A Coefficient Test for Quintic Permutation Polynomials Over Integer Rings

For selecting appropriate permutation polynomials (PPs) in practical applications, it is necessary to know the coefficients of the polynomial since a brute-force exhaustive search is impractical when the number of PPs is large. Previous results give the conditions on the coefficients of a polynomial of degree up to four so that it is a PP modulo a given positive integer. For polynomials of degree higher than four, we only know the conditions so that they are PPs modulo a power of two. In [13] all PPs of degree no more than six are generated using an algorithm based on normalized PPs, two previous important theorems about PPs and the Chinese remainder theorem. In this paper, we propose a coefficient test for quintic permutation polynomials (5-PPs) over integer rings which, unlike the algorithm from [13], allows to decide directly whether a polynomial of degree five or less is PP. Using the proposed coefficient test, the coefficients of PPs modulo a given positive integer can be obtained in a desired order, which is tractable in computer processing.