Score:0 Crypto

# Question on notation of random variables in probability ensembles 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.

Score:1 Crypto An instance of the random variable (,) is considered as the encryption algorithm.

I believe an instance of the variable $$X(a,n)$$ would usually refer to an encryption of a corresponding input $$a$$. ($$X(a,n)$$ is the encryption of $$a$$ with security parameter $$n$$)

Intuitively, this definition says that if you are given an element from either $$X$$ or $$Y$$, it is hard to distinguish which where it came from.

The distinguisher $$D$$ is given either an element from $$X$$ or $$Y$$, and the probability $$|\operatorname{Pr}[D(X(a, n))=1]-\operatorname{Pr}[D(Y(a, n))=1]|$$ indicates the ability for the distinguisher $$D$$ to distinguish these. Consider, as an example, if $$|\operatorname{Pr}[D(X(a, n))=1]-\operatorname{Pr}[D(Y(a, n))=1]| = 1,$$ what would this imply about $$D$$, $$X$$ and $$Y$$?  