Cyclotomic orthomorphisms of finite fields (Q1759888)

A mapping \(f\) of a group onto itself is called \textit{orthomorphism} if both \(x\mapsto f(x)\) and \(x \mapsto f(x)-x\) define bijections, and \(f\) is called a \textit{strong complete mapping} if, in addition, \(x\mapsto f(x)+x\) defines a bijection too. Such mappings have applications in cryptography, coding theory and combinatorics (for instance in the construction of designs and of mutually orthogonal sets of Latin squares). Furthermore, for primes p, strong complete mappings of \((\mathbb{F}_p,+)\) correspond to toroidal-p-queens solutions (cf.\ \textit{J. Bell} and \textit{B. Stevens} [Discrete Math. 309, No. 1, 1--31 (2009; Zbl 1228.05002)]). A useful class of orthomorphism is built by those which are constructed using cyclotomy in finite fields (see e.g. \textit{A. B. Evans} [Congr. Numerantium 101, 97--107 (1994; Zbl 0864.11064)] and \textit{H. Niederreiter} and \textit{A. Winterhof} [Discrete Math. 295, No. 1--3, 161--171 (2005; Zbl 1078.11068)]): If \(q=km+1\) is a prime power and \(\gamma\) a generator of the multiplicative group \(\mathbb{F}_q^{\ast}\), a \textit{cyclotomic mapping of index} \(k\) is a mapping \(f:\mathbb{F}_q \rightarrow \mathbb{F}_q\) defined by \(f(0)=0\) and \(f(x)=a_rx\) if \(x\in \gamma^r<\gamma^k>\) for some \(a_0, a_1, \ldots a_{k-1}\in \mathbb{F}_q\). In the present paper, the authors determine the number of cyclotomic orthomorphisms of \(\mathbb{F}_q\) of index \(3,4,5\) and \(6\) for certain classes of prime powers \(q\) and count the number of cyclotomic mappings of index \(2\) which are strong complete mappings of \(\mathbb{F}_p\) for primes \(p\).