It is known that the set of rotation symmetric Boolean functions has many functions with various useful properties for cryptography. This study shows how to construct some families of rotation symmetric functions which are balanced or plateaued. The authors also consider vectorial Boolean functions [that is, maps from GF (2) n to GF (2) m ] which are k ‐rotation symmetric and they give two infinite families of such functions which are permutations with the maximum possible algebraic degree. The families of functions that they give provide a source, which can be searched for functions with other useful cryptographic properties.

In this paper, we consider constructions of rotation symmetric bent functions, which are of the forms: fc(x) = Σi=1m-1ci(Σj=0n-1xjxi+j) + cm(Σj=0m-1xjxm+j) and ft(x) = Σi=0n-1(xixt+ixm+i+ xixt+i) + Σi=0m-1xixm+i, where n = 2m, ciϵ {0,1} (the subscript u of xuin the previous expressions is taken as u modulo n). For each case, a necessary and sufficient condition is obtained. To the best of our knowledge, this class of cubic rotation symmetric bent functions is the first example of an infinite class of nonquadratic rotation symmetric bent functions.

In this paper, we consider the conjecture that σ2t+1l-1,2tare the only nonlinear balanced elementary symmetric Boolean functions wheretandlare positive integers. We prove ifn=2t+1l-1,lodd and 2t+1\nmidd, σn,dis balanced if and only ifd=2k, 1 ≤k≤t. Our results verify most cases of the conjecture forn≡ 3 (mod 4) .