Bent函数的构造

The Construction of Bent Functions

摘要: Bent函数是非线性度达到最高的一类具有偶数个变量的布尔函数。一个熟知的结果是：f是n = 2k元Bent函数的充分必要条件是f由两个谱支撑互不相交的2k − 1元半Bent函数的级联而成。提出一个方法：用2^k−1个标号互不相同的仿射函数，沿子空间级联，可得到2k − 1元半Bent函数，从而可以构造2k元Bent函数。并证明：当k = 1，2时，用这种方法可以得到所有的n = 2，4元Bent函数。

Abstract: Bent functions are a class of Boolean functions with even number of variables that have the highest degree of nonlinearity. A well-known result is that the sufficient and necessary condition for f to be a Bent function with n = 2k variables is that f is a concatenation of two semi Bent functions with 2k − 1 variables whose spectral support are disjoint. A method is proposed: by concatenating 2^k−1 affine functions with different labels along a given subspace, we can construct a semi Bent function with 2k − 1 variables, and Bent functions with 2k variables can be obtained accordingly. We prove that: when k = 1, 2, all the Bent functions with 2 or 4 variables can be obtained in this way.