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Irreducible factors of a class of permutation polynomials. (English) Zbl 07174374
The authors present results on the degrees of the irreducible factors of a permutation polynomial of the finite field \({\mathbb F}_q\) of \(q\) elements.
More precisely, they study permutation polynomials \(f_n(x)\) recursively defined by \[f_0(x)=ax+a_0,\quad f_i(x)=f_{i-1}^{d_i}(x)+a_i,~i=1,\ldots,n,\] where \(a\in {\mathbb F}_q^*\), \(a_0,\ldots,a_n\in {\mathbb F}_q\) and \(d_1,\ldots,d_n\ge 2\) with \(\gcd(d_i,q-1)=1\), \(i=1,\ldots,n\). Note that each permutation polynomial is of this form with \(d_i=q-2\), \(i=1,\ldots,n\), and some \(n\) by a well-known result of L. Carlitz [Proc. Am. Math. Soc. 11, 456–459 (1960; Zbl 0095.03003)].
The first main result (Theorem 2.2) is the following. Let \(d=\mathrm{lcm}(d_1,\ldots,d_n)\) and assume \(\gcd(d,q)=1\). Then the degree of each irreducible factor \(Q(x)\) of \(F_n(x)\) is a divisor of \(d_1 d_2 \cdots d_{n-1}\mathrm{ord}_d(q)\).
Moreover, (Theorem 2.3) the degree of \(Q(x)\) is either \(1\) or divisible by ord\(_\ell(q)\) for some prime divisor \(\ell\) of \(d\). Several additional results are proved.
The results of this paper enable one to produce families of permutation polynomials of large degrees, where possible degrees of their irreducible factors are known. See also the follow-up paper of the same authors [ “ Permutation polynomials and factorization”, Cryptogr. Commun. 12, No. 5, 913–934 (2020; doi:10.1007/s12095-020-00446-y)], in particular, in view of applications.
MSC:
11T06 Polynomials over finite fields
11T55 Arithmetic theory of polynomial rings over finite fields
12E20 Finite fields (field-theoretic aspects)
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