Testing k-wise and almost k-wise independence

In this work, we consider the problems of testing whether a distribution over {0, 1}n is k-wise (resp. ( , k)-wise) independent using samples drawn from that distribution. For the problem of distinguishing k-wise independent distributions from those that are -far from k-wise independence in statistical distance, we upper bound the number of required samples by ~O(nk /2 ) and lower bound it by (n k-1 2 /) (these bounds hold for constant k, and essentially the same bounds hold for general k). To achieve these bounds, we use Fourier analysis to relate a distribution's distance from k-wise independence to its biases, a measure of the parity imbalance it induces on a set of variables. The relationships we derive are tighter than previously known, and may be of independent interest. To distinguish ( , k)-wise independent distributions from those that are -far from ( , k)-wise independence in statistical distance, we upper bound the number of required samples by O `k log n 2 2 ? and low...

• Type: Conference
• Year: 2007
• Where: STOC
• Authors: Noga Alon, Alexandr Andoni, Tali Kaufman, Kevin Matulef, Ronitt Rubinfeld, Ning Xie
• Keywords: Algorithms, K-wise Independence, NSF Graduate Fellowship, STOC 2007, Tel Aviv University