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置换或置换多项式在密码学中的重要性是众所周知的。作为置换及其应用的数学基础,本文讨论的是有限域上的置换多项式。首先给出了一些判定准则,如Hermite准则,然后讨论了几类特殊的置换多项式,如Dickson多项式。其次,对置换多项式类和对称群的关系进行了研究。接下来,探讨了置换多项式和例外多项式的关系,这里应用了有限域上方程理论。最后,将置换多项式推广到了多个未定元的情形,当然,也讨论了涉及到的正交多项式系,并对前面的结果作了推广。
The importance of permutation or permutation in cryptography is well known. As a mathematical basis for the substitution and its applications, this article discusses the permutation polynomials over finite fields. First, some criteria are given, such as the Hermite criterion, and then some special permutation polynomials, such as Dickson polynomials, are discussed. Secondly, the relationship between permutation polynomials and symmetric groups is studied. Next, the relationship between permutation polynomials and exceptional polynomials is explored, where the finite field on-the-loop theory is applied. Finally, we extend the permutation polynomial to several undetermined cases. Of course, the orthogonal polynomials involved are also discussed, and the previous results are generalized.