On the lower bounds of the second order nonlinearities of some Boolean functions

The r  th order nonlinearity of a Boolean function is an important cryptographic criterion in analyzing the security of stream as well as block ciphers. It is also important in coding theory as it is related to the covering radius of the Reed–Muller code R(r,n)R(r,n). In this paper we deduce the lower bounds of the second order nonlinearities of the following two types of Boolean functions:1.fλ(x)=Tr1n(λxd) with d=22r+2r+1d=22r+2r+1 and λ∈F2n∗, where n=6rn=6r.2.f(x,y)=Tr1t(xy2i+1), where x,y∈F2t,n=2t,n⩾6x,y∈F2t,n=2t,n⩾6 and i   is an integer such that 1⩽i