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$\mathbb Z_n^*$ is a mathematical notation for the multiplicative group of integers modulo $n$. In other words, it is the set of integers that are relatively prime to $n$, all taken modulo $n$ (excluding zero). The $*$ symbol is commonly used for denoting "the set of elements in the multiplicative group", which in this case means "the set of elements that ...

The $1^k$ is a formalism that's only there to make the theoreticians happy. You can safely ignore it. When you actually implement the cryptosystem, you don't try to pass the string $1^k$; instead, you pass $k$, the security parameter (a representation of how much cryptographic strength is desired from the key generation algorithm). I wish I could leave it ...