关于GF(q)上的完全非线性函数和广义Bent函数

给出了一般有限域上广义bent函数一个较弱的定义，并考虑了它和完全非线性函数的关系.证明了 元 值逻辑函数 是 上的完全非线性函数当且仅当对任意的 ， 是 上的广义bent函数，同时说明了已有的及本文提出的广义bent函数定义的异同点，并给出一个是广义bent函数但不是完全非线性函数的例子.结果表明在我们的定义下，一般有限域和剩余类环上的完全非线性函数和广义bent函数的研究是一致的.其次建立了 和它的分量函数的谱值的对应关系，进而证明了 是 上的完全非线性函数当且仅当它的分量函数 是 维向量广义bent函数.

On Perfect Nonlinear Function and Generalized Bent function Over

A weak definition of generalized bent function over finite fields is presented. And then the relation between Perfect nonlinear functions and generalized bent functions is studied. We prove that a q-ary logic function over （ ）is a perfect nonlinear function iff is a generalized bent function for each non-zero element in . The difference between several versions of generalized bent function’s definitions over finite fields is discussed. Relations between spectrum of and that of its component functions are also presented. Furthermore we prove that is nonlinear perfect function over iff its component function is - dimension vector generalized bent function.