The main contributions of the second part of this dissertation are focused on the cryptographic properties of logical functions over finite field , with the help of the properties of trace functions , and that of p - polynomials , as well as the permutation theory over finite field : the new definition of chrestenson linear spectrum is given and the relation between the new chrestenson linear spectrum and the chrestenson cyclic spectrum is presented , followed by the inverse formula of logical function over finite field ; the distribution for linear structures of the logical functions over finite field is discussed and the complete construction of logical functions taking on all vectors as linear structures is suggested , which leads to the conception of the extended affine functions over finite field , whose cryptographic properties is similar to that of the affine functions over field gf ( 2 ) and prime field fp ; the relationship between the degeneration of logical functions and the linear structures , the degeneration of logical functions and the support of chrestenson spectrum , as well as the relation between the nonlinearity and the linear structures are discussed ; using the relation of the logical functions over finite field and the vector logical functions over its prime field , we reveal the relationship between the perfect nonlinear functions over finite field and the vector generalized bent functions over its prime field ; the existence or not of the perfect nonlinear functions with any variables over any finite fields is offered , and some methods are proposed to construct the perfect nonlinear functions by using the balanced p - polynomials over finite field 重新定义了有限域上逻辑函数的chrestenson线性谱，考察了新定义的chrestenson线性谱和原来的chrestenson循环谱的关系，并利用一组对偶基给出了有限域上逻辑函数的反演公式；给出了有限域上随机变量联合分布的分解式，并利用随机变量联合分布的分解式对有限域上逻辑函数的密码性质进行了研究；给出了有限域上逻辑函数与相应素域上向量逻辑函数的关系，探讨了它们之间密码性质的联系，如平衡性，相关免疫性，扩散性，线性结构以及非线性度等；讨论了有限域上逻辑函数各类线性结构之间的关系，并给出了任意点都是线性结构的逻辑函数的全部构造，由此引出了有限域上的“泛仿射函数”的概念；考察了有限域上逻辑函数的退化性与线性结构的关系、退化性与chrestenson谱支集的关系；给出了有限域逻辑函数非线性度的定义，利用有限域上逻辑函数的非线性度与相应素域上向量逻辑函数非线性度的关系，考察了有限域上逻辑函数的非线性度与线性结构的关系；利用有限域上逻辑函数与相信息工程大学博士学位论文应素域上向量逻辑函数的关系，揭示了有限域上的广义bent函数与相应素域上的广义bent函数的关系，以及有限域上的完全非线性函数与相应素域上向量广义bent函数之间的关系;给出了任意有限域上任意。
