# Attack against factorization of $p-1$ of $\mathbb{Z}_p^*$ group

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?

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