# why do we take computational distinguishability over ensembles

In the Cornell lecture notes, computational indistinguishability is defined as

Definition 69.4 (Computational Indistinguishability). Let $$\{X_n\}_n$$ and $$\{Y_n\}_n$$ be ensembles where $$X_n$$,$$Y_n$$ are distributions over $$\{0, 1\}^{l(n)}$$ for some polynomial $$l(·)$$. We say that $$\{X_n\}_n$$ and $$\{Y_n\}_n$$ are computationally indistinguishable if for all non-uniform p.p.t. $$D$$ (called the “distinguisher”), there exists a negligible function $$e(·)$$ such that $$∀n ∈ N$$

$$Pr[t ← X_n, D(t) = 1] − Pr[t ← Y_n, D(t) = 1] < e(n)$$

My question is, why do we need ensembles of distribution? Why not just define it over two distributions $$X$$ and $$Y$$?

Why not just define it over two distributions $$X$$ and $$Y$$?
If you were to only consider two fixed distributions, there would be no scaling with the security parameter (and thus the involved lengths) possible. This is why we use ensembles, parametrized by the value of the security parameter $$n$$.