Rotation Symmetric Functions Research Articles

Maximum Entropy Closure Relation for Higher Order Alignment and Orientation Tensors Compared to Quadratic and Hybrid Closure

A closure relation expresses the fourth order orientation tensor as a function of the second order one. Two well-known closure relations, the hybrid closure and the maximum entropy closure, are compared in the case of a rotation symmetric orientation distribution function. The maximum entropy closure predicts a positive fourth order parameter in the whole range of the second order parameter, whereas the hybrid closure results in negative fourth order parameters for small values of the second order one. For the maximum entropy closure quadratic fit polynomials are presented. For a general distribution without rotation symmetry, the expression for the entropy is exploited to derive an explicit form for the maximum entropy distribution. Lowest order approximation of this distribution function leads to simple closure forms for the fourth order alignment tensor and also for higher order alignment tensors.

Several classes of even-variable 1-resilient rotation symmetric Boolean functions with high algebraic degree and nonlinearity

Recent research shows that the class of rotation symmetric Boolean functions is potentially rich in functions of cryptographic significance. In this paper, some classes of 2m-variable (m is an odd integer) 1-resilient rotation symmetric Boolean functions are got, whose nonlinearity and algebraic degree are studied. For the first time, we obtain 2m-variable 1-resilient rotation symmetric Boolean functions having high nonlinearity and optimal algebraic degree. In addition, we obtain a class of non-linear rotation symmetric 1-resilient function for every n≥5, and a class of quadratic rotation symmetric (k−1)-resilient function of n=3k variables, where k is an integer.

Using Inclusion / Exclusion to find Bent and Balanced Monomial Rotation Symmetric Functions

Abstract There are many cryptographic applications of Boolean functions. Recently, research has been done on monomial rotation symmetric (MRS) functions which have useful cryptographic properties. In this paper we use the inclusion/exclusion principle to construct formulas for the weights of two subclasses of MRS functions: degree d short MRS functions and d-functions. From these results we classify bent and balanced functions of these forms.

On the existence and non-existence of some classes of bent–negabent functions

In this paper we investigate different questions related to bent–negabent functions. We first take an expository look at the state-of-the-art research in this domain and point out some technical flaws in certain results and fix some of them. Further, we derive a necessary and sufficient condition for which the functions of the form $${\mathbf{x}}\cdot \pi ({\mathbf{y}})\oplus h({\mathbf{y}})$$ [Maiorana–McFarland ( $${\mathcal {M}}$$ )] is bent–negabent, and more generally, we study the non-existence of bent–negabent functions in the $${\mathcal {M}}$$ class. We also identify some functions that are bent–negabent. Next, we continue the recent work by Mandal et al. (Discrete Appl Math 236:1–6, 2018) on rotation symmetric bent–negabent functions and show their non-existence in larger classes. For example, we prove that there is no rotation symmetric bent–negabent function in $$4p^k$$ variables, where p is an odd prime. We present the non-existence of such functions in certain classes that are affine transformations of rotation symmetric functions. Keeping in mind the existing literature, we correct here some technical issues and errors found in other papers and provide some novel results.

A generic construction of rotation symmetric bent functions

<p style='text-indent:20px;'>Rotation symmetric bent functions are a special class of Boolean functions, and their construction is of theoretical and practical interest. In this paper, we propose a generic construction of rotation symmetric bent functions by modifying the support of a known class of quadratic rotation symmetric bent functions, which generalizes some earlier works. Moreover, many infinite classes of rotation symmetric bent functions with maximal algebraic degree can be easily obtained from our construction.</p>

Affine equivalence for quadratic rotation symmetric Boolean functions

Let $$f_n(x_0, x_1, \ldots , x_{n-1})$$ denote the algebraic normal form (polynomial form) of a rotation symmetric (RS) Boolean function of degree d in $$n \ge d$$ variables and let $$wt(f_n)$$ denote the Hamming weight of this function. Let $$(0, a_1, \ldots , a_{d-1})_n$$ denote the function $$f_n$$ of degree d in n variables generated by the monomial $$x_0x_{a_1} \ldots x_{a_{d-1}}.$$ Such a function $$f_n$$ is called monomial rotation symmetric (MRS). It was proved in a 2012 paper that for any MRS $$f_n$$ with $$d=3,$$ the sequence of weights $$\{w_k = wt(f_k):~k = 3, 4, \ldots \}$$ satisfies a homogeneous linear recursion with integer coefficients. This result was gradually generalized in the following years, culminating around 2016 with the proof that such recursions exist for any rotation symmetric function $$f_n.$$ Recursions for quadratic RS functions were not explicitly considered, since a 2009 paper had already shown that the quadratic weights themselves could be given by an explicit formula. However, this formula is not easy to compute for a typical quadratic function. This paper shows that the weight recursions for the quadratic RS functions have an interesting special form which can be exploited to solve various problems about these functions, for example, deciding exactly which quadratic RS functions are balanced.

A Note on Rotation Symmetric S-boxes

This paper is twofold. The first is devoted to study a class of quadratic rotation symmetric S-boxes (RSSBs) which was presented by Gao G, et al., Constructions of quadratic and cubic rotation symmetric bent functions, IEEE Transactions on Information Theory, vol. 58, no. 7, pp. 4908–4913, 2012, by decomposing a class of cubic rotation symmetric bent functions. The authors obtain its nonlinearity and differential uniformity of such class of S-boxes. In particular, the compositional inversion of the class of rotation symmetric S-boxes is also presented. Then the authors introduce a steepest-descent-like search algorithm for the generation of RSSBs. The algorithm finds 5,6,7,8-bit RSSBs with very good cryptographic properties which can be applied in designing cryptographical algorithms.

Constructing 1-resilient rotation symmetric functions over <inline-formula><tex-math id="M1">\begin{document}$ {\mathbb F}_{p} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ {q} $\end{document}</tex-math></inline-formula> variables through special orthogonal arrays

Based on the relation between resilient functions and large sets of orthogonal arrays, two classes of $ q $-variable 1-resilient rotation symmetric functions(RSFs) over $ {\mathbb F}_{p} $ are constructed. The first class of 1-resilient functions is obtained with the help of a Latin square with maximum cycle, and the second class of 1-resilient functions is constructed via switching the rotation symmetric orbits of the former class. Firstly, an efficient method to construct OA$ (pq, q, p, 1) $ is presented via solving a linear equation system. Secondly, we propose some schemes to construct more $ q $-variable 1-resilient RSFs by modifying the $ l $-value support tables of the known $ q $-variable 1-resilient RSFs. In addition, two examples are given to demonstrate our constructions. It seems that the $ q $-variable 1-resilient RSFs constructed by our methods can not be constructed by earlier constructions.

A new construction of rotation symmetric bent functions with maximal algebraic degree

In this paper, for any even integer \begin{document}$ n = 2m\ge4 $\end{document} , a new construction of \begin{document}$ n $\end{document} -variable rotation symmetric bent function with maximal algebraic degree \begin{document}$ m $\end{document} is given as \begin{document}$ f(x_0,x_1\cdots,x_{n-1}) = \bigoplus\limits_{i = 0}^{m-1}(x_ix_{m+i})\oplus \bigoplus\limits_{i = 0}^{n-1}(x_ix_{i+1}\cdots x_{i+m-2} \overline{x_{i+m}} ), $\end{document} whose dual function is \begin{document}$ \widetilde{f}(x_0,x_1\cdots,x_{n-1}) = \bigoplus\limits_{i = 0}^{m-1}(x_ix_{m+i})\oplus \bigoplus\limits_{i = 0}^{n-1}(x_ix_{i+1}\cdots x_{i+m-2} \overline{x_{i+n-2}} ), $\end{document} where \begin{document}$ \overline{x_{i}} = x_{i}\oplus 1 $\end{document} and the subscript of \begin{document}$ x $\end{document} is modulo \begin{document}$ n $\end{document} .

Constructions of rotation symmetric bent functions with high algebraic degree

Rotation symmetric (RotS) bent functions have attracted much attention due to their cryptographic significance in the last few years. However, few known constructions of RotS bent functions with degree higher than 3 are available in the literature. In this paper, we first introduce a systematic construction of n-variable RotS bent functions for n=2m=4pr, p≥2, r≥1. The new RotS bent functions possess high degree p+1. For a fixed n, the number of such functions is ∑p|m2,p>122p−2 and it leads to a great increase in the number of known RotS bent functions. In addition, using any n-variable RotS bent function with degree p+1 we constructed, we deduce a new class of n-variable RotS bent functions with degree ranging from p+1 to m.

Recursions associated to trapezoid, symmetric and rotation symmetric functions over Galois fields

Rotation symmetric Boolean functions are invariant under circular translation of indices. These functions have very rich cryptographic properties and have been used in different cryptosystems. Recently, Thomas Cusick proved that exponential sums of rotation symmetric Boolean functions satisfy homogeneous linear recurrences with integer coefficients. In this work, a generalization of this result is proved over any Galois field. That is, exponential sums over Galois fields of some rotation symmetric polynomials satisfy linear recurrences with integer coefficients. In the particular case of F2, an elementary method is used to obtain explicit recurrences for exponential sums of some of these functions. The concept of trapezoid Boolean function is also introduced and it is showed that the linear recurrences that exponential sums of trapezoid Boolean functions satisfy are the same as the ones satisfied by exponential sums of the corresponding rotations symmetric Boolean functions. Finally, it is proved that exponential sums of trapezoid and symmetric polynomials also satisfy linear recurrences with integer coefficients over any Galois field Fq. Moreover, the Discrete Fourier Transform matrix and some Complex Hadamard matrices appear as examples in some of our explicit formulas of these recurrences.

Weight Recursions for Any Rotation Symmetric Boolean Functions

Let $f_{n}(x_{1}, x_{2}, \ldots, x_{n})$ denote the algebraic normal form (polynomial form) of a rotation symmetric Boolean function of degree $d$ in $n \geq d$ variables and let $wt(f_{n})$ denote the Hamming weight of this function. Let $(1, a_{2}, \ldots, a_{d})_{n}$ denote the function $f_{n}$ of degree $d$ in $n$ variables generated by the monomial $x_{1}x_{a_{2}} \ldots x_{a_{d}}$ . Such a function $f_{n}$ is called monomial rotation symmetric (MRS). It was proved in a 2012 paper that for any MRS $f_{n}$ with $d=3$ , the sequence of weights $\{w_{k} = wt(f_{k}):\,\,k = 3, 4, \ldots \}$ satisfies a homogeneous linear recursion with integer coefficients. In this paper, it is proved that such recursions exist for any rotation symmetric function $f_{n}$ ; such a function is generated by some sum of $t$ monomials of various degrees. A Mathematica program is available on arxiv.org which explicitly computes the homogeneous linear recursion for the weights, given any rotation symmetric $f_{n}$ . The reader who is only interested in finding some recursions can use the program and not be concerned with the details of the rather complicated proofs in this paper.

Several infinite families of <i>p</i>-ary weakly regular bent functions

As an optimal combinatorial object, bent functions have been an interesting research object due to their important applications in cryptography, coding theory, and sequence design. The characterization and construction of bent functions are challenging problems in general. The objective of this paper is to present a construction of p-ary weakly regular bent functions from known weakly regular bent functions. This generalizes some earlier constructions of Boolean bent functions and p-ary bent functions, and produces several infinite families of p-ary weakly regular bent functions from known ones. Some infinite families of p-ary rotation symmetric bent functions are obtained as well.

Two infinite classes of rotation symmetric bent functions with simple representation

In the literature, few n-variable rotation symmetric bent functions have been constructed. In this paper, we present two infinite classes of rotation symmetric bent functions on $${\mathbb {F}}_2^{n}$$ of the two forms: where $$n=2m$$ , $${\upgamma }(X_0,X_1,\ldots , X_{m-1})$$ is any rotation symmetric polynomial, and $$m/\textit{gcd}(m,t)$$ is odd. The class (i) of rotation symmetric bent functions has algebraic degree ranging from 2 to m and the other class (ii) has algebraic degree ranging from 3 to m. Moreover, the two classes of rotation symmetric bent functions are disjoint.

Construction of rotation symmetric bent functions with maximum algebraic degree

Construction of rotation symmetric bent functions with maximum algebraic degree

Systematic Constructions of Rotation Symmetric Bent Functions, 2-Rotation Symmetric Bent Functions, and Bent Idempotent Functions

Rotation symmetric bent functions and their generation two-rotation symmetric bent functions are two classes of cryptographically significant Boolean functions. However, few constructions have been presented in the literature, which either have restriction on integer $n$ or have algebraic degree no more than 4. In this paper, for any even integer $n\ge 4$ , three classes of bent functions are presented respectively. Most notably, the proposed $n$ -variable rotation symmetric bent functions and two-rotation symmetric bent functions can have any possible algebraic degree ranging from 2 to $n/2$ . Besides, we obtain bent idempotent functions with the maximal algebraic degree $n/2$ .

Constructions of p‐variable 1‐resilient rotation symmetric functions over GF(p)

AbstractRotation symmetric Boolean functions have been extensively studied in the recent years because of their applications in cryptography. In this study, a novel method to construct p‐variable 1‐resilient rotation symmetric functions over GF(p) is proposed based on a Latin square with maximum cycle structure, which is not required to solve any equation system. And a lower bound on the number of p‐variable 1‐resilient rotation symmetric functions is given. At last, an equivalent characterization of p‐variable 1‐resilient rotation symmetric functions over GF(p) is demonstrated, as a direct corollary, the number of p‐variable 1‐resilient rotation symmetric functions is represented by all the solutions of the equation system. Copyright © 2017 John Wiley & Sons, Ltd.

Investigations on cubic rotation symmetric bent functions

We identify an infinite class of cubic rotation symmetric bent functions and prove that these functions do not have affine derivatives. Some experimental results concerning rotation symmetric bent functions in 6, 8, 10 and 12 variables are also included.

Idempotent and p-potent quadratic functions: distribution of nonlinearity and co-dimension

The Walsh transform $$\widehat{Q}$$Q^ of a quadratic function $$Q:{\mathbb F}_{p^n}\rightarrow {\mathbb F}_p$$Q:FpnźFp satisfies $$|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}$$|Q^(b)|ź{0,pn+s2} for all $$b\in {\mathbb F}_{p^n}$$bźFpn, where $$0\le s\le n-1$$0≤s≤n-1 is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class $$\mathcal {C}_1$$C1 is defined for arbitrary n as $$\mathcal {C}_1 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{\lfloor (n-1)/2\rfloor }a_ix^{2^i+1})\;:\; a_i \in {\mathbb F}_2\}$$C1={Q(x)=Trn(źi=1ź(n-1)/2źaix2i+1):aiźF2}, and the larger class $$\mathcal {C}_2$$C2 is defined for even n as $$\mathcal {C}_2 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{(n/2)-1}a_ix^{2^i+1}) + \mathrm{Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in {\mathbb F}_2\}$$C2={Q(x)=Trn(źi=1(n/2)-1aix2i+1)+Trn/2(an/2x2n/2+1):aiźF2}. For an odd prime p, the subclass $$\mathcal {D}$$D of all p-ary quadratic functions is defined as $$\mathcal {D} = \{Q(x) = \mathrm{Tr_n}(\sum _{i=0}^{\lfloor n/2\rfloor }a_ix^{p^i+1})\;:\; a_i \in {\mathbb F}_p\}$$D={Q(x)=Trn(źi=0źn/2źaixpi+1):aiźFp}. We determine the generating function for the distribution of the parameter s for $$\mathcal {C}_1, \mathcal {C}_2$$C1,C2 and $$\mathcal {D}$$D. As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case $$p > 2$$p>2, the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed---Muller codes corresponding to $$\mathcal {C}_1$$C1 and $$\mathcal {C}_2$$C2 in terms of a generating function.

Counting equivalence classes for monomial rotation symmetric Boolean functions with prime dimension

Recently much progress has been made on the old problem of determining the equivalence classes of Boolean functions under permutation of the variables. In this paper we prove an asymptotic formula for the number of equivalence classes under permutation for degree d monomial rotation symmetric (MRS) functions, in the cases where d ? 3 is arbitrary and the number of variables n is a prime. Our counting formula has two main terms and an error term; this is the first instance of such a detailed result for Boolean function equivalence classes which is valid for arbitrary degree and infinitely many n. We also prove an exact formula for the count of the equivalence classes when d = 5; this extends previous work for d = 3 and 4.