TR20-148 Authors: Lijie Chen, Roei Tell

Publication: 29th September 2020 20:09

Downloads: 375

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Extending the classical ``hardness-to-randomness'' line-of-works, Doron et al. (FOCS 2020) recently proved that derandomization with near-quadratic time overhead is possible, under the assumption that there exists a function in $\mathcal{DTIME}[2^n]$ that cannot be computed by randomized SVN circuits of size $2^{(1-\epsilon)\cdot n}$ for a small $\epsilon$.

In this work we extend their inquiry and answer several open questions that arose from their work. Our main result is that *derandomization with almost no time overhead is possible*, under a plausible hypothesis. Specifically, we show that probabilistic algorithms that run in time $T(n)$ can be deterministically simulated in time $n\cdot T(n)^{1+\epsilon}$, under a hypothesis that is formally incomparable to the one of Doron et al., but is arguably more standard: We assume that there exist non-uniformly secure one-way functions, and that for $\delta=\delta(\epsilon)$ and $k=k_T(\epsilon)$ there exists a problem in $\mathcal{DTIME}[2^{k\cdot n}]$ that is hard for algorithms that run in time $2^{(k-\delta)\cdot n}$ and use $2^{(1-\delta)\cdot n}$ bits of advice. We also show that the latter hypothesis (or, more accurately, a relaxation of it that we use) is in fact necessary to obtain the derandomization conclusion if one relies on a PRG construction (as is indeed our approach).

For sub-exponential time functions $T(n)=2^{n^{o(1)}}$ we further improve the derandomization time to $n^{1+\epsilon}\cdot T(n)$, under a mildly stronger hypothesis. We also show that *the multiplicative time overhead of $n$ is essentially optimal*, conditioned on a counting version of the non-deterministic strong exponential-time hypothesis (i.e., on $\# NSETH$). Nevertheless, we show that *in the average-case setting a faster derandomization is possible*: Under hypotheses similar to the ones in our main result, we show that for every $L\in\mathcal{BPTIME}[n^k]$ there exists a deterministic algorithm $A_L$ running in time $n^{\epsilon}\cdot n^{k}$ such that for every distribution $\mathcal{D}$ over $\{0,1\}^n$ samplable in time $n^k$ it holds that $\Pr_{x\sim\mathcal{D}}[A_L(x)=L(x)]\ge1-n^{-\omega(1)}$.

Lastly, we present an alternative proof for the result of Doron et al. using a *proof paradigm that is both considerably simpler and more general*; in fact, we show how to simplify the analysis of any construction that ``extracts randomness from a pseudoentropic string''. We use this simpler proof to extend their result, deducing a mildly slower derandomization (i.e., with cubic or quadratic overhead) from weaker hardness assumptions (i.e., for SVN circuits that do not use randomness).