K-wise Independence and -biased K-wise Indepedence Proof: Sketch for the Forward Direction, Recall That ^
نویسنده
چکیده
Conversely, suppose all the low-order Fourier coe cients are 0. Since D̂(100 0) = Pr[x1 = 0] Pr[x1 = 1] = 0, x1 is equally likely to be 0 or 1, and similarly for the other xi's. Now let pa1a2 = Pr[x1x2 = a1a2]. Since x1, x2, and x1 x2 are all unbiased, we get the equations p00 + p01 = p00 + p10 = p00 + p11 = 1=2. Furthermore, p00 + p01 + p10 + p11 = 1. From these equations, we deduce that all of the pa1a2 's are 1=4, and similarly for any pair xi and xj. Proceeding inductively in this manner shows that D is k-wise independent.
منابع مشابه
K-wise Independence and -biased K-wise Indepedence
1 Deenitions Consider a distribution D on n bits x = x 1 x n. D is k-wise independent ii for all sets of k indices S = fi 1 x ik = a 1 a k ] = 1 2 k : The idea is that if we restrict our attention to any k positions in x, no matter how many times we sample from D, we cannot distinguish D from the uniform distribution over n bits. We can get a Fourier interpretation of k-wise independence by vie...
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