On connection between affine splitting of a Boolean function and its algebraic, combinatorial and cryptographic properties

In this paper, the following results are obtained: 1) for an affine splitting of a Boolean function – an upper bound of algebraic degree; 2) for a dual bent function – some sufficient conditions to be affine splitting, and 3) for any Boolean function with a non-trivial subspace of the linear structures – an upper bound of nonlinearity. Besides, the following assertions are proved: 1) affine splitting is an invariant of complete affine group; 2) if a bent function is normal or weakly normal, then its dual function is normal or weakly normal respectively; 3) the coefficients of the incomplete Walsh–Hadamard transformation of a bent function and of its dual function are the same for zero values of variables; 4) a relation connecting the squares of the Walsh–Hadamard coefficients of a function over cosets of a subspace with the squares of the coefficients of the incomplete Walsh–Hadamard transformation of this function.