Because of Euler's Theory, we can state that \(g^x \pmod p\) is equal to \(g^{x \pmod {(p-1)}} \pmod p\), and that \(g^x \pmod p\) will not be equal to \(g^{x \pmod {p}}\) if \(x\) is greater than \(p\). In the following \(p\), \(q\), \(p'\) and \(q'\) are all prime numbers. If so we have the following truths:
\(g^x \pmod p \neq g^{x \pmod {p}} \pmod p\), where \(x \ge p\)
\(g^x \pmod p \equiv g^{x \pmod {(p-1)}} \pmod p\)
\(g^x \pmod n \equiv g^{x \pmod {(p-1)(q-1)}} \pmod n\) and where \(n=pq\).
\(g^x \pmod n \equiv g^{x \pmod {(p'q')}} \pmod n\) and where \(n=pq\), \(p=2p'+1\) and \(q=2q'+1\).