The Nonlinearity Lower Bounds on the Second Order of Cubic Monomial Boolean Functions

This paper investigates cubic monomial Boolean functions fμ(x)=Tr(μxd) with n variables,where d=2i+2j+1,μ∈GF(2n)*,and nij.The known results show that the Boolean functions fμ(x) has good lower bounds on the second nonlinearity for n2i.This paper firstly studies all lower bounds on the nonlinearity of the derivatives of fμ(x),then the lower bounds on the second order nonlinearity of fμ(x) for n≤2i are given.The results show that the lower bounds on the second order nonlinearity of fμ(x) for n≤2i are tighter than that of fμ(x) for n2i.Therefore,whether n2i or n≤2i,the Boolean functions fμ(x) can resist quadratic or linear approximation attacks.