How strong is Nisanʼs pseudo-random generator?

We study the resilience of the classical pseudo-random generator (PRG) of Nisan (1992) [6] against space-bounded machines that make multiple passes over the input. 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^kn passes over the input, using a seed of length log^k^^^'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^O^(^1^) passes even for a seed of length 2^@Q^(^l^o^g^n^). The observations made in this note outline a more general approach in understanding the difficulty of derandomizing BPNC^1.