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It is said that for the group $\mathbb{Z}_p^*$, the factorization of $p-1$, is critical.

If $p-1$ has some small factors $q_1, q_2, q_3, q_4$, then when we transmit $g^x \bmod p$ where $g$ is a generator of this group, the attacker can derive $x \bmod q_1q_2q_3q_4$

How does it happen?

Can someone please provide a practical example, say for $p = 29$, where $p$ is prime but $p-1$ has a bunch of small factors?

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    $\begingroup$ Have you looked at the Pohling-Hellman algorithm? Any remaining question after that? $\endgroup$ – fgrieu Jan 23 at 14:59
  • $\begingroup$ You can easily find example. Take a look at this answer for instance. $\endgroup$ – Faulst Jan 25 at 18:15

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