On the distribution of the power generator

We present a new method to study the power generator of pseudorandom numbers modulo a Blum integer m. This includes as special cases the RSA generator and the Blum–Blum–Shub generator. We prove the uniform distribution of these, provided that the period t ≥ m3/4+δ with fixed δ > 0 and, under the same condition, the uniform distribution of a positive proportion of the leftmost and rightmost bits. This sharpens and generalizes previous results which dealt with the RSA generator, provided the period t ≥ m23/24+δ . We apply our results to deduce that the period of the binary sequence of the rightmost bit has exponential length.