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 , z 2 , … , z 2 n − 1 ) over GF ( 2 n ) , leading to another RS Boolean function. The trace representation of f ′ is directly deduced from the algebraic normal form of f, but we show that f and 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 ′ is not (2) f ′ is bent and f is not (3) f and f ′ are both bent (we show that this is always the case for quadratic functions and we also investigate cubic functions).