Simple Constructions of Almost k‐wise Independent Random Variables

Abstract

Abstract We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o (1)) (log log n + k /2 + log k + log 1/ϵ), where ϵ is the statistical difference between the distribution induced on any k bit locations and the uniform distribution. This is asymptotically comparable to the construction recently presented by Naor and Naor (our size bound is better as long as ϵ < 1/( k log n )). An additional advantage of our constructions is their simplicity.