Specific permutation polynomials over finite fields

Classes of Permutation Polynomials Based on Cyclotomy and an Additive Analogue

“…One way to get explicit examples satisfying the conditions of this result is as follows: if B = x q/p + x q/p 2 + • • • + x p + x and A ∈ F p [x], then A(B(x)) = B(A(x)), so f permutes F q if and only if A permutes ker B and A(x) + B(g(x)) permutes im B = F p . In case g is a constant (in F q ) times a polynomial over F p , this becomes (a slight generalization of) [26,Thm. 6].…”

“…More generally, any polynomial of the form f (x) := x r h(x (q−1)/d ) induces a mapping on F q modulo d-th powers, so testing whether f permutes F q reduces to testing whether the induced mapping on cosets Date: October 30, 2018. I thank José Marcos for sending me preliminary versions of his paper [26], and for encouraging me to develop consequences of his ideas while his paper was still under review.…”

“…In the recent preprint [26], Marcos gives five constructions of permutation polynomials. His first two constructions are new classes of permutation polynomials having the above cyclotomic form.…”

“…The idea of the analogy is that T (x) induces a homomorphism F q → F p , just as x (q−1)/d induces a homomorphism from F * q to its subgroup of d-th roots of unity. The fourth construction in [26] is a variant of the third for polynomials of the form L(x) + h(T (x))(L(x) + c), and the fifth construction replaces T (x) with other symmetric functions in x q/p , x q/p 2 , . .…”

“…In this paper I present rather more general versions of the first four constructions from [26], together with simplified proofs. I can say nothing new about the fifth construction from [26], although that construction is quite interesting and I encourage the interested reader to look into it.…”