A note on ex-tracting randomness from Santha-Vazirani sources

Santha and Vazirani [SV86] proposed the notion of a semi-random source of bits (also known as Santha-Vazirani sources), and proved that it is impossible to (deterministically) extract an almost-uniform random bit from such a source (while it is possible given several independent Santha-Vazirani sources). We provide a simpler and more transparent proof of their impossibility result. Our proof has the advantage of applying to a natural strengthening of Santha-Vazirani sources. Moreover, our proof technique has been used in [DOPS04] to obtain impossibility results on doing cryptography with semi-random sources. A source of length n is a random variable X taking values in {0, 1}n. We will denote the individual bits of X by X = X1 · · ·Xn. If Y is a random variable taking values in {0, 1}, the bias of Y is |Pr[Y = 0]− Pr[Y = 1]|, i.e. the smallest δ such that (1− δ)/2 ≤ Pr[Y = 0] ≤ (1 + δ)/2. Definition 1 For δ ∈ [0, 1], a source X of length n is a Santha-Vazirani (SV) source (or semirandom source) with bias δ if for every i ∈ [n] and every x1, . . . , xi ∈ {0, 1}, the bias of Xi conditioned on X1 = x1, . . . , Xi−1 = xi−1 is at most δ. That is, 1− δ 2 ≤ Pr[Xi = xi|X1 = x1, . . . , Xi−1 = xi−1] ≤ 1 + δ 2 . ∗Most of this research was performed while at AT&T Labs Research. Florham Park, NJ, and while visiting the Institute for Advanced Study, Princeton, NJ. Research was supported in part by US-Israel Binational Science Foundation Grant 2002246. †Supported by US-Israel BSF Grant 2002246, NSF grant CCR-0133096, and ONR Grant N00014-04-1-0478. URL: http://eecs.harvard.edu/~salil. ‡Add acks.