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中文摘要：
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n元q值逻辑函数f是GF（q）上的完全非线性函数当且仅当对任意的β∈GF（q）^＊,βf是GF（q）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,并给出了一个是广义Bent函数但不是完全非线性函数的例子. 结果表明,一般有限域和剩余类环上的完全非线性函数与广义Bent函数的研究是一致的. 其次建立了f和它的分量函数谱值的对应关系,进而证明了f是GF（q）上的完全非线性函数,当且仅当它的分量函数（f1,f2,...,fm）是m维向量广义Bent函数.
英文摘要：
A weak definition of generalized Bent function over finite fields is proposed. And the relation between perfect nonlinear functions and generalized Bent functions is studied. It is proved that a qary logic function f over GF（q） （q = p^m） is a perfect nonlinear function iff βf is a generalized Bent function for each non - zero element β in GF（q）. The common ground and difference between several versions of generalized Bent function＇s definitions over finite fields is discussed. An example which is generalized Bent function but not perfect nonlinear function is also presented. By the proposed definition, it is shown that the study of generalized Bent function and perfect nonlinear function over finite fields and residue class rings are consistent. Relations between spectrum of f and that of its component functions are also presented. Furthermore it is proved that f is nonlinear perfect function over GF（q） iff its component function （f1, f2,…, fm） is m-dimension vector generalized bent function.
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