4/5/2023 0 Comments 19 permute 3![]() However, the problem of determining the compositional inverse of known classes of permutation polynomial seems to be an even more complicated problem. ![]() The construction of permutation polynomials over finite fields is an old and difficult problem that continues to attract interest due to their applications in various area of mathematics. Find inverse polynomial of known classes of permutation polynomials over F q. Therefore, it is both interesting and important to find more explicit classes of permutation polynomials. ![]() Find new classes of permutation polynomials of F q.Īlthough several classes of permutation polynomials have been found in recent years, but, an explicit and unified characterization of permutation polynomials is not known and seems to be elusive today. There are so many open problems and conjectures on permutation polynomials over finite fields but here we are listing few of them. Very little is known concerning which polynomials are permutation polynomials, despite the attention of numerous authors. For f, g ∈ F q x we have f α = g α for all α ∈ F q if and only if f x ≡ g x mod x q − x.ĭue to the finiteness of the field, the followings are the equivalent conditions for a polynomial to be a permutation polynomial.ĭefinition 2.The polynomial f ∈ F q xis a permutation polynomial of F qif and only if one of the following conditions holds:į x = ahas a solution in F qfor each a ∈ F q į x = ahas a unique solution in F qfor each a ∈ F q.ġ.3 Open problems on permutation polynomials Two polynomials represent the same function if and only if they are the same by reduction modulo x q − x, according to the following result. Given any arbitrary function ϕ : F q → F q, the unique polynomial g ∈ F q xwith deg g < qrepresenting ϕcan be found by the formula g x = ∑ c ∈ F q ϕ c 1 − x − c q − 1, see (, Chapter 7). First it will be convenient to define permutation polynomial over a finite field.ĭefinition 1.A polynomial f x ∈ F q xis said to be a permutation polynomial over F qfor which the associated polynomial function c ↦ f cia a permutation of F q, that is, the mapping from F qto F qdefined by fis one–one and onto.įinite fields are polynomially complete, that is, every mapping from F qinto F qcan be represented by a unique polynomial over F q. In this section, we collect some basic facts about permutation polynomials over a finite field that will be frequently used throught the chapter.
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