P = BPP if E requires exponential circuits: Derandomizing the XOR lemma

Yao showed that the XOR of independent random instances of a somewhat hard Boolean problem becomes almost completely unpredictable. In this paper we show that, in non-uniform settings, total independence is not necessary for this result to hold. We give a pseudo-random generator which produces n instances of a problem for which the analog of the XOR lemma holds. Combining this generator with the results of [25, 6] gives substantially improved results for hardness vs randomness tradeoffs. In particular, we show that if any problem in E = DTIME(2^O(n)) has circuit complexity 2^Ω(n), then P = BPP. Our generator is a combination of two known ones - the random walks on expander graphs of [1, 10, 19] and the nearly disjoint subsets generator of [23, 25]. The quality of the generator is proved via a new proof of the XOR lemma which may be useful for other direct product results.