Almost k-wise independence versus k-wise independence

Authors:
Noga Alon;Oded Goldreich;Yishay Mansour
Affiliations:
Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat-Aviv, Israel;Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel;School of Computer Science, Tel Aviv University, Ramat-Aviv, Israel
Venue:
Information Processing Letters
Year:
2003

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Abstract

We say that a distribution over {0,1}n is (ε,k)-wise independent if its restriction to every k coordinates results in a distribution that is ε-close to the uniform distribution. A natural question regarding (ε, k)-wise independent distributions is how close they are to some k-wise independent distribution. We show that there exist (ε, k)-wise independent distributions whose statistical distance is at least nO(k) ċ ε from any k-wise independent distribution. In addition, we show that for any (ε,k)-wise independent distribution there exists some k-wise independent distribution, whose statistical distance is nO(k) ċ ε.