On Exponential-time Hypotheses, Derandomization, and Circuit Lower Bounds

The Exponential-Time Hypothesis (ETH) is a strengthening of the ĝ conjecture, stating that 3-SAT on n variables cannot be solved in (uniform) time 2^ϵċn, for some ϵ > 0. In recent years, analogous hypotheses that are "exponentially strong"forms of other classical complexity conjectures (such as ĝ ĝ., or coĝ ) 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 ĝ., can be simulated on "average-case"in deterministic (nearly-)polynomial-time (i.e., in time 2^Õ(log(n)) = n^{loglog(n)O(1)}). The derandomization relies on a conditional construction of a pseudorandom generator with near-exponential stretch (i.e., with seed length Õ(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 ĝ., is completely equivalent to circuit lower bounds against ĝ.,°, 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.Last, we show that disproving certain exponential-time hypotheses requires proving breakthrough circuit lower bounds. In particular, if CircuitSAT for circuits over n bits of size poly(n) can be solved by probabilistic algorithms in time 2^n/polylog(n), then ĝ.,ĝ.,° does not have circuits of quasilinear size.