The Lower Bounds on the Second Order Nonlinearity of Bent Functions and Semi-bent Functions

This paper studies the lower bounds on the second order nonlinearity of bent functions and semi-bent functionsf(x,y) with n+1variables, where xGF(2n), yGF(2). Firstly, the values of the nonlinearity of the2n-1 derivatives of the Boolean function f(x,y) are given. Then, the tight lower bounds on the other2n derivatives of f(x,y) are deduced. Furthermore, the tight lower bounds on the second order nonlinearity off(x,y)are presented. The derived bounds are better than the existing general ones. The results show that these functionsf(x,y) have higher second order nonlinearity, and can resist the quardratic and affine approximation attacks.