关于Budaghyan-Carlet APN六项范数的存在性
On existence of Budaghyan-Carlet APN hexanomials
摘要：
Budaghyan和Carlet在一个域上构造了一族几乎完全非线性（APN）六项多项式，其中元素为r^2，项的次数为r+1、s+1、rs+1、rs+r、rs+s和r+s，其中r = 2^m，s = 2^n，GCD（m，n）=1。该结构需要一个技术条件，这在有限数量的例子中得到了经验验证。Bracken、Tan和Tan（arXiv：1110.3177 [cs.it]）证明了当m = 2或4（mod 6）时，条件成立。在这篇文章中，我们证明了Budaghyan和Carlet的构造产生了对所有m和n的APN多项式。在GCD（m，n）= k >= 1的情况下，Budaghyan和Carlet证明了六项多项式的非零导数是从F_{r^2}到F_{r^2}的2^k-1映射，只要同样的技术条件成立。我们证明他们的建设产生六项与此微分性质的所有m和n。
英文摘要：
Budaghyan and Carlet constructed a family of almost perfect nonlinear (APN) hexanomials over a field with r^2 elements, and with terms of degrees r+1, s+1, rs+1, rs+r, rs+s, and r+s, where r = 2^m and s = 2^n with GCD(m,n)=1. The construction requires a technical condition, which was verified empirically in a finite number of examples. Bracken, Tan, and Tan (arXiv:1110.3177 [cs.it]) proved the condition holds when m = 2 or 4 (mod 6). In this article, we prove that the construction of Budaghyan and Carlet produces APN polynomials for all m and n. In the case where GCD(m,n) = k >= 1, Budaghyan and Carlet showed that the nonzero derivatives of the hexanomials are 2^k-to-one maps from F_{r^2} to F_{r^2}, provided the same technical condition holds. We prove their construction produces hexanomials with this differential property for all m and n.
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