  This paper mainly studies problems about so called "permutation polynomials
modulo $m$", polynomials with integer coefficients that can induce bijections
over Z_m={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.
