On exponential-time hypotheses, derandomization, and circuit lower bounds: Extended abstract

The Exponential-Time Hypothesis (ETH) is a strengthening of the mathcal{P} neq mathcal{NP} conjecture, stating that 3-SAT on n variables cannot be solved in (uniform) time 2{epsilon cdot n}, for some epsilon > 0. In recent years, analogous hypotheses that are'exponentially-strong' forms of other classical complexity conjectures (such as mathcal{NP} not subseteq mathcal{BPP} or co mathcal{NP} not subseteq mathcal{NP}) have also been introduced, and have become widely influential. In this work, we focus on the interaction of exponential-time hypotheses with the fundamental and closely-related questions of derandomization and circuit lower bounds. We show that even relatively-mild variants of exponential-time hypotheses have far-reaching implications to derandomization, circuit lower bounds, and the connections between the two. Specifically, we prove that: 1)The Randomized Exponential-Time Hypothesis (rETH) implies that mathcal{BPP} can be simulated on'average-case' in deterministic (nearly-)polynomial-time (i.e., in time 2{tilde{O}(log(n))}=n{log ! log(n){O(1)}}). The derandomization relies on a conditional construction of a pseudorandom generator with near-exponential stretch (i.e., with seed length tilde{O}(log(n))); this significantly improves the state-of-the-art in uniform'hardness-to-randomness' results, which previously only yielded pseudorandom generators with sub-exponential stretch from such hypotheses. 2)The Non-Deterministic Exponential-Time Hypothesis (NETH) implies that derandomization of mathcal{BPP} is completely equivalent to circuit lower bounds against mathcal{E}, and in particular that pseudorandom generators are necessary for derandomization. In fact, we show that the foregoing equivalence follows from a very weak version of NETH, and we also show that this very weak version is necessary to prove a slightly stronger conclusion that we deduce from it. Lastly, we show that disproving certain exponential-time hypotheses requires proving breakthrough circuit lower bounds. In particular, if CireuitSAT for circuits over n bits of size poly(n) can be solved by probabilistic algorithms in time 2n/polylog(n), then mathcal{BP} epsilon does not have circuits of quasilinear size.