An ( n k ) -bit-fixing source is a distribution on n bit strings, that is fixed on n − k of the coordinates, and jointly uniform on the remaining k bits. Explicit constructions of bit-fixing extractors by Gabizon, Raz and Shaltiel [SICOMP 2006] and Rao [CCC 2009], extract (1 − o (1)) k bits for k = polylo g ( n ) , almost matching the probabilistic argument. Intriguingly, unlike other well-studied sources of randomness, a result of Kamp and Zuckerman [SICOMP 2006] shows that, for any k , some small portion of the entropy in an ( n k ) -bit-fixing source can be extracted. Although the extractor does not extract all the entropy, it does extract (1 2 − o (1)) log ( k ) bits.

In this paper we prove that when the entropy k is small enough compared to n , this exponential entropy-loss is unavoidable. More precisely, one cannot extract more than log ( k ) 2 + O (1) bits from ( n k ) -bit-fixing sources. The remaining entropy is inaccessible, information theoretically. By the Kamp-Zuckerman construction, this negative result is tight.

Our impossibility result also holds for what we call zero-fixing sources. These are bit-fixing sources where the fixed bits are set to 0 . We complement our negative result, by giving an explicit construction of an ( n k ) -zero-fixing extractor, that outputs ( k ) bits, even for k = polyloglo g ( n ) . Furthermore, we give a construction of an ( n k ) -bit-fixing extractor, that outputs k − O (1) bits, for entropy k = ( 1 + o (1)) log log n , with running-time n O (( log log n ) 2 ) .