Notes on Complexity Theory

Put differently, let D = {x1, . . . , x`} and consider the random variables Yi = hK(xi) (where K is uniform). If H is pairwise independent then each Yi is uniformly distributed, and moreover the random variables Y1, . . . , Y` are pairwise independent; i.e., for any i 6= j the random variables Yi and Yj are independent. We show a simple construction of a pairwise-independent family for D = R = F, where F is any finite field. Setting F = GF (2n), and viewing strings of length n as field elements, we obtain a construction with D = R = {0, 1}n. By truncating the output, we obtain a construction with D = {0, 1}n and R = {0, 1}` for any n ≥ `. By padding the input with 0s, we obtain a construction for any ` ≥ n. Fix D = R = F and let H = {ha,b}a,b∈F where ha,b(x) = ax + b. We claim that H is pairwise independent. Indeed, fix any distinct x, x′ ∈ F and any y, y′ ∈ F, and consider the probability (over choice of a, b) that