If \( a^k \equiv 1 \), then \( (a^{-1})^k \equiv 1 \) and conversely, so both have the same order. Setting the order of \( a^{-1} \) equal to \( \varphi(n) \) is a common error.
The multiplicative order of \( a \) modulo \( n \), \( \operatorname{ord}_n(a) \), is the smallest positive exponent that returns \( a \) to the unit class modulo \( n \).