DLP - method for calculating order

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The DLP is defined as finding $x$, for:

$$g^x=h\pmod{p}$$

Where $p$ is prime

1) I think that to calculate the multiplicative order you take $g$ and $p$, and find $m$, where $m$ is the minimum positive integer s.t. $g^m=1\pmod{p}$.

2) Another method is that $m=order(Z_p^*)=p-1$

Question: Which method is correct?

I then check if $m$ can be wrote in form $q^e$, where $q$ is prime and $e \ge 1$.

I see that $q$ and $e$ can be used to calculate $x$.

I'm following the algorithm "Proposition 2.34" in Introduction to Mathematical Cryptography by Jones, Silverman et al