A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions

We study more in detail the relationship between rotation symmetric (RS) functions and idempotents, in univariate and bivariate representations, and deduce a construction of bent RS functions from semi-bent RS functions. We deduce the first infinite classes found of idempotent and RS bent functions of algebraic degree more than 3. We introduce a transformation from any RS Boolean function f   over GF(2)n into the idempotent Boolean function f′(z)=f(z,z2,…,z2n−1)f′(z)=f(z,z2,…,z2n−1) over GF(2n)GF(2n), leading to another RS Boolean function. The trace representation of f′f′ is directly deduced from the algebraic normal form of f, but we show that f   and f′f′, which have the same algebraic degree, are in general not affinely equivalent to each other. We exhibit infinite classes of functions f such that (1) f   is bent and f′f′ is not (2) f′f′ is bent and f is not (3) f   and f′f′ are both bent (we show that this is always the case for quadratic functions and we also investigate cubic functions).