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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 F〓; 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n元完全非线性函数存在性与否的完整证明，并利用有限域上平衡的p-多项式的性质给出了有限域上完全非线性函数的一些基本构造方法。

And Adaiah the son of Jeroham, the son of Pashur, the son of Malchijah, and Maasiai the son of Adiel, the son of Jahzerah, the son of Meshullam, the son of Meshillemith, the son of Immer

9：12 有玛基雅的曾孙，巴施户珥的孙子，耶罗罕的儿子亚大雅，又有亚第业的儿子玛赛。

And Adaiah the son of Jeroham, the son of Pashur, the son of Malchijah, Maasiai the son of Adiel, the son of Jahzerah, the son of Meshullam, the son of Meshillemith, the son of Immer

9：12 有玛基雅的曾孙，巴施户珥的孙子，耶罗罕的儿子亚大雅，又有亚第业的儿子玛赛。

And Adaiah the son of Jeroham, the son of Pashhur, the son of Malchijah; and Maasai the son of Adiel, the son of Jahzerah, the son of Meshullam, the son of Meshillemith, the son of Immer

9：12 有玛基雅的曾孙、巴施户珥的孙子、耶罗罕的儿子亚大雅；又有亚第业的儿子玛赛；亚第业是雅希细拉的儿子，雅希细拉是米书兰的儿子，米书兰是米实利密的儿子，米实利密是音麦的儿子。

12 Adaiah son of Jeroham, the son of Pashhur, the son of Malkijah; and Maasai son of Adiel, the son of Jahzerah, the son of Meshullam, the son of Meshillemith, the son of Immer.

亚第业是雅希细拉的儿子、雅希细拉是米书兰的儿子、米书兰是米实利密的儿子、米实利密是音麦的儿子。

And Adaiah the son of Jeroham, the son of Pashur, the son of Malchijah, and Maasiai the son of Adiel, the son of Jahzerah, the son of Meshullam, the son of Meshillemith, the son of Immer

亚第业是雅希细拉的儿子、雅希细拉是米书兰的儿子、米书兰是米实利密的儿子、米实利密是音麦的儿子。

And Adaiah the son of Jeroham, the son of Pashur, the son of Malchijah, and Maasiai the son of Adiel, the son of Jahzerah, the son of Meshullam, the son of Meshillemith, the son of Immer

012 有玛基雅的曾孙、巴施户珥的孙子、耶罗罕的儿子亚大雅，又有亚第业的儿子玛赛；亚第业是雅希细拉的儿子；雅希细拉是米书兰的儿子；米书兰是米实利密的儿子；米实利密是音麦的儿子。

Adaiah son of Jeroham, the son of Pashhur, the son of Malkijah; and Maasai son of Adiel, the son of Jahzerah, the son of Meshullam, the son of Meshillemith, the son of Immer.

有玛基雅的曾孙，巴施户珥的孙子，耶罗罕的儿子亚大雅，又有亚第业的儿子玛赛。

Adaiah son of Jeroham, the son of Pashhur, the son of Malkijah; and Maasai son of Adiel, the son of Jahzerah, the son of Meshullam, the son of Meshillemith, the son of Immer.

有玛基雅的曾孙、巴施户珥的孙子、耶罗罕的儿子亚大雅，又有亚第业的儿子玛赛；亚第业是雅希细拉的儿子；雅希细拉是米书兰的儿子；米书兰是米实利密的儿子；米实利密是音麦的儿子。

Adaiah, son of Jeroham, son of Pashhur, son of Malchijah; Maasai, son of Adiel, son of Jahzerah, son of Meshullam, son of Meshillemith, son of Immer.

此外，有耶洛罕的儿子阿达雅；耶洛罕是帕市胡尔的儿子，帕市胡尔是玛耳基雅的儿子；还有阿狄耳的儿子玛赛；阿狄耳是雅赫则辣的儿子，雅赫则辣是默叔蓝的儿子，默叔蓝是默史肋米特的儿子，默史肋米特是依默尔的儿子

We are forbidden from giving money to these shoeless, skinny children, as this will perpetuate begging.

向光着脚，骨瘦如柴的孩子施舍是不可以的，因为要招来无休无止的乞讨。

It would be hard not to fall victim to some sharpeyed policeman.

要想不被某个目光锐利的警察抓到，谈何容易。