О свойствах бент-функций, построенных по некоторой бентфункции с помощью подпространств

Bent functions are maximally nonlinear Boolean functions with an even number of variables. They were introduced by Rothaus in 1976. For their own sake as interesting combinatorial objects, but also because of their relations to coding theory (Reed-Muller codes) and applications in cryptography (design of stream ciphers), they have attracted a lot of research, specially in the last 15 years. The class of bent functions contains a subclass of functions, introduced by Youssef and Gong in 2001, the so-called hyper-bent functions whose properties are still stronger and whose elements are still rarer than bent functions. Bent and hyper-bent functions are not classified. A complete classification of these functions is elusive and looks hopeless. So, it is important to design constructions in order to know as many of (hyper)-bent functions as possible. This paper is devoted to the constructions of bent and hyper-bent Boolean functions in polynomial forms. We survey and present an overview of the constructions discovered recently. We extensively investigate the link between the bentness property of such functions and some exponential sums (involving Dickson polynomials).

Binary bent functions have a strictly specified number of non-zero values. In the same way, ternary bent functions satisfy certain requirements on the elements of their value vectors. These requirements can be used to specify six classes of ternary bent functions. Classes are mutually related by encoding of function values. Functions within a class are mutually related by permutation of elements in their function vectors. Given a basic ternary bent function, other functions in the same class can be constructed by permutation matrices having a block structure similar to that of the factor matrices appearing in the Good-Thomas decomposition of Cooley-Tukey Fast Fourier transform and related algorithms. Conversion of function vectors into matrices or equivalent matrix-valued vectors of smaller length leads to the redistribution of space complexity of related manipulation algorithms. In this matrix representations, construction of other bent functions from a given known bent function is performed by using either properties of bent functions or by manipulation of such representations in terms of FFT-like permutation matrices of dimensions smaller than the length of the function vector of the initial bent functions.

<p style='text-indent:20px;'>This paper first gives out basic background and some definitions and propositions for Fourier matrix and bent function. Secondly we construct an standard orthogonal basis by the eigenvectors of the corresponding Fourier matrix. At last the diagonalization work of Fourier matrix is completed and some theorems about them are proved.

Rotation Symmetric Boolean Functions –; Count and Cryptographic Properties

We introduce the notion of a bent function on a finite nonabelian group which is a natural generalization of the well-known notion of bentness on a finite abelian group due to Logachev, Salnikov and Yashchenko. Using the theory of linear representations and noncommutative harmonic analysis of finite groups we obtain several properties of such functions similar to the corresponding properties of traditional abelian bent functions.