  Permutation polynomials over a ring of modulo $2^w$ are compatible with
digital computers and digital signal processors, and so they are in particular
expected to be useful for cryptography and pseudo random number generator. In
general, the period of the polynomial should be long in such fields. In this
paper, we derive the necessary and sufficient condition which specify
one-stroke polynomials which are permutation polynomials whose periods are
maximized.
