Decomposing self-dual bent functions

Abstract

Bent functions are Boolean functions in even number of variables that have maximal nonlinearity. They have flat Walsh–Hadamard spectrum and are of interest for their applications in algebra, coding theory and cryptography. A bent function is called self-dual if it coincides with its dual bent function. In current work we study the decomposition of the form $\left( f_0,f_1,\ldots ,f_{2^k-1}\right)$ of the vector of values of a self-dual bent function, formed by the concatenation of $2^k$ Boolean functions $f_j$ in $n-k$ variables. We treat the cases $k=1,2$. Based on a spectral characterization, we introduce a notion of self-dual near-bent function in odd number of variables and prove that there exists a one-to-one correspondence between the notions of self-duality for even and odd number of variables. As a result the characterization for the decomposition $\left( f_0,f_1\right)$ is obtained. For the decomposition $f=\left( f_0,f_1,f_2,f_3\right)$ we prove that if sign vectors of subfunctions $f_j$ are linearly dependent, then all these subfunctions are bent. We prove that for ${n\geqslant 6}$ the converse does not hold, that is the obtained condition is sufficient only. These results are also generalized for the case of an arbitrary bent function. Three new iterative constructions of self-dual bent functions are proposed. One of them allows to build a class of self-dual bent functions which cannot be decomposed into the concatenation of four bent functions. Based on the constructions a new iterative lower bound on the cardinality of the set of self-dual bent functions is obtained.