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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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