What is the difference between, and security of $Z_p$ 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.