Statistically close

The variation distance of two distributions $X$ and $Y$ over a finite domain $D$, (often referred to as statistical difference or statistical distance 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 $\{X_{k}\}_{k\in \mathbb {N} }$ and $\{Y_{k}\}_{k\in \mathbb {N} }$ are statistically close if $\Delta (X_{k},Y_{k})$ is a negligible function in $k$.

See also

• Zero-knowledge proof
• Randomness extractor