On the Possibility of Basing Cryptography on EXP≠ BPP

Liu and Pass (FOCS’20) recently demonstrated an equivalence between the existence of one-way functions (OWFs) and mild average-case hardness of the time-bounded Kolmogorov complexity problem. In this work, we establish a similar equivalence but to a different form of time-bounded Kolmogorov Complexity—namely, Levin’s notion of Kolmogorov Complexity—whose hardness is closely related to the problem of whether EXP≠ BPP. In more detail, let Kt(x) denote the Levin-Kolmogorov Complexity of the string x; that is, Kt(x)=minΠ∈{0,1}∗,t∈N{|Π|+⌈logt⌉:U(Π,1t)=x}, where U is a universal Turing machine, and U(Π, 1 ^t) denotes the output of the program Π after t steps, and let MKtP denote the language of pairs (x, k) having the property that Kt(x) ≤ k. We demonstrate that: MKtP∉ Heur_negBPP (i.e., MKtP is infinitely-often two-sided error mildly average-case hard) iff infinitely-often OWFs exist.MKtP∉ Avg_negBPP (i.e., MKtP is infinitely-often errorless mildly average-case hard) iff EXP≠ BPP. Thus, the only “gap” towards getting (infinitely-often) OWFs from the assumption that EXP≠ BPP is the seemingly “minor” technical gap between two-sided error and errorless average-case hardness of the MKtP problem. As a corollary of this result, we additionally demonstrate that any reduction from errorless to two-sided error average-case hardness for MKtP implies (unconditionally) that NP≠ P. We finally consider other alternative notions of Kolmogorov complexity—including space-bounded Kolmogorov complexity and conditional Kolmogorov complexity—and show how average-case hardness of problems related to them characterize log-space computable OWFs, or OWFs in NC⁰.