Rotation Symmetric Bent Functions Research Articles

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.

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} .

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

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.

On homogeneous rotation symmetric bent functions

In this paper, we present partial results towards the conjectured nonexistence of homogeneous rotation symmetric bent functions having degree > 2.

On affine (non)equivalence of Boolean functions

In this paper we construct a multiset S(f) of a Boolean function f consisting of the weights of the second derivatives of the function f with respect to all distinct two-dimensional subspaces of the domain. We refer to S(f) as the second derivative spectrum of f. The frequency distribution of the weights of these second derivatives is referred to as the weight distribution of the second derivative spectrum. It is demonstrated in this paper that this weight distribution can be used to distinguish affine nonequivalent Boolean functions. Given a Boolean function f on n variables we present an efficient algorithm having O(n22n ) time complexity to compute S(f). Using this weight distribution we show that all the 6-variable affine nonequivalent bents can be distinguished. We study the subclass of partial-spreads type bent functions known as PS ap type bents. Six different weight distributions are obtained from the set of PS ap bents on 8-variables. Using the second derivative spectrum we show that there exist 6 and 8 variable bent functions which are not affine equivalent to rotation symmetric bent functions. Lastly we prove that no non-quadratic Kasami bent function is affine equivalent to Maiorana–MacFarland type bent functions.