On bent and semi-bent quadratic Boolean functions

The maximum-length sequences, also called m-sequences, have received a lot of attention since the late 1960s. In terms of linear-feedback shift register (LFSR) synthesis they are usually generated by certain power polynomials over a finite field and in addition are characterized by a low cross correlation and high nonlinearity. We say that such a sequence is generated by a semi-bent function. Some new families of such function, represented by f(x)=Σ_i=1^(n-1)2/c_iTr(x(2ⁱ)+1), n odd and c_i∈F₂, have recently (2002) been introduced by Khoo et al. We first generalize their results to even n. We further investigate the conditions on the choice of c_i for explicit definitions of new infinite families having three and four trace terms. Also, a class of nonpermutation polynomials whose composition with a quadratic function yields again a quadratic semi-bent function is specified. The treatment of semi-bent functions is then presented in a much wider framework. We show how bent and semi-bent functions are interlinked, that is, the concatenation of two suitably chosen semi-bent functions will yield a bent function and vice versa. Finally, this approach is generalized so that the construction of both bent and semi-bent functions of any degree in certain range for any n≥7 is presented, n being the number of input variables.