Paper 2021/1076

Hardness of KT Characterizes Parallel Cryptography

Hanlin Ren and Rahul Santhanam

Abstract

A recent breakthrough of Liu and Pass (FOCS'20) shows that one-way functions exist if and only if the (polynomial-)time-bounded Kolmogorov complexity, ${\mathrm{K}}^t$, is bounded-error hard on average to compute. In this paper, we strengthen this result and extend it to other complexity measures: * We show, perhaps surprisingly, that the $\mathrm{K}\mathrm{T}$ complexity is bounded-error average-case hard if and only if there exist one-way functions in *constant parallel time* (i.e., ${\mathsf{N}\mathsf{C}}^0$). This result crucially relies on the idea of *randomized encodings*. Previously, a seminal work of Applebaum, Ishai, and Kushilevitz (FOCS'04; SICOMP'06) used the same idea to show that $$-computable one-way functions exist if and only if logspace-computable one-way functions exist. * Inspired by the above result, we present randomized average-case reductions among the $$-versions and logspace-versions of $$ complexity, and the $$ complexity. Our reductions preserve both bounded-error average-case hardness and zero-error average-case hardness. To the best of our knowledge, this is the first reduction between the $$ complexity and a variant of $$ complexity. * We prove tight connections between the hardness of $$ complexity and the hardness of (the hardest) one-way functions. In analogy with the Exponential-Time Hypothesis and its variants, we define and motivate the *Perebor Hypotheses* for complexity measures such as $$ and $$. We show that a Strong Perebor Hypothesis for $$ implies the existence of (weak) one-way functions of near-optimal hardness $$. To the best of our knowledge, this is the first construction of one-way functions of near-optimal hardness based on a natural complexity assumption about a search problem. * We show that a Weak Perebor Hypothesis for $$ implies the existence of one-way functions, and establish a partial converse. This is the first unconditional construction of one-way functions from the hardness of $$ over a natural distribution. * Finally, we study the average-case hardness of $$. We show that it characterizes cryptographic pseudorandomness in one natural regime of parameters, and complexity-theoretic pseudorandomness in another natural regime.