On the distribution of the power generator (Q2723530)

Let \(e\geq 2\), \(m\geq 1\) and \(v\) be integers such that \(\gcd(v,m)= 1\). Then the sequence \((u_n)\) is defined by NEWLINE\[NEWLINEu_n= u_{n-1}^e\pmod m, \quad 0\leq u_n\leq m-1,\;n=1,2,\dotsNEWLINE\]NEWLINE with the initial state \(u_0= v\). Two important special cases are given when \(\gcd(e,\varphi(m))= 1\) where \(\varphi(m)\) is the Euler function, the RSA generator, and the case \(e=2\), the Blum-Blum-Schub generator. The main result is to show that these generators are uniformly distributed when the period \(t> m^{3/4+\delta}\) with fixed \(\delta> 0\).