characterization, we obtain an infinite family of APN functions for $ n=2m $ with $m$ being an odd positive integer: $ f(x)=a{rm Tr}^{n}_{m}(bx^3)+a^q{rm Tr}^{n}_{m}(b^3x^9) $, where $ ain mathbb{F}_{2^n}$ such that $ a+a^q eq 0 $ and $ b $ is a non-cube in $ mathbb{F}_{2^n} $.
We derive two sufficient conditions for a function of a Markov random field (MRF) on a given graph to be a MRF on the same graph. The first condition is information-theoretic and parallels a recent information-theoretic characterization of lumpabilit
y of Markov chains. The second condition, which is easier to check, is based on the potential functions of the corresponding Gibbs field. We illustrate our sufficient conditions at the hand of several examples and discuss implications for practical applications of MRFs. As a side result, we give a partial characterization of functions of MRFs that are information-preserving.
In this paper, we propose a two-dimensional (2D) joint transmit array interpolation and beamspace design for planar array mono-static multiple-input-multiple-output (MIMO) radar for direction-of-arrival (DOA) estimation via tensor modeling. Our under
lying idea is to map the transmit array to a desired array and suppress the transmit power outside the spatial sector of interest. In doing so, the signal-tonoise ratio is improved at the receive array. Then, we fold the received data along each dimension into a tensorial structure and apply tensor-based methods to obtain DOA estimates. In addition, we derive a close-form expression for DOA estimation bias caused by interpolation errors and argue for using a specially designed look-up table to compensate the bias. The corresponding Cramer-Rao Bound (CRB) is also derived. Simulation results are provided to show the performance of the proposed method and compare its performance to CRB.