Abstract Boolean functions used in symmetric-key cryptosystems must have high second-order nonlinearity to withstand several known attacks and some potential attacks which may exist but are not yet efficient and might be improved in the future. The second-order nonlinearity of Boolean functions also plays an important role in coding theory, since the maximal second-order nonlinearity of all Boolean functions in n variables equals the covering radius of the Reed–Muller code with length 2 n and order r . It is well-known that providing a tight lower bound on the second-order nonlinearity of a general Boolean function with high algebraic degree is a hard task, excepting a few special classes of Boolean functions. In this paper, we improve the lower bounds on the second-order nonlinearity of three classes of Boolean functions of the form f i ( x ) = Tr 1 n ( x d i ) in n variables for i = 1 , 2 and 3, where Tr 1 n denotes the absolute trace mapping from F 2 n to F 2 and d i ’s are of the form (1) d 1 = 2 m + 1 + 3 and n = 2 m , (2) d 2 = 2 m + 2 m + 1 2 + 1 , n = 2 m with odd m , and (3) d 3 = 2 2 r + 2 r + 1 + 1 and n = 4 r with even r .

中文翻译：

---

改进三类布尔函数的二阶非线性下界

摘要 对称密钥密码系统中使用的布尔函数必须具有较高的二阶非线性才能抵御几种已知的攻击和一些可能存在但尚未有效并可能在未来改进的潜在攻击。布尔函数的二阶非线性在编码理论中也起着重要作用，因为 n 个变量中所有布尔函数的最大二阶非线性等于长度为 2 n 且阶数为 r 的 Reed-Muller 码的覆盖半径。众所周知，为具有高代数次数的一般布尔函数的二阶非线性提供严格的下界是一项艰巨的任务，除了少数特殊类别的布尔函数。在本文中，