Standard Notation

$:=$: Used to declare/initialize the contents of a variable (unless they are output by an algorithm, in which case we use $\gets$), e.g., $y := g^x$
$\gets$: Initialize the contents of a variable to the output of an algorithm. e.g., $c \gets \mathsf{Enc}(pk, x)$.

$\stackrel{\$}{\gets}$ or $\stackrel{R}{\gets}$: sample uniformly at random. For example, $x \stackrel{\$}{\gets} \mathcal{X}$ means to sample $x$ uniformly at random from the set (or distribution) $\mathcal{X}$.
$\{0,1\}^*$ and $\{0,1\}^n$: The set of binary strings of arbitrary length and length $n$, respectively.
$[n]$: The set of integers $\{1, 2, \dots, n\}$.
$\mathcal{A}$: Adversary
$\mathbb{G}$: A group
i.i.d.: Independent and identically distributed
$\mathsf{td}$: Trapdoor
w.h.p.: With high probability.
$\mathbb{Z}_n$: The additive group of integers modulo $n$, namely $\{0,\dots,n-1\}$. This is a cyclic group, and in fact also a ring). When $n$ is a prime $p$, all non-identity elements are generators of the group. In that case, the set is also a finite field (since all nonzero elements have multiplicative inverses).
$\mathbb{Z}_n^\times$ or $\mathbb{Z}_n^*$: The multiplicative subgroup of $\mathbb{Z}_n$, which consists of all elements of $\mathbb{Z}_n$ that are coprime with $n$. This means that they all have multiplicative inverses modulo $n$ and thus this group is a commutative multiplicative group. By the definition of Euler's totient function, this means that the order of the group is $\lvert \mathbb{Z}_n^* \rvert = \phi(n)$.
When $n$ is a prime $p$, $\lvert \mathbb{Z}_p^* \rvert = \phi(p) = p-1$ is not prime (for $p>3$). Thus, computing the order of any element in the group is efficient if the factorization of $p-1$ is known.
For more number theory/algebra basics, see David Wu's factsheet.