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Formally, a block cipher is a family of permutations, indexed by the key. More specifically, let $P$ be the set of all permutations (shufflings of elements as you put it) on the set of $n$ bit strings, i.e. the Symmetric Group $Sym(\{0,1\}^n)$. The $n$-bit block cipher $B$ is a subset of $P$. The key specifies which element of the subset $B$ is to be used ...