How strong is Nisanʼs pseudo-random generator?

We study the resilience of the classical pseudo-random generator (PRG) of Nisan [Nis92] against space-bounded machines that make multiple passes over the input. Our motivation comes from the derandomization of BPNC. Observe that if for every log-space machine that reads its input n times there is a PRG fooling this machine, then in particular we fool NC circuits. Nisan’s PRG is known to fool log-space machines that read the input once. We ask what are the limits of this PRG regarding log-space machines that make multiple passes over the input. We show that for every constant k Nisan’s PRG fools log-space machines that make log n passes over the input, using a seed of length log ′ n, for some k′ > k. We complement this result by showing that in general Nisan’s-PRG cannot fool log-space machines that make n passes even for a seed of length 2 √ log . The observations made in this note outline a more general approach in understanding the difficulty of derandomizing BPNC.