Improving lower bounds on the second-order nonlinearity of three classes of Boolean functions

Abstract

Boolean functions used in symmetric-key cryptosystems must have high second-order nonlinearity to withstand several known attacks and some potential attacks which may exist but are not yet efficient and might be improved in the future. The second-order nonlinearity of Boolean functions also plays an important role in coding theory, since the maximal second-order nonlinearity of all Boolean functions in n variables equals the covering radius of the Reed–Muller code with length 2n and order r. It is well-known that providing a tight lower bound on the second-order nonlinearity of a general Boolean function with high algebraic degree is a hard task, excepting a few special classes of Boolean functions. In this paper, we improve the lower bounds on the second-order nonlinearity of three classes of Boolean functions of the form fi(x)=Tr1n(xdi) in n variables for i=1,2 and 3, where Tr1n denotes the absolute trace mapping from F2n to F2 and di’s are of the form (1) d1=2m+1+3 and n=2m, (2) d2=2m+2m+12+1, n=2m with odd m, and (3) d3=22r+2r+1+1 and n=4r with even r.