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I need to find out $\log_g {-1}$ in $\mathbb{Z}_n$ where $n$ is an odd prime and $g$ is a primitive root mod $n$. How do I do that?

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    Try squaring $-1$.2010-12-20

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Seems you are looking for $x$ such that $g^x = -1$.
$x = \frac{n-1}{2}$ seems to work because $\phi(n)=n-1$ and g is primitive root and $g^{\phi(n)}=1$