Statistically Close

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10/7/2007
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The variation distance of two distributions X {\displaystyle X} and Y {\displaystyle Y} over a finite domain D {\displaystyle D} , (often referred to as statistical difference or statistical distance in cryptography) is defined as Δ ( X , Y ) = 1 2 ∑ α ∈ D | Pr [ X = α ] − Pr [ Y = α ] | {\displaystyle \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 ∈ N {\displaystyle \{X_{k}\}_{k\in \mathbb {N} }} and { Y k } k ∈ N {\displaystyle \{Y_{k}\}_{k\in \mathbb {N} }} are statistically close if Δ ( X k , Y k ) {\displaystyle \Delta (X_{k},Y_{k})} is a negligible function in k {\displaystyle k} .
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