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We present a general technique for obtaining permutation polynomials over a finite field from permutations of a sub-field. By applying this technique to the simplest classes of permutation polynomials on the subfield, we obtain several new families of permutation polynomials. Some of these have the additional property that both f (x) and f (x) + x induce… (More)

Suppose x m +ax n is a permutation polynomial over F p , where p > 5 is prime and m > n > 0 and a ∈ F * p. We prove that gcd(m−n, p−1) / ∈ {2, 4}. In the special case that either (p − 1)/2 or (p − 1)/4 is prime, this was conjectured in a recent paper by Masuda, Panario and Wang.

For a fixed prime p, we consider the set of maps Z/pZ → Z/pZ of the form a → Tn(a), where Tn(x) is the degree-n Chebyshev polynomial of the first kind. We observe that these maps form a semigroup, and we determine its size and structure.

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