General weak random sources

The following model for a weak random source is considered. The source is asked only once for R bits, and the source outputs an R-bit string such that no string has probability more than 2 ^-δR of being output. for some fixed δ>0. A pseudorandom generator that runs in time n^{O(log n)} and simulates RP using as a seed a string from such a source is exhibited. Under the generalized Paley graph conjecture, a generator that runs in polynomial time and simulates RP is given, as well as a different generator that produces almost perfectly random bits at a rate arbitrarily close to optimal using as seeds strings from a constant number of independent weak random sources