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Consider a randomized algorithm with runtime $n$, which succeeds with high probability. The algorithm uses $O(n)$ uniformly random bits.

Now it is given that we can replace these uniformly random bits by $k$-wise independent bits, where $k$ depends on $n$ (say $k = \sqrt{n}$). Can we then reduce the number of random bits to $\tilde{O}(k)$ while keeping a runtime $\tilde{O}(n)$?

Clearly we could use k-wise independent bit generators (such as a random polynomial). However, naively implementing this would cost $\tilde{O}(k)$ steps per call, potentially increasing the runtime to $\tilde{O}(n k)$.

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Yes. You can generate a random polynomial of degree $k$, then evaluate this polynomial at $n$ different points in $\tilde{O}(n)$ time using the DFT (the DFT lets you evaluate a polynomial of degree $n$ at $n$ different points in $\tilde{O}(n)$ time).

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  • $\begingroup$ Is this easy to see for $n$ arbitrary points? Anyway a formal proof seems to be given in the paper "Fast modular transforms via division" by Moenck and Borodin (SWAT '72). $\endgroup$ – smapers Sep 25 at 13:27
  • $\begingroup$ @smapers, no, it doesn't work for arbitrary points (as far as I know). But for this application you don't need to evaluate the polynomial at arbitrary points; it suffices to evaluate it at any $n$ points that are convenient. If you found a citation, I encourage you to write an answer of your own providing a reference to this result. $\endgroup$ – D.W. Sep 25 at 17:03
  • $\begingroup$ You need to evaluate the polynomial at $n$ points of the field $F_p$ for some prime $p \geq n$ (so $n$ distinct integers). Am I correct that DFT only allows to evaluate the polynomial at the $n$ roots of unity? $\endgroup$ – smapers Sep 26 at 6:58
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    $\begingroup$ @smapers, Yes. The DFT works over finite fields, too; see e.g. en.wikipedia.org/wiki/…. $\endgroup$ – D.W. Sep 26 at 15:58

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