Bent Functions Research Articles

Self-Cleaning Bending Sensors Based on Semitransparent ZnO Nanostructured Films.

The design of multifunctional nanostructured materials is the key to the development of smart wearable devices. For instance, nanostructures endowed with both piezoelectric and photocatalytic activities could well be the workhorse for solar-light-driven self-cleaning wearable sensors. In this work, a simple strategy for the assembly of a flexible, semitransparent piezophotocatalytic system is demonstrated by leveraging rational wet chemistry synthesis of ZnO-based nanosheets/nanoflowers (NSs/NFs) under basic pH conditions onto flexible ITO/PET supports. A KMnO4 pretreatment before the ZnO synthesis (seeded ZnO) allows for the control of the density, size, and orientation of the NSs/NFs systems compared to the systems produced in the absence of seeding (seedless ZnO). The electrical response of the sensors is extracted at a 1 V bias as a function of bending in the interval between 0 and 90°, being the responsivity toward bending significantly enhanced by the KMnO4 treatment effect. The photocatalytic activity of the sensors is analyzed in aqueous solution (methylene blue, 25 μM) by a solar simulator, resulting in similar values between seedless and seeded ZnO. Upon bending the sensor, the photocatalytic activity of seedless ZnO is almost unaffected, whereas that of seeded ZnO is improved by about 25%. The sensor's reusability and repeatability are tested in up to three different cycles. These results open up the way toward the seamless integration of bending sensitivity and photocatalysis into a single device.

Nonexistence results of generalized bent functions from [formula omitted] to [formula omitted

New nonexistence results of generalized bent functions from Z2n to Zm are presented. In particular, we prove that there exists no generalized bent function from Z24 to Zm for every odd m.

Criteria for maximal nonlinearity of a function over a finite field

Abstract An n-place function over a field with q elements is called maximally nonlinear if it has the greatest nonlinearity among all such functions. Criteria and necessary conditions for maximal nonlinearity are obtained, which imply that, for even n, the maximally nonlinear functions are bent functions, but, for q > 2, the known families of bent functions are not maximally nonlinear. For an arbitrary finite field, a relationship between the Hamming distances from a function to all affine mappings and the Fourier spectra of the nontrivial characters of the function are found.

Explicit infinite families of bent functions outside the completed Maiorana–McFarland class

During the last five decades, many different secondary constructions of bent functions were proposed in the literature. Nevertheless, apart from a few works, the question about the class inclusion of bent functions generated using these methods is rarely addressed. Especially, if such a “new” family belongs to the completed Maiorana–McFarland ({{{mathcal {M}}}{{mathcal {M}}}}^#) class then there is no proper contribution to the theory of bent functions. In this article, we provide some fundamental results related to the inclusion in {{{mathcal {M}}}{{mathcal {M}}}}^# and eventually we obtain many infinite families of bent functions that are provably outside {{{mathcal {M}}}{{mathcal {M}}}}^#. The fact that a bent function f is in/outside {{{mathcal {M}}}{{mathcal {M}}}}^# if and only if its dual is in/outside {{{mathcal {M}}}{{mathcal {M}}}}^# is employed in the so-called 4-decomposition of a bent function on {mathbb {F}}_2^n, which was originally considered by Canteaut and Charpin (IEEE Trans Inf Theory 49(8):2004–2019, 2003) in terms of the second-order derivatives and later reformulated in (Hodžić et al. in IEEE Trans Inf Theory 65(11):7554–7565, 2019) in terms of the duals of its restrictions to the cosets of an (n-2)-dimensional subspace V. For each of the three possible cases of this 4-decomposition of a bent function (all four restrictions being bent, semi-bent, or 5-valued spectra functions), we provide generic methods for designing bent functions provably outside {{{mathcal {M}}}{{mathcal {M}}}}^#. For instance, for the elementary case of defining a bent function h(textbf{x},y_1,y_2)=f(textbf{x}) oplus y_1y_2 on {mathbb {F}}_2^{n+2} using a bent function f on {mathbb {F}}_2^n, we show that h is outside {{{mathcal {M}}}{{mathcal {M}}}}^# if and only if f is outside {{{mathcal {M}}}{{mathcal {M}}}}^#. This approach is then generalized to the case when two bent functions are used. More precisely, the concatenation f_1||f_1||f_2||(1oplus f_2) also gives bent functions outside {{{mathcal {M}}}{{mathcal {M}}}}^# if f_1 or f_2 is outside {{{mathcal {M}}}{{mathcal {M}}}}^#. The cases when the four restrictions of a bent function are semi-bent or 5-valued spectra functions are also considered and several design methods of constructing infinite families of bent functions outside {{{mathcal {M}}}{{mathcal {M}}}}^# are provided.

More constructions of n-cycle permutations

n-cycle permutations with small n have the advantage that their compositional inverses are efficient in terms of implementation. They can be also used in constructing Bent functions and designing codes. Since the AGW Criterion was proposed, the permuting property of several forms of polynomials has been studied. In this paper, characterizations of several types of n-cycle permutations are investigated. Three criteria for n-cycle permutations of the form xh(λ(x)), h(ψ(x))φ(x)+g(ψ(x)) and g(xqi−x+δ)+bx with general n are provided. We demonstrate these criteria by providing explicit constructions. For the form of xrh(xs), several new explicit triple-cycle permutations are also provided. Finally, we also consider triple-cycle permutations of the form xt+cTrqm/q(xs) and provide one explicit construction. Many of our constructions are both new in the n-cycle property and the permutation property.

Several classes of new weakly regular bent functions outside $$\mathcal{R}\mathcal{F}$$, their duals and some related (minimal) codes with few weights

Several classes of new weakly regular bent functions outside $$\mathcal{R}\mathcal{F}$$, their duals and some related (minimal) codes with few weights

Maximally nonlinear functions over finite fields

Abstract An n-place function over a field F q $ \mathbf {F}_q $ with q elements is called maximally nonlinear if it has the largest nonlinearity among all q-valued n-place functions. We show that, for even n=2, a function is maximally nonlinear if and only if its nonlinearity is q n − 1 ( q − 1 ) − q n 2 − 1 $ q^{n-1}(q - 1) - q^{\frac n2-1} $ ; for n=1, the corresponding criterion for maximal nonlinearity is q − 2. For q > 2 $ q \gt 2 $ and even n=2, we describe the set of all maximally nonlinear quadratic functions and find its cardinality. In this case, all maximally nonlinear quadratic functions are quadratic bent functions and their number is smaller than the halved number of the bent functions.

The solution to an open problem on the bentness of Mesnager's functions

Let n=2m. In the present paper, we study the binomial Boolean functions of the formfa,b(x)=Tr1n(ax2m−1)+Tr12(bx2n−13), where m is an even positive integer, a∈F2n⁎ and b∈F4⁎. We show that fa,b is a bent function if the Kloosterman sumKm(a2m+1)=1+∑x∈F2m⁎(−1)Tr1m(a2m+1x+1x) equals 4, thus settling an open problem of Mesnager. The proof employs tools including computing Walsh coefficients of Boolean functions via multiplicative characters, divisibility properties of Gauss sums, and graph theory.

A study of binary decision diagram characteristics of bent Boolean functions

Bent Boolean functions exist only for an even number of variables, moreover, they are unbalanced. Therefore, they are used in coding theory and in many areas of computer science. General form of bent functions is still unknown. One way of representing Boolean functions is with a reduced ordered binary decision diagram (ROBDD). The strength of ROBDDs is that they can represent Boolean functions data with a high level of redundancy in a compact form, as long as the data is encoded in such a way that the redundancy is exposed. This paper investigates characteristics of bent functions with focus on their ROBDD parameters. Decision diagram experimental framework has been used for implementation of a program for calculation of the ROBDD parameters. The results presented in this paper are intended to be used to create methods for the construction of bent functions using a ROBDD as a data structure from which the bent functions can be discovered.

Beam bending and Λ-fractional analysis

<abstract> <p>Since the global stability criteria for Λ-fractional mechanics have been established, the Λ-fractional beam bending problem is discussed within that context. The co-existence of the phase phenomenon is revealed, allowing for elastic curves with non-smooth curvatures. The variational bending problem in the Λ-fractional space is considered. Global minimization of the total energy function of beam bending is necessarily applied. The variational Euler-Lagrange equation yields an equilibrium equation of the elastic curve, with the simultaneous possible corners being expressed by Weierstrass-Erdmann corner conditions.</p> </abstract>

Vandermonde sets, hyperovals and Niho bent functions

<p style='text-indent:20px;'>We consider relationships between Vandermonde sets and hyperovals. Hyperovals are Vandermonde sets, but in general, Vandermonde sets are not hyperovals. We give necessary and sufficient conditions for a Vandermonde set to be a hyperoval in terms of power sums. Therefore, we provide purely algebraic criteria for the existence of hyperovals. Furthermore, we give necessary and sufficient conditions for the existence of hyperovals in terms of <inline-formula><tex-math id="M1">\begin{document}$ g $\end{document}</tex-math></inline-formula>-functions, which can be considered as an analog of Glynn's Theorem for <inline-formula><tex-math id="M2">\begin{document}$ o $\end{document}</tex-math></inline-formula>-polynomials. We also get some important applications to Niho bent functions.</p>

Derivatives of bent functions in connection with the bent sum decomposition problem

Derivatives of bent functions in connection with the bent sum decomposition problem

A further study on the construction methods of bent functions and self-dual bent functions based on Rothaus’s bent function

A further study on the construction methods of bent functions and self-dual bent functions based on Rothaus’s bent function

Modulation of the TCO/MoOx Front Contact Enables >21% High-Efficiency Dopant-Free Silicon Solar Cells

Highly efficient dopant-free silicon solar cell requires effective electron- and hole-selective contacts. We have recently reported a remarkable efficiency of 21.3% for dopant-free silicon solar cells with a low work function and transparent and conductive MgFxOy electron extraction. The transparent conductive oxide (TCO) films and molybdenum oxide (MoOx) front hole contact also play significant roles in the high efficiency of dopant-free solar cells. In this study, zinc-doped indium oxide (IZO), tin-doped indium oxide (ITO), and titanium-doped indium oxide (ITiO) have been deposited via magnetron sputtering at room temperature. The work function and optoelectrical properties of TCO films and energy band bending together with the MoOx layer are investigated. The impact of this band bending on the photoelectrical performances of devices is analyzed in detail as a function of the MoOx/TCO work function mismatch. The optimized IZO film presents the highest work function (5.21 eV) and conductivity (2140 S/cm), resulting in reduced barrier height at the MoOx/TCO interface and enhanced hole-selective transport. The light-soaking treatment is introduced to repair the sputtered damage during the IZO deposition. Finally, an impressive cell efficiency of 21.0%, together with Voc, FF, and Jsc of 712.4 mV, 76.4%, and 38.6 mA/cm2, respectively, shows great potential and provides an effective solution to obtain highly efficient dopant-free silicon solar cells.

Quadratic almost bent functions—Their partial characterization and design in the spectral domain

Quadratic almost bent (AB) functions are characterized by the property that the duals of their component functions are bent functions. We prove that these duals are also quadratic and illustrate that these bent duals may give rise to vectorial bent functions (in certain cases having a maximal output dimension). A necessary and sufficient condition for ensuring bentness of the linear combinations of quadratic bent duals is provided. Moreover, we provide a rather detailed analysis related to the structure of quadratic AB functions in the spectral domain, more precisely with respect to their Walsh supports, their intersection and restrictions of these bent duals to suitable subspaces. In particular, we completely determine the intersection of Walsh supports of the coordinate (semi-bent) functions for Gold AB mappings. We also provide the design of quadratic AB functions in the spectral domain by identifying (using computer simulations) suitable sets of bent dual functions. For instance, when n = 7 , this approach provides several AB functions which are not CCZ-equivalent to Gold functions.

Transparency Order and Cross-Correlation Analysis of Boolean Functions

Transparency order is considered to be a cryptographically significant property that characterizes the resistance of S-boxes in opposition to differential power analysis attacks. The S-box having low transparency order is more resistant to these attacks. Until now, little attempts have been noticed to examine theoretically the transparency order and its relationship with other cryptographic properties. All constructions associated with transparency order are relying on search algorithms. In this paper, we discuss the new interpretation of bent functions in terms of their transparency order. Using the concept of vector concatenation and correlation characteristics, we find the transparency order of Boolean functions. The notion of complementary transparency order is given. For a pair of Boolean functions, we interpret complementary transparency order by their Walsh-Hadamard transform. We establish a relationship of transparency order with cross-correlation for a pair of Boolean functions. We find a relationship of transparency order with <img src=image/13428361_01.gif>−variable decomposition bent functions. We generalize the bounds on sum-of-squares of autocorrelation in terms of transparency order of Boolean functions using Walsh-Hadamard spectra. Further the transparency order of a function fulfilling the propagation criterion about a linear subspace is evaluated.

On the Electronic Structure of 2H-MoS2: Correlating DFT Calculations and In-Situ Mechanical Bending on TEM.

We present a systematic density functional theory study to determine the electronic structure of bending 2H-MoS2 layers up to 75° using information from in-situ nanoindentation TEM observations. The results from HOMO/LUMO and density of states plots indicate a metallic transition from the typical semiconducting phase, near Fermi energy level (EF) as a function of bending, which can mainly occur due to bending curvatures inducing a stretching and contracting of sulfur-sulfur chemical bonds located mostly over basal (001)-plane; furthermore, molybdenum ions play a major role in such transitions due to reallocation of their metallic d-character orbitals and the creation of “free electrons”, possibly having an overlap between Mo-dx2-y2 and Modz2 orbitals. This research on the metallic transition of 2H-MoS2 allows us to understand the high catalytic activity for MoS2 nanostructures as extensively reported in the literature.

Several classes of bent functions over finite fields

Inspired by the works of Mesnager (IEEE Trans Inf Theory 60(7):4397–4407, 2014) and Tang et al. (IEEE Trans Inf Theory 63(10):6149–6157, 2017), we study a class of bent functions of the form $$f(x)=g(x)+F(f_1(x),f_2(x),\ldots ,$$ $$f_{\tau }(x))$$ , where g(x) is a function from $$\mathbb {F}_{p^{n}}$$ to $${\mathbb {F}}_{p}$$ , $$f_i(x)=\prod _{j=1}^{\kappa _i}{\text {Tr}}(u_{ij}x)$$ for $$1\le i\le \tau $$ , $$\kappa _i\ge 1$$ and $$F(x_1,\ldots ,x_{\tau })$$ is a reduced polynomial in $${\mathbb {F}}_{p}[x_1,\ldots ,x_{\tau }]$$ . As a consequence, we obtain a generic result on the Walsh transform of f(x) in terms of g(x) and characterize the bentness of f(x) for the cases $$F(x_1,\ldots ,x_{\tau })$$ without and with restrictions respectively, which enables us to generalize some earlier works and derive new bent functions from known ones. In addition, we study the construction of bent functions f(x) when g(x) is not bent for the first time and present a class of bent functions from non-bent Gold functions.

Linear codes and incidence structures of bent functions and their generalizations

In this paper, we consider further applications of (n,m)-functions for the construction of 2-designs. For instance, we provide a new application of the extended Assmus-Mattson theorem, by showing that linear codes of certain APN functions with the classical Walsh spectrum support 2-designs. With this result, we give several sufficient conditions for an APN function with the classical Walsh spectrum to be CCZ-inequivalent to a quadratic one. On the other hand, we use linear codes and combinatorial designs in order to study important properties of (n,m)-functions. In particular, we provide a characterization of a quadratic Boolean bent function by means of the 2-transitivity of its automorphism group. Finally, we give a new design-theoretic characterization of (n,m)-plateaued and (n,m)-bent functions and provide a coding-theoretic as well as a design-theoretic interpretation of the extendability problem for (n,m)-bent functions.

Robust Code Constructions Based on Bent Functions and Spline Wavelet Decomposition

The paper investigates new robust code constructions based on bent functions and spline–wavelet transformation. Implementation of bent functions in code construction increases the probability of error detection in the data channel and cryptographic devices. Meanwhile, the use of spline wavelet theory for constructing the codes gives the possibility to increase system security from the actions of an attacker. Presented constructions combine spline-wavelets functions and bent functions. Developed robust codes, compared to existing ones, have a higher parameter of the maximum error masking probability. Illustrated codes ensure the security of transmitted information. Some of the granted constructions were implemented on FPGA.