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Say we are given the following cyclic group $G$ which is generated by $g$ element of $\mathbb{Z}_p$. In this case $g=13$ and $p=23$ so $13$ element of $\mathbb{Z}_{23}$.

If there exists two random $g$ values in the cyclic group $G$, how does one go about computing the discrete log in $G$?

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Say $g=8$. Then since $8\cong {13^2}\pmod{23}$, we say the discrete log of $8$ with respect to the base $13$ mod $23$ is $2$.

You can write $2=\operatorname {ind}_{13}8\pmod{23}$. (Gauß called it the index.)

I chose an easy example. These can be difficult to compute, which must be why they are used in cryptography.

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