The lower bounds on the second order nonlinearity of three classes of Boolean functions with high nonlinearity

The r  th order nonlinearity of Boolean functions is an important cryptographic criterion associated with some attacks on stream and block ciphers. It is also very useful in coding theory, since it is related to the covering radii of Reed–Muller codes. This paper tightens the lower bounds of the second order nonlinearity of three classes of Boolean functions in the form f(x)=tr(xd)f(x)=tr(xd) in n   variables, where (1) d=2m+1+3d=2m+1+3 and n=2mn=2m, or (2) d=2m+2m+12+1, n=2mn=2m and m   is odd, or (3) d=22r+2r+1+1d=22r+2r+1+1 and n=4rn=4r.