关于二次非线性度达最大值的布尔函数的研究

THE RESEARCH ON BOOLEAN FUNCTIONS WITH MAXIMUM QUADRATIC NONLINEARITY

在密码学中 ,为抵抗二次逼近引入了二次bent函数、二阶Walsh谱与二次非线性度的概念 ,并得到了n元布尔函数的二次非线性度的最大值为 2 n -1-2 n/ 2 -1.二次bent函数的二次非线性度达到了这一最大值 .因此 ,二次bent函数既可以抵抗线性逼近又可以抵抗二次逼近攻击 ,是具有优良密码学特性的函数 .但本文利用矩阵运算、向量的内积运算及汉明重量证明了这类函数实际上是不存在的 .

To resist quadratic approach, the quadratic bent functions, 2 nd order Walsh spectrum and quadratic nonlinearty are presented. The maximum value of quadratic nonlinearty of Boolean function which has n variates is 2 n-1 -2 n/ 2-1 and the quadratic bent function's quadratic nonlinearty is equal to this value. So this kind of functions not only can resist linear approach but also quadratic approach. They have good cryptographic properties. But there are not any quadratic bent functions in the world. With operations of matrix, product of vectors and Hamming weight, this conclusion is proved in this paper.