Constructions of Quadratic and Cubic Rotation Symmetric Bent Functions

In this paper, we consider constructions of rotation symmetric bent functions, which are of the forms: f(c)(x) = Sigma(m-1)(i=1) c(i)(Sigma(n-1)(j=0) x(j)x(i+j)) + c(m) (Sigma(m-1)(j=0) x(j)x(m+j)) and, f(t)(x) = Sigma(n-1)(i=0) (x(i)x(t)+(i)x(m+i) + x(i)x(t+i)) + Sigma(m-1)(i=0) x(i)x(m+i) where, n = 2m, c(i) is an element of {0, 1} (the subscript u of x(u) in 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.