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$\mathbb{Z}^*_p$ vs $\mathbb{Z}^*_{p-1}$ vs $\mathbb{Z}^*_{p^2}$ vs $\mathbb{Z}^+_{p^2}$

I know $p$ is the value. The value create must be coprime to $p$. Does that mean that the value create must be coprime to $p-1$? What about the rest?

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The multiplicative group $\mathbb{Z}/n\mathbb{Z}^\times$ has order $\phi(n)$ and is cyclic for $n$ prime. The additive group $\mathbb{Z}/n\mathbb{Z}^+$ has order $n$ and is always cyclic. Because $\phi(p) = p-1$, if you pick the orders appropriately, they will be isomorphic:

$$\mathbb{Z}/p\mathbb{Z}^\times \cong \mathbb{Z}/(p-1)\mathbb{Z}^+$$

The isomorphism $\sigma$ permits you to write $\sigma(a \times b) = \sigma(a)+\sigma(b)$, which has a very intuitive interpretation as a logarithm in the multiplicative group.

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