What is the difference between, and security of $Z_p$ and $F_p$? While reading some papers, I found that sometimes $Z_p$ was used but sometimes $F_p$ was used ($p$ is prime). Usually, the author will only use their subgroups like this: choose a big prime factor of $p-1$ , denote $q$, and set $G=\langle g \rangle$. 

My questions are :


Why do we only use the subgroup while $Z_p$ OR $F_p$? Or maybe we can choose a $p$ such that $p-1$ is prime and now if $Z_p^*$ or $F_p^*$ can be directly used?
What's the difference of using $Z_p$ and  $F_p$?
You should read the Wikipedia article on finite fields. For each prime $p$ and for each $n &gt;0$, there is a unique field of order $p^n$ (up to isomorphism). This field is usually denoted $F_{p^n}$.
Now, for $n = 1$, the field $F_p$ can be identified with the set $\mathbf{Z}/p\mathbf{Z}$ of  integers modulo $p$, which is also sometimes denoted $\mathbf{Z}_p$. However, the notation $\mathbf{Z}_p$ is also used in number theory to denote the ring of $p$-adic integers and this may create a potential confusion.

The multiplicative group of every finite field is cyclic, that is, is generated by a single element. This multiplicative group is often used in cryptography.