Existing proofs that deduce BPL=L from circuit lower bounds convert randomized algorithms into deterministic algorithms with large constant overhead in space. We study space-bounded derandomization with minimal footprint, and ask what is the minimal possible space overhead for derandomization.
We show that $BPSPACE[S] \subseteq DSPACE[c \cdot S]$ for $c \approx 2$, assuming space-efficient cryptographic PRGs, and, either: (1) lower bounds against bounded-space algorithms with advice, or: (2) lower bounds against certain uniform compression algorithms.
Under additional assumptions regarding the power of catalytic computation, in a new setting of parameters that was not studied before, we are even able to get $c \approx 1$.

Our results are constructive: Given a candidate hard function (and a candidate cryptographic PRG) we show how to transform the randomized algorithm into an efficient deterministic one.
This follows from new PRGs and targeted PRGs for space-bounded algorithms, which we combine with novel space-efficient evaluation methods. A central ingredient in all our constructions is hardness amplification reductions in logspace-uniform $TC^0$, that were not known before.