Results on Constructions of Rotation Symmetric Bent and Semi-bent Functions

In this paper, we introduce a class of cubic rotation symmetric (RotS) functions and prove that it can yield bent and semi-bent functions. To the best of our knowledge, this is the second primary construction of an infinite class of nonquadratic RotS bent functions which could be found and the first class of nonquadratic RotS semi-bent functions. We also study a class of idempotents (giving RotS functions through the choice of a normal basis of \(GF(2^n)\) over \(GF(2)\)). We derive a characterization of the bent functions among these idempotents and we relate their precise determination to a problem studied in the framework of APN functions. Incidentally, the proofs of bentness given here are useful for a paper studying a construction of idempotents from RotS functions, entitled “A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions” by the same authors, to appear in the journal JCT series A.