  In this paper, we focus on analyzing the period distribution of the inversive
pseudorandom number generators (IPRNGs) over finite field $({\rm
Z}_{N},+,\times)$, where $N>3$ is a prime. The sequences generated by the
IPRNGs are transformed to 2-dimensional linear feedback shift register (LFSR)
sequences. By employing the generating function method and the finite field
theory, the period distribution is obtained analytically. The analysis process
also indicates how to choose the parameters and the initial values such that
the IPRNGs fit specific periods. The analysis results show that there are many
small periods if $N$ is not chosen properly. The experimental examples show the
effectiveness of the theoretical analysis.
