Statistically close

The variation distance of two distributions $X$ and $Y$ over a finite domain $D$, (often referred to as statistical difference ^[1] or statistical distance^[2] in cryptography) is defined as

$\Delta(X,Y)=\frac{1}{2} \sum _{\alpha \in D} | \Pr[X=\alpha] - \Pr[Y=\alpha] |$.

We say that two probability ensembles $\{X_k\}_{k\in\N}$ and $\{Y_k\}_{k\in\N}$ are statistically close if $\Delta(X_k,Y_k)$ is a negligible function in $k$.

References

1. Goldreich, Oded (2001). Foundations of Cryptography: Basic Tools (1st ed.). Berlin: Cambridge University Press. p. 106. ISBN 0-521-79172-3.
2. Reyzin, Leo. (Lecture Notes) Extractors and the Leftover Hash Lemma

See also

• Zero-knowledge proof
• Randomness extractor