On the Possibility of Basing Cryptography on $\mathsf{EXP}\ne \mathsf{BPP}$

Abstract

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 $\mathsf{EXP}\ne \mathsf{BPP}$. In more detail, let Kt(x) denote the Levin-Kolmogorov Complexity of the string x; that is, $Kt(x) = \min _{{\varPi }\in \{0,1\}^*, t \in \mathbb {N}}\{|{\varPi }| + \lceil \log t \rceil : U({\varPi }, 1^t) = x\}$, where U is a universal Turing machine, and $U({\varPi },1^t)$ denotes the output of the program $\varPi$ after t steps, and let $\mathsf{MKtP}$ denote the language of pairs (x, k) having the property that $Kt(x) \le k$. We demonstrate that:

• $\mathsf{MKtP}\notin \mathsf{Heur}_{\mathsf{neg}}\mathsf{BPP}$ (i.e., $\mathsf{MKtP}$ is infinitely-often two-sided error mildly average-case hard) iff infinitely-often OWFs exist.

• $\mathsf{MKtP}\notin \mathsf{Avg}_{\mathsf{neg}}\mathsf{BPP}$ (i.e., $\mathsf{MKtP}$ is infinitely-often errorless mildly average-case hard) iff $\mathsf{EXP}\ne \mathsf{BPP}$.

Thus, the only “gap” towards getting (infinitely-often) OWFs from the assumption that $\mathsf{EXP}\ne \mathsf{BPP}$ is the seemingly “minor” technical gap between two-sided error and errorless average-case hardness of the $\mathsf{MKtP}$ problem.

As a corollary of this result, we additionally demonstrate that any reduction from errorless to two-sided error average-case hardness for $\mathsf{MKtP}$ implies (unconditionally) that $\mathsf{NP}\ne \mathsf{P}$.

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 $\mathsf{NC}^{0}$.

R. Pass—Supported in part by NSF Award SATC-1704788, NSF Award RI-1703846, AFOSR Award FA9550-18-1-0267, and a JP Morgan Faculty Award. This material is based upon work supported by DARPA under Agreement No. HR00110C0086. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government or DARPA.