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) = Σ_i=1^m-1 c_i(Σ_j=0^n-1 x_jx_i+j) + c_m(Σ_j=0^m-1 x_jx_m+j) and f_t(x) = Σ_i=0^n-1 (x_ix_t+ix_m+i + x_ix_t+i) + Σ_i=0^m-1 x_ix_m+i, where n = 2m, c_i ϵ {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.