On the cycle structure of permutation polynomials

Let m, n be positive integers such that m > 1 divides n. In this paper, we introduce a special class of piecewise-affine permutations of the finite set [1, n] := {1, . . . , n} with the property that the reduction (mod m) of m consecutive elements in any of its cycles is, up to a cyclic shift, a fixed permutation of [1, m]. Our main result provides the cycle decomposition of such permutations. We further show that such permutations give rise to permutations of finite fields. In particular, we explicitly obtain classes of permutation polynomials of finite fields whose cycle decomposition and its inverse are explicitly given.

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“…where κ(x) is given by Eq. (6). From the same corollary, the latter has exactly ϕ(d)N 1 solutions x = mx 0 ∈ [1, n].…”

Section: Cycle Decompositionmentioning

confidence: 83%

“…However, there is no study on their cycle decomposition. It is worth mentioning that, for only few families of permutation polynomials, we know the cycle decomposition without needing to describe the whole permutation; namely, monomials [1], Mbius maps [6], Dickson polynomials [10] and certain linearized polynomials [11,13].…”

Section: Introductionmentioning

confidence: 99%

Permutations from an arithmetic setting

Reis

Ribas

2020

Discrete Mathematics

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“…Theorem 4.3 shows that any permutation of I k can be viewed as a map f → P * f with P ∈ G k and, in particular, there exist permutations P ∈ G k for which G(P, I k ) comprises a full cycle (i.e., all the elements of I k lie in the same orbit): in this case, we have µ * k (P ) = |I k |. However, the construction of such permutations seems to be out of reach: in fact, even the construction of permutations of finite fields with a full cycle is not completely known [6]. Having this in mind, it would be interesting to obtain permutations P for which µ * k (P ) is reasonable large.…”

Section: Iterated Construction Of Irreducible Polynomialsmentioning

confidence: 99%

The dynamics of permutations on irreducible polynomials

Reis

Wang

2020

Finite Fields and Their Applications

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We study degree preserving maps over the set of irreducible polynomials over a finite field. In particular, we show that every permutation of the set of irreducible polynomials of degree k over Fq is induced by an action from a permutation polynomial of F q k with coefficients in Fq. The dynamics of these permutations of irreducible polynomials of degree k over Fq, such as fixed points and cycle lengths, are studied. As an application, we also generate irreducible polynomials of the same degree by an iterative method. * q , the multiplicative order of α is defined by ord(α) := min{d > 0 | α d = 1}. The degree deg(α) of α ∈ F q over F q is defined as the degree of the minimial polynomial of α over F q . If a, n are positive integers such that gcd(a, n) = 1, then ord n a := min{r > 0 | a r ≡ 1 (mod n)}. In the following theorem, we present some well-known results on the multiplicative order of elements in finite fields.

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“…Consequently, as pointed out in [4], any permutation f of F q can be represented by a polynomial of the form…”

Section: Introductionmentioning

confidence: 99%

Complete mappings and Carlitz rank

Işık

Topuzoğlu

Winterhof

2016

Des. Codes Cryptogr.

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The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any d ≥ 2 and any prime p > (d 2 − 3d + 4) 2 there is no complete mapping polynomial inFor arbitrary finite fields F q , we give a similar result in terms of the Carlitz rank of a permutation polynomial rather than its degree. We prove that if n < ⌊q/2⌋, then there is no complete mapping in F q [x] of Carlitz rank n of small linearity. We also determine how far permutation polynomials f of Carlitz rank n < ⌊q/2⌋ are from being complete, by studying value sets of f + x. We provide examples of complete mappings if n = ⌊q/2⌋, which shows that the above bound cannot be improved in general.

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On the cycle structure of permutation polynomials

Cited by 40 publications

References 7 publications

Permutations from an arithmetic setting

Permutations from an arithmetic setting

The dynamics of permutations on irreducible polynomials

Complete mappings and Carlitz rank

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