Definition of Polynomial-Time Indistinguishability

Question

We call two ensembles $X$ and $Y$ indistinguishable in polynomial time if for every probabilistic polynomial-time algorithm $D$ and for every positive polynomial $p(\cdot)$, and all sufficiently large n's we have $$|Pr[D(X_n,1^n)=1]-Pr[D(Y_n,1^n)=1]| < \frac{1}{p(n)}$$.

One question I didn't confront with at the beginning is, does the definition imply that $|X_n|=|Y_n|$?

After a little bit of thinking I came to the following conclusion which I want to verify. $X_n$ can be distributed over $\{0,1\}^{poly_1(n)}$ while $Y_n$ is distributed over $\{0,1\}^{poly_2(n)}$. For sufficiently large $n$ the polynomials grow monotonously and one could use the length as a distinguishing attack. We know there exists a distinguisher that makes by coincidence use of the very exact same polynomial $p_1$ or $p_2$ and can easily compare the lengths of the input with its hardwired polynomial. We would require that an $N$ exists such that $|X_n|=|Y_n|$ is true for all $n>N$ (which would imply the polynomials to be equal), else we will never be able to find such ensembles.

Answer 1

One can have $|X_n|\neq |Y_n|$, or even $|X_n| = f(n) \neq \infty = |Y_n|$ for some function $f(n)$. This occurs in practice, namely in lattice-based cryptography, where $X_n$ is some (ideal) discrete Gaussian, while $Y_n$ is some approximation to it of finite support (so one can sample from it in constant-time). For a discrete gaussian of mean $\mu$ and standard deviation $\sigma$, one can replace it with that same gaussian conditioned to being in the interval $[\mu-10\sigma, \mu+10\sigma]$ or something.

What I think you're missing is that being "distinguishable" vs "indistinguishable" is not a binary. If one ignores pseudorandomness for a bit (and talks about purely statistics), then often the total variation distance (or statistical distance)

$$\Delta(X, Y) = \frac{1}{2}\sum_{x\in\mathsf{supp}(X)\cup\mathsf{supp}(Y)}|\Pr[X = x]-\Pr[Y = x]|.$$

This (essentially) measures how hard it is to distinguish $X$ and $Y$. For example, if $\Delta(X,Y)\leq \epsilon$, then it takes $O(1/\epsilon^2)$ samples from an unknown distribution to reliably (say with probability close to 1) distinguish between the cases of the samples coming from $X$ or $Y$. So if $\epsilon \approx 2^{-n}$, one can say they cannot be distinguished with polynomially many samples.

There are other times that one may want to say it is hard to distinguish distributions that have $|X_n|\neq |Y_n|$. For example, let $X_n$ be uniform over all functions $F: \{0,1\}^n\to\{0,1\}^n$. Let $Y_n$ be uniform over all invertible functions with this domain/range. Then $|X_n|\neq |Y_n|$, but one can bound the statistical distance between these distributions provided one gets at most $q$ queries (this is called "PRP-PRF Switching Lemma").