Motivation for the definition of statistical distance

Statistical distance is a widely used measure in cryptography for comparing two distributions. One can define various other measures for capturing the differences between two distributions, but why do we prefer to use statistical distance? What is the physical significance of statistical distribution definition, which goes as follows:

Let $X_1$ and $X_2$ be random variables with domains $D_1$ and $D_2$, then the statistical distance between $X_1$ and $X_2$ is defined as:

$\delta(X_1,X_2) = \dfrac{1}{2}\sum\limits_{x \in D_1 \cup D_2}|Pr(X_1 = x) - Pr(X_2 = x)|$

I found this definition in the second chapter of the textbook titled "Secure Multiparty Computation and Secret Sharing".

2 Answers

Statistical indistinguishability (i.e., negligible statistical distance) implies computational indistinguishability, but is typically easier to prove since it does not require a computational argument with a reduction and all that stuff.

Of course, it is not always possible to prove computational indistinguishability in that way; statistical indistinguishability is a stronger, information-theoretic notion.

An operational meaning of the statistical distance between two random variables defined by $$\delta(X_1,X_2) = \dfrac{1}{2}\sum\limits_{x \in D_1 \cup D_2}\left|Pr(X_1 = x) - Pr(X_2 = x)\right|$$ is that an optimal function $f$ distinguishing between $X_1$ and $X_2$ can be defined on $D_1 \cup D_2$ via $$f(z)=\left\{ \begin{array}{lcl} X_1 \quad&if~~& Pr(X_1=z)>Pr(X_2=z)\ & & \\ X_2 & &else \end{array} \right.$$ where $z$ is the observed output from an unknown source that is distributed as $X_1$ or $X_2$. This function correctly identifies the distribution with probability $\delta(X_1,X_2).$ If there are points where the two probabilities are equal, $f$ can uniformly answer $X_1$ or $X_2.$