Every Boolean bent function f can be written either as a concatenation f = f1|| f2 of two complementary semi-bent functions f1, f2

Every Boolean bent function$f$can be written either as a concatenation$f=f_{1}\Vert f_{2}$of two complementary semi-bent functions$f_{1}, f_{2}$; or as a concatenation$f=f_{1}\Vert f_{2}\Vert f_{3}\Vert f_{4}$of four Boolean functions$f_{1}, f_{2}, f_{3}, f_{A}$, all of which are simultaneously bent, semi-bent, or 5-valued spectra-functions. In this context, it is essential to ask: When does a bent concatenation$f$(not) belong to the completed Maiorana-McFarland class${\mathcal{M}}^{\# }{?}$In this article, we answer this question completely by providing a full characterization of the structure of$\mathcal{M}$-subspaces for the concatenation of the form$f=f_{1}\Vert f_{2}$and$f=f1\Vert f2\Vert f\mathrm{s}\Vert f4$, which allows us to specify the necessary and sufficient conditions so that$f$is outside$\mathcal{M} \neq$. Based on these conditions, we propose several explicit design methods of specifying bent functions outside$\mathcal{M}^{\#}$in the special case when$f=g\Vert h\Vert g\Vert (h+1)$, where$g$and$h$are bent functions.

Abstract In this article, we identify certain instances of bent functions, constructed using the so-called $$P_\tau $$ P τ property, that are provably outside the completed Maiorana–McFarland ( $${\mathcal{M}\mathcal{M}}^\#$$ M M # ) class. This also partially answers an open problem in posed by Kan et al. (IEEE Trans Inf Theory, https://doi.org/10.1109/TIT.2022.3140180 , 2022). We show that this design framework (using the $$P_\tau $$ P τ property), can provide instances of bent functions that are outside the known classes of bent functions, including the classes $${\mathcal{M}\mathcal{M}}^\#$$ M M # , $${{\mathcal {C}}},{{\mathcal {D}}}$$ C , D and $${{\mathcal {D}}}_0$$ D 0 , where the latter three were introduced by Carlet in the early nineties. We provide two generic methods for identifying such instances, where most notably one of these methods uses permutations that may admit linear structures. For the first time, a set of sufficient conditions for the functions of the form $$h(y,z)=Tr(y\pi (z)) + G_1(Tr_1^m(\alpha _1y),\ldots ,Tr_1^m(\alpha _ky))G_2(Tr_1^m(\beta _{k+1}z),\ldots ,Tr_1^m(\beta _{\tau }z))+ G_3(Tr_1^m(\alpha _1y),\ldots ,Tr_1^m(\alpha _ky))$$ h ( y , z ) = T r ( y π ( z ) ) + G 1 ( T r 1 m ( α 1 y ) , … , T r 1 m ( α k y ) ) G 2 ( T r 1 m ( β k + 1 z ) , … , T r 1 m ( β τ z ) ) + G 3 ( T r 1 m ( α 1 y ) , … , T r 1 m ( α k y ) ) to be bent and outside $${\mathcal{M}\mathcal{M}}^\#$$

An optimized implementation of S-boxes has a significant impact on the performance of cryptographic primitives. SAT-based methods can find optimal implementations for moderately sized S-boxes but their efficiency decreases when handling complex S-boxes. To improve the efficiency of the implementations, we propose two different methods, namely OR-encoding and IF-encoding, to encode the implementations of S-boxes. Furthermore, we also simplify the encoding of the outputs of logic gates and introduce new SAT-based search methods to optimize the implementations of S-boxes. Finally, to get a better trade-off between the search results (optimized implementations of S-boxes) and the search efficiency (in terms of time complexity), an encoding scheme using local solutions is proposed. Compared to the previous methods, our algorithms are relatively simple and more efficient. For instance, when a serial software implementation is considered, then the S-boxes of Sycon, ASCON, and the$\chi $function in Xoodyak, require 6, 1, and 2 fewer programming instructions, respectively, than the best known methods. Similar improvements are obtained for hardware implementations of S-boxes in some cryptographic primitives (e.g. LBlock, RECTANGLE, PRESENT/PHOTON-Beetle, TWINE, and ASCON), with the saving of gate equivalent (GE) that range from 1.67GE to 5.34GE compared to the current best implementations. Furthermore, our model can be applied to 6-bit, 7-bit, and 8-bit S-boxes, when the considered S-boxes are of low complexity.

In this article, we provide the first systematic analysis of bent functions f on Fn2 in the Maiorana-McFarland class M regarding the origin and cardinality of their M-subspaces, i.e., vector subspaces such that for any two elements a, b from this subspace, the second-order derivative DaDbf is the zero function on Fn2. By imposing restrictions on permutations π of Fn/2 2, we specify the conditions so that Maiorana-McFarland bent functions f(x, y) = x · π(y) + h(y) admit a unique M-subspace of dimension n/2. On the other hand, we show that permutations π with linear structures give rise to Maiorana-McFarland bent functions that do not have this property. In this way, we contribute to the classification of Maiorana-McFarland bent functions, since the number of M-subspaces of a fixed dimension is invariant under equivalence. Additionally, we give several generic methods of specifying permutations π so that f ∈ M admits a unique M-subspace. Most notably, using the knowledge about M-subspaces, we show that using the bent 4-concatenation of four suitably chosen Maiorana-McFarland bent functions on Fn−2 2, one can in a generic manner generate bent functions on Fn2 outside the completed Maiorana-McFarland class M# for any even n ≥ 8. Remarkably, with our construction methods, it is possible to obtain inequivalent bent functions on F82 not stemming from the two primary classes, the partial spread class PS and M. In this way, we contribute to a better understanding of the origin of bent functions in eight variables, since only a small fraction of about 276 bent functions stems from PS and M, whereas their total number on F82 is approximately 2106.

In this article, we propose several generic methods for constructing minimal linear codes over the prime field Fp. The first construction uses the direct sum of an arbitrary functionf: Fpr→ Fpand a bent functiong: Fps→ Fpto induce minimal codes with parameters [pr+s- 1,r+s+ 1] and minimum distance larger thanpr(p- 1)(ps-1-ps/2-1). For the first time, we provide a general construction of linear codes from a subclass of non-weakly regular plateaued functions, which partially answers an open problem posed by Li and Mesnager. The second construction deals with a bent functiong: Fpm→ Fpand a suitable subspace of derivatives ofg, i.e., functions of the formg(y+a) -g(y) for somea∈ F*pm. We also provide a sound generalization of the recently introduced concept of non-covering permutations. Some important structural properties of this class of permutations are derived in this context. The most remarkable observation is that the class of non-covering permutations includes all APN power permutations (characterized by having two-to-one derivatives). Finally, the last construction combines the previous two methods (direct sum, non-covering permutations and subspaces of derivatives), using a bent function in the Maiorana-McFarland class, to construct minimal codes (even those violating the Ashikhmin-Barg bound) with larger dimensions. This last method proves to be highly flexible since it can lead to several non-equivalent codes, depending to a great extent on the choice of the underlying non-covering permutation.

Abstract 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 ( $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M 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 $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # and eventually we obtain many infinite families of bent functions that are provably outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # . The fact that a bent function f is in/outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # if and only if its dual is in/outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # is employed in the so-called 4-decomposition of a bent function on $${\mathbb {F}}_2^n$$ 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 $$(n-2)$$ ( 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 $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # . For instance, for the elementary case of defining a bent function $$h(\textbf{x},y_1,y_2)=f(\textbf{x}) \oplus y_1y_2$$ h ( x , y 1 , y 2 ) = f ( x ) ⊕ y 1 y 2 on $${\mathbb {F}}_2^{n+2}$$ F 2 n + 2 using a bent function f on $${\mathbb {F}}_2^n$$ F 2 n , we show that h is outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # if and only if f is outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$ M M # . This approach is then generalized to the case when two bent functions are used. More precisely, the concatenation $$f_1||f_1||f_2||(1\oplus f_2)$$ f 1 | | f 1 | | f 2 | |

In the mid‐sixties, Rothaus introduced the notion of bent function and later presented a secondary construction of bent functions (building new bent functions from already defined ones), called Rothaus’ construction. In Zhang et al. 2017 (‘Constructing bent functions outside the Maiorana–Mcfarland class using a general form of Rothaus,’ IEEE Transactions on Information Theory , 2017, vol. 63, no. 8, pp. 5336–5349.’) provided two constructions of bent functions using a general form of Rothaus and showed that the obtained classes lie outside the completed Maiorana–McFarland ( ) class. In this study, the authors propose two similar methods for constructing bent functions outside the completed class but with significantly simplified sufficient conditions compared to those in Zhang et al. 2017. These simplified conditions do not induce any serious restrictions on the choice of permutations used in the construction apart from a simple requirement on their algebraic degree and the request that the component functions of one permutation do not admit linear structures. This enables us to generate a huge class of bent functions lying outside the completed class. Even more importantly, they prove that the new classes of bent functions are affine inequivalent to the bent functions in Zhang et al. 2017.

Encryption algorithm has an important application in ensuring the security of the Internet of Things. Boolean function is the basic component of symmetric encryption algorithm, and its many cryptographic properties are important indicators to measure the security of cryptographic algorithm. This paper focuses on the sum-of-squares indicator of Boolean function; an upper bound and a lower bound of the sum-of-squares on Boolean functions are obtained by the decomposition Boolean functions; some properties and a search algorithm of Boolean functions with the same autocorrelation (or cross-correlation) distribution are given. Finally, a construction method to obtain a balanced Boolean function with small sum-of-squares indicator is derived by decomposition Boolean functions. Compared with the known balanced Boolean functions, the constructed functions have the higher nonlinearity and the better global avalanche characteristics property.

The design of plateaued functions over GF(2)n, also known as 3-valued Walsh spectra functions (taking the values from the set {0, ±2Γ(n+s/2)1}), has been commonly approached by specifying a suitable algebraic normal form which then induces this particular Walsh spectral characterization. In this paper, we consider the reversed design method which specifies these functions in the spectral domain by specifying a suitable allocation of the nonzero spectral values and their signs. We analyze the properties of trivial and nontrivial plateaued functions (as affine inequivalent distinct subclasses), which are distinguished by their Walsh support Sf (the subset of GF(2)n having the nonzero spectral values) in terms of whether it is an affine subspace or not. The former class exactly corresponds to partially bent functions and admits linear structures, whereas the latter class may contain functions without linear structures. A simple sufficient condition on Sf , which ensures the nonexistence of linear structures, is derived and some generic design methods of nontrivial plateaued functions without linear structures are given. The extended affine equivalence of plateaued functions is also addressed using the concept of dual of plateaued functions. Furthermore, we solve the problem of specifying disjoint spectra (non)trivial plateaued functions of maximal cardinality whose concatenation can be used to construct bent functions in a generic manner. This approach may lead to new classes of bent functions due to large variety of possibilities to select underlying duals that define these disjoint spectra plateaued functions. An additional method of specifying affine in equivalent plateaued functions, obtained by applying a nonlinear transform to their input domain, is also given.

In this paper, we give an efficient method for constructing a large set of disjoint spectra functions without linear structures, which are not equivalent to partially linear functions. This positively answers the open problem [“how to construct a large set of disjoint spectra functions which are not (linearly equivalent to) partially linear functions” raised by Zhang and Xiao]. At the same time, this significantly extends a recent result of Zhang, where a method of specifying four disjoint spectra functions was given. It is demonstrated that such sets can be utilized in the design of highly nonlinear resilient functions. In the second part, based on a generalization of the indirect sum method, we give an alternative approach for designing sets of disjoint spectra functions of even larger cardinality than already given ones, but these functions then admit linear structures. In addition, it is shown that using suitable initial functions in the generalized indirect sum method we can specify highly nonlinear resilient Boolean functions (in odd number of input variables n) whose nonlinearity in many cases exceeds the current best known values. Moreover, we design some balanced functions (for odd n) that also achieve the highest nonlinearity known.

The minimum degree (resp. algebraic degree ) of a Boolean permutation is the minimum (resp. maximum) algebraic degree of all the non‐zero linear combinations of its coordinate functions. In this study, the authors concentrate on the design of Boolean permutations with optimal minimum degree . First, they present a novel method for optimising the minimum degrees of known Boolean permutations. Second, they show that the Boolean permutations, which are obtained by optimising Boolean permutations without optimal algebraic degree , have optimal minimum degree. At last, it is shown that their method generates an infinite class of involutions with optimal minimum degree.

In the mid 1960s, Rothaus proposed the so-called “most general form” of constructing new bent functions by using three (initial) bent functions whose sum is again bent. In this paper, we utilize a special case of Rothaus construction when two of these three bent functions differ by a suitably chosen characteristic function of an n/2-dimensional subspace. This simplification allows us to treat the induced bent conditions more easily, also implying the possibility to specify the initial functions in the partial spread class and most notably to identify several instances of the so-called non-normal bent functions. Affine inequivalent bent functions within this class are then identified using a suitable selection of initial bent functions within the partial spread class (stemming from the complete Desarguesian spread). It is also shown that when the initial bent functions belong to the class D, then, under certain conditions, the constructed functions provably do not belong to the completed Maiorana-McFarland class. We conjecture that our method potentially generates an infinite class of non-normal bent functions (all tested ten-variable functions are non-normal but unfortunately they are weakly normal) though there are no efficient computational tools for confirming this.

The problem of constructing bent-negabent functions that do not belong to the completed Maiorana-McFarland class emerges implicitly through a series of construction methods proposed recently. These approaches manage to optimize the algebraic degree of bent-negabent functions, but all of the constructed bent-negabent functions belong to the completed Maiorana-McFarland class. In this paper, we use the indirect sum construction (proposed by Carlet in 2004) for constructing the bent-negabent functions that are not provably contained in this class, which is the first significant attempt in this direction. To achieve this, we first provide a class of bent functions with certain desirable properties that does not belong to this class and demonstrate the existence of the class members. Then, embedding these functions in the framework of the indirect sum construction, we are able to specify sufficient conditions for bent-negabent functions not being contained in the completed Maiorana-McFarland class.

ABSTRACT In this paper, we concentrate on the design of 1‐resilient Boolean functions with desirable cryptographic properties. Firstly, we put forward a novel secondary construction to obtain 1‐resilient functions. Next, we present the relationships between the properties of these constructed 1‐resilient functions and that of the initial functions. Based on the construction and a class of bent functions on n variables, we can obtain a class of ( n + 3)‐variable 1‐resilient non‐separable cryptographic functions with a high algebraic immunity, whose nonlinearity is equal to the bent concatenation bound 2 n + 2 − 2 ( n + 2)/2 . Furthermore, we propose a set of 1‐resilient non‐separable functions on odd number of variables with an optimal algebraic degree, a high algebraic immunity, and a high nonlinearity. Copyright © 2011 John Wiley & Sons, Ltd.

This paper investigated the regional land cover classification based on multi-temporal MODIS data. The study area lies in Northeastern China, where there are diverse and relative homogeneous land cover types. Through experiment, NDVI time-series data can be used to distinguish the woody (perennial) and herbaceous (annual), vegetation and non-vegetation categories depending on the seasonal differences. Grassland and cropland (one-crop-per-year), needle-leaf deciduous forest and broad leaf deciduous forest have similar phenological characteristics easy to be confused. We add the LST (land surface temperature) data to resolve this problem. But built-up area and bare land must depend on further information to be divided. Validated results with 363 ground truth filed samples; the result shows that the temperature-vegetation index (TVI) includes more information. The overall land cover classification accuracies with NDVI and TVI are 62.26% and 71.63% respectively. Based on this study, we concluded that TVI is more sensitive to land cover than NDVI, and MODIS data has its strength in the regional land cover mapping.