Cryptographic hardness under projections for time-bounded Kolmogorov complexity

Abstract

A version of time-bounded Kolmogorov complexity, denoted KT, has received attention in the past several years, due to its close connection to circuit complexity and to the Minimum Circuit Size Problem MCSP. Essentially all results about the complexity of MCSP hold also for MKTP (the problem of computing the KT complexity of a string). Both MKTP and MCSP are hard for SZK (Statistical Zero Knowledge) under BPP-Turing reductions; neither is known to be NP-complete.Recently, some hardness results for MKTP were proved that are not (yet) known to hold for MCSP. In particular, MKTP is hard for DET (a subclass of P) under nonuniform ≤mNC0 reductions. In this paper, we improve this, to show that MKTP‾ is hard for the (apparently larger) class NISZKL under not only ≤mNC0 reductions but even under projections. Also MKTP‾ is hard for NISZK under ≤mP/poly reductions. Here, NISZK is the class of problems with non-interactive zero-knowledge proofs, and NISZKL is the non-interactive version of the class SZKL that was studied by Dvir et al.As an application, we provide several improved worst-case to average-case reductions to problems in NP, and we obtain a new lower bound on MKTP (which is currently not known to hold for MCSP).