, 03.12.2019 01:20 hoopstarw4438

# :pairwise independence. suppose we want to generate n pairwise independent random variables in the range {0, 1, 2, . . , m âˆ’ 1}. we will assume that n and m are powers of 2 and let n = {0, 1} n and m = {0, 1} m (hence n = log n and m = log m). we saw a scheme in the lecture using mn bits. here we will revisit that scheme in a different way and then see how it can be made more randomness-efficient. pick a uniformly random matrix a âˆˆ {0, 1} mÃ—n and a random vector b âˆˆ {0, 1} m . then for a vector v âˆˆ {0, 1} n , set xv = av + b mod 2 (by this we mean component wise mod 2). (a) suppose we pick a and b uniformly at random. show that under this scheme, for all w âˆˆ {0, 1} n where w 6= 0 and for all Î³ âˆˆ {0, 1} m , pa[aw = Î³ mod 2] = 1 2m . why does this guarantee that xu and xv are independent for u 6= v and u 6= 0, v 6= 0?

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