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On monomial complete permutation polynomials. (English) Zbl 1372.11107
Summary: We investigate monomials $$a x^d$$ over the finite field with $$q$$ elements $$\mathbb{F}_q$$, in the case where the degree $$d$$ is equal to $$\frac{q - 1}{q' - 1} + 1$$ with $$q = (q')^n$$ for some $$n$$. For $$n = 6$$ we explicitly list all $$a$$’s for which $$a x^d$$ is a complete permutation polynomial (CPP) over $$\mathbb{F}_q$$. 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 $$a x^d$$ is a CPP. For odd $$n$$, we show that if $$q$$ is large enough with respect to $$n$$ then $$a x^d$$ cannot be a CPP over $$\mathbb{F}_q$$, unless $$q$$ is even, $$n \equiv 3(\operatorname{mod} 4)$$, and the trace $$\operatorname{Tr}_{\mathbb{F}_q / \mathbb{F}_{q'}}(a^{- 1})$$ is equal to 0.

##### MSC:
 11T06 Polynomials over finite fields 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 94A60 Cryptography
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