Permutation Polynomials modulo m

This paper mainly studies problems about so called “permutation polynomials modulo m”, polynomials with integer coefficients that can induce bijections over Zm = {0, · · · , m−1}. The necessary and sufficient conditions of permutation polynomials are given, and the number of all permutation polynomials of given degree and the number induced bijections are estimated. A method is proposed to determine all equivalent polynomials from the induced polynomial function, which can be used to determine all equivalent polynomials that induce a given bijection. A few problems have not been solved yet in this paper and left for open study. Note: After finishing the first draft, we noticed that some results obtained in this paper can be proved in other ways (see Remark 2). In this case, this work gives different and independent proofs of related results.