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...