Further results on permutation polynomials from trace functions
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Further results on permutation polynomials from trace functions Danyao Wu1 · Pingzhi Yuan2 Received: 20 February 2020 / Revised: 30 July 2020 / Accepted: 19 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract For a prime p and positive integers m, n, let 𝔽q be a finite field with q = pm elements and 𝔽qn be an extension of 𝔽q . Let h(x) be a polynomial over 𝔽qn satisfying the follow(x)◦h(x) = 𝜏(x)◦Trnm (x) ; (ii) For any s ∈ 𝔽q , h(x) is injective ing conditions: (i) Trnm m m nm −1 on Trm (x) (s), where 𝜏(x) is a polynomial over 𝔽q . For b, c ∈ 𝔽q , 𝛿 ∈ 𝔽qn , and positive integers i, j, d with q ≡ ±1 (mod d) , we propose a class of permutation polynomials of the form 1+ b(Trnm m (x) + 𝛿)
i(qn −1) d
1+ + c(Trnm m (x) + 𝛿)
j(qn −1) d
+ h(x)
over 𝔽qn by employing the Akbary–Ghioca–Wang (AGW) criterion in this paper. Accordingly, we also present the permutation polynomials of the form 1+ b(Trnm m (x) + 𝛿)
i(qn −1) d
+ h(x)
by letting c = 0 and choosing some special i, which covered some known results of this form. Keywords Finite field · Polynomial · Permutation polynomial Mathematics Subject Classification 11C08 · 12E10
* Danyao Wu [email protected] Pingzhi Yuan [email protected] 1
School of Computer Science and Technology, Dongguan University of Technology, Dongguan 523808, China
2
School of Mathematics, South China Normal University, Guangzhou 510631, China
13
Vol.:(0123456789)
D. Wu, P. Yuan
1 Introduction Let 𝔽q be the finite field with q = pm elements and 𝔽qn be an extension of 𝔽q , where the prime p is the character of 𝔽q and m, n are positive integers. Let 𝔽qn [x] be the ring of polynomials of one variable over 𝔽qn . A polynomial f (x) ∈ 𝔽qn [x] is called a permutation polynomial of 𝔽qn if the associated polynomial function f ∶ c ↦ f (c) from 𝔽qn into 𝔽qn is a permutation of 𝔽qn . Permutation polynomials over finite fields have significant applications in coding theory, cryptography, combinatorial designs and so on. For more details of the recent advances and contributions to the area, the reader is referred to [4–6, 8, 10] and the references therein. Zeng et al. [9] investigated the permutation behaviour of the polynomials of the form
(Trnm (x) + 𝛿)s + L(x) m (x) is the trace function over 𝔽2nm , where k, m, n ∈ ℤ+ , s = k(q ± 1) + 1, 𝛿 ∈ 𝔽2nm , Trnm m (x) + x from 𝔽2mn to 𝔽2m and L(x) = Trnm or x. The main method was to determine the m number of solutions of some equations over finite fields. Recently, two classes of permutation polynomials of the form 2m 2m k s k s1 k s2 x + (Tr2m m (x) + 𝛿) and x + (Trm (x) + 𝛿) + (Trm (x) + 𝛿)
over 𝔽22m were presented [7] based on the AGW criterion, where m, k, s, s1 , s2 ∈ ℤ+ and 𝛿 ∈ 𝔽22m . Inspired by the work of [7, 9], we propose a class of permutation polynomials of the form 1+ b(Trnm m (x) + 𝛿)
i(qn −1) d
1+ + c(Trnm m (x) + 𝛿)
j(qn −1) d
+ h(x)
(1)
over 𝔽qn by employing the AGW criterion as well, where i, j, d, m ∈ ℤ+ with q = pm , q ≡ ±1 (mod d), b, c ∈ 𝔽q , 𝛿 ∈ 𝔽qn , and h(x) is a polynomial over 𝔽qn
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