Considering \(a^m \mod n = 1\), \(GDC(a, n) = 1\), then there must exist for \(m = \phi(n)\) (from Euler’s Theorem#) but may be smaller and once powers reach \(m\), cycle will repeat. If the smallest is \(m = \phi(n)\) then \(a\) is called a Primitive Root.
If \(p\) is prime, then successive powers of \(a\) “generate” the group \(\mod p\). However, it is relatively hard to find Primitive Root.
Note: I’ve no idea what the hell I’m talking about.