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 ≤ NC 0 m reductions. In this paper, we improve this, to show that MKTP is hard for the (apparently larger) class NISZK L under not only ≤ NC 0 m reductions but even under projections. Also MKTP is hard for NISZK under ≤ P / poly m reductions. Here, NISZK is the class of problems with non-interactive zero-knowledge proofs, and NISZK L is the non-interactive version of the class SZK L 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 ).