摘要:
该文研究了形如f(x,y)的n+1变元bent函数和半bent函数的二阶非线性度，其中xGF(2n), yGF(2)。首先给出了f(x,y)的2n-1个导数非线性度的精确值；然后推导出了函数f(x,y)的其余2n个导数的非线性度紧下界。进而给出了f(x,y)的二阶非线性度的紧下界。通过比较可知所得下界要优于现有的一般结论。结果表明f(x,y)具有较高的二阶非线性度，可以抵抗二次函数逼近和仿射逼近攻击。
Abstract:
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.
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