Inverses for Fourth-Degree Permutation Polynomials Modulo 32Ψ or 96Ψ, with Ψ as a Product of Different Prime Numbers Greater than Three

In this paper, we address the inverse of a true fourth-degree permutation polynomial (4-PP), modulo a positive integer of the form 32kLΨ, where kL∈{1,3} and Ψ is a product of different prime numbers greater than three. Some constraints are considered for the 4-PPs to avoid some complicated coefficients’ conditions. With the fourth- and third-degree coefficients of the form k4,fΨ and k3,fΨ, respectively, we prove that the inverse PP is (I) a 4-PP when k4,f∈{1,3} and k3,f∈{1,3,5,7} or when k4,f=2 and (II) a 5-PP when k4,f∈{1,3} and k3,f∈{0,2,4,6}.