Determining the number of true different permutation polynomials of degrees up to five by Weng and Dong algorithm

When Is the Number of True Different Permutation Polynomials Equal to 0?

“…We have particularized the algorithm from [11] for permutation polynomials under ZF sufficient conditions, and we obtained the number of null polynomials and the quantities C p,d−PPs,ZF required in the algorithm. We have also obtained the form of N's prime factorization such that the number of td PPs of any degree, fulfilling ZF sufficient conditions, is equal to zero.…”

“…. , N − 1 of integers modulo N. Below, we give the algorithm given in [11] for determining the number of td PPs of degree up to five based on the WD algorithm.…”

“…In [8,9], the method from [7] was used to determine the number of td CPPs, fourth-degree PPs (4-PPs), and fifth-degree PPs (5-PPs) under Zhao and Fan (ZF) sufficient conditions given in [10]. In [11], an algorithm to determine the number of td PPs of degrees up to five, based on the Weng and Dong (WD) algorithm from [12], was given.…”

“…The paper is structured as follows. In Section 2, we recall the algorithm from [11], which is based on the WD algorithm [12]. In Section 3, we obtain a necessary condition so that the number of td PPs of a certain degree is equal to 0 (Lemma 1).…”

“…Using the result from Lemma 1 in Sections 3.1-3.4, we obtain the form of N's prime factorization so that the number of td QPPs, CPPs, 4-PPs, and 5-PPs is equal to 0, respectively. In Section 4, we obtain the number of null polynomials and the quantities required in the algorithm from [11] to determine the number of any degree td PPs fulfilling ZF sufficient conditions. Then, in Theorem 1, we obtain the prime factorization of N so that the number of any degree td PPs fulfilling ZF sufficient conditions is equal to 0.…”