Let's consider this definition of computational indistinguishability.
Computational indistinguishability. A probability ensemble $X=\{X(a, n)\}_{a \in\{0,1\}^{*} ; n \in \mathbb{N}}$ is an infinite sequence of random variables indexed by $a \in\{0,1\}^{*}$ and $n \in \mathbb{N}$. In the context of secure computation, the value $a$ will represent the parties' inputs and $n$ will represent the security parameter. Two probability ensembles $X=\{X(a, n)\}_{a \in\{0,1\}^{*} ; n \in \mathbb{N}}$ and $Y=\{Y(a, n)\}_{a \in\{0,1\}^{*} ; n \in \mathbb{N}}$ are said to be computationally indistinguishable, denoted by $X \stackrel{c}{\equiv} Y$, if for every non-uniform polynomial-time algorithm $D$ there exists a negligible function $\mu(\cdot)$ such that for every $a \in\{0,1\}^{*}$ and every $n \in \mathbb{N}$, $$ |\operatorname{Pr}[D(X(a, n))=1]-\operatorname{Pr}[D(Y(a, n))=1]| \leq \mu(n) $$
From my understanding $D$ is the distinguishing algorithm, e.g. the adversary in security proofs. An instance of the random variable $X(a,n)$ is considered as the encryption algorithm. However from my understanding only the output of the encryption algorithm, e.g. the ciphertext, is passed to $D$. For people coming from mathematical background this is a bit confusing because random variable is a function $X:Ω \rightarrow Ε$ where $Ω$ is the σ-algebra of the events space and $E$ is a measurable space.
Can someone help me clarify the notation and the definition that is used? Thanks in advance.
