A trigonometric approach for Chebyshev polynomials over finite fields (Q2811684)

The well-known Chebyshev polynomials over the real or complex field are classified in four families. The first two kinds have analogues over finite fields, generally referred to as Dickson polynomials. \textit{Q. Wang} and \textit{J. L. Yucas} [Finite Fields Appl. 18, No. 4, 814--831 (2012; Zbl 1266.11128)] have introduced more kinds of Dickson polynomials as linear combinations of those of the first and second kind.NEWLINENEWLINEThe paper under review introduces Chebyshev polynomials of the third and fourth kind over finite fields using tools of trigonometric flavour. Elaborating on work of the third author et al. [``Trigonometry in finite fields and a new Hartley transform'', in: Proceedings of the IEEE International Symposium on Infomation Theory, ISIT '98. New York: IEEE (1998; \url{doi:10.1109/ISIT.1998.708898})], the present authors study basic properties of hyperbolic trigonometric functions over finite fields of odd characteristic. Relations thus obtained allow them to define four kinds of Chebyshev polynomials. Recurrence relations satisfied by the newly introduced polynomials coincide, on the one hand, with those encountered in the classical context and, on the other hand, with those derived by non-trigonometric approaches for Dickson polynomials of the first and second kind. They are useful in the study of periodicity and symmetry properties of Chebyshev polynomials over finite fields.NEWLINENEWLINEA whole section is devoted to the investigation of permutation properties of these polynomials. The emphasis is put on polynomials of the second kind. The authors show a different way to derive known sufficient conditions for Chebyshev polynomials of the second kind to permute a finite field of odd characteristic. Sufficient conditions under which Chebyshev polynomials of the third or fourth kind permute the coefficients field are also pointed out. The necessity of the latter conditions has been verified by computer for characteristic \(p\leq 31\) and base field \(\mathbb F_{p^2}\).NEWLINENEWLINEFor the entire collection see [Zbl 1314.11002].