In this paper, we construct several new classes of complete permutation monomials a −1 x d over a finite field F q n with exponents d=[Formula presented]+1, [Formula presented]+1, and [Formula presented]+1, respectively, where q=p k is a power of a prime number p. Our approach uses the AGW criterion (the multiplicative case) together with Dickson permutation polynomials and a class of exceptional polynomials respectively. One of our results confirms Conjecture 4.18 by G. Wu, N. Li, T. Helleseth, Y. Zhang in [42] under the assumption that the characteristic p is primitive modulo a prime number n+1. Moreover, we show that Conjecture 4.18 is false in general using our approach and a counterexample is provided. We also re-confirm Conjecture 4.20 in [42] that was proved recently in [24], and extend some of these recent results to more general n's and more general a's.

Complete permutation polynomials, Finite fields, Monomials, Permutation polynomials
Finite Fields and their Applications
School of Mathematics and Statistics

Feng, X. (Xiutao), Lin, D. (Dongdai), Wang, L. (Liping), & Wang, Q. (2019). Further results on complete permutation monomials over finite fields. Finite Fields and their Applications, 57, 47–59. doi:10.1016/j.ffa.2019.01.003