In constructing a SHVZK simulator for a sigma protocol I am working on I have encountered some fairly basic questions, but ones which are not often discussed in textbooks and papers - consider the two following probability distributions:
- $X_n$ - The uniform distribution on $\{1, \dots n\}=[n]$
- $Y_n$ - The uniform distribution on $[n] \backslash \{a\}$, for some fixed integer $a \in [n]$.
The distributions are not perfectly indistinguishable because an unbounded distinguisher could keep sampling elements from each distribution until it is clear which element is missing from the second distribution.
The distributions are statistically indistinguishable with statistical distance \begin{align*} \Delta(X_n,Y_n) &= \frac{1}{2}\sum_{x \in [n]}{|\Pr[X_n=x] - \Pr[Y_n = x] |} \\ &= \frac{1}{2}(\sum_{x \in [n]\backslash\{a\}}{\bigg|\frac{1}{n} - \frac{1}{n-1}\bigg|}) + \frac{1}{2}(\frac{1}{n} - 0)\\ &= \frac{1}{2}(n-1)\frac{1}{n(n-1)} + \frac{1}{2n}\\ &= \frac{1}{n} \end{align*} If $n = \exp(\lambda)$, the distributions are statistically indistinguishable since $\frac{1}{\exp(\lambda)} \leq \text{negl}(\lambda)$.
Now, the question, which I feel I've somewhat answered already (see answer below), but was not that clear to me from my readings:
Q: Intuitively, I've heard statistical indistinguishability been described as the following: "An unbounded distinguisher cannot distinguish the two distributions" (For example, this post), but this seems false in the example above?
