 
Summary: Simple Constructions of Almost kwise Independent
Random Variables
Noga Alon
Oded Goldreich
Johan H°astad
RenŽe Peralta§
February 22, 2002
Abstract
We present three alternative simple constructions of small proba
bility 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
), where is the statistical dif
ference 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 < 1/(k log n)). An additional advantage of our
constructions is their simplicity.
Keywords: Probabilistic computation, removing randomness, shiftregister se
quences, small probability spaces.
