We investigate monomials axd over the finite field with q elements Fq, in the case where the degree d is equal to +1 with q=(q′)n for some n. For n=6 we explicitly list all a's for which axd is a complete permutation polynomial (CPP) over Fq. Some previous characterization results by Wu et al. for n=4 are also made more explicit by providing a complete list of a's such that axd is a CPP. For odd n, we show that if q is large enough with respect to n then axd cannot be a CPP over Fq, unless q is even, n≡3(mod4), and the trace TrFjavax.xml.bind.JAXBElement@54673199/Fjavax.xml.bind.JAXBElement@7a30de8d(a−1) is equal to 0.
Bartoli, D., Giulietti, M., Zini, G. (2016). On monomial complete permutation polynomials. FINITE FIELDS AND THEIR APPLICATIONS, 41, 132-158 [10.1016/j.ffa.2016.06.005].
On monomial complete permutation polynomials
Zini, G
2016
Abstract
We investigate monomials axd over the finite field with q elements Fq, in the case where the degree d is equal to +1 with q=(q′)n for some n. For n=6 we explicitly list all a's for which axd is a complete permutation polynomial (CPP) over Fq. Some previous characterization results by Wu et al. for n=4 are also made more explicit by providing a complete list of a's such that axd is a CPP. For odd n, we show that if q is large enough with respect to n then axd cannot be a CPP over Fq, unless q is even, n≡3(mod4), and the trace TrFjavax.xml.bind.JAXBElement@54673199/Fjavax.xml.bind.JAXBElement@7a30de8d(a−1) is equal to 0.
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