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How can I use these properties to attack the DHKE? I know that the order of $a$ is always $2$ for $a = P - 1$ in $Z_p$. The subgroups generated by a will be $\{1,a\}$.

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closed as off-topic by e-sushi, cygnusv, yyyyyyy, poncho, SEJPM Jun 20 '15 at 20:48

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If the order of $a=-1\mod{p}$ and $b=a^{e}\mod{p}$, then $b=\pm{1}\mod{p}$.
If $b=1\mod{p}$, the discrete logarithm is $0$.
If $b=-1\mod{p}$, the discrete logarithm is $1$.

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