Toward Basing Cryptography on the Hardness of EXP

Abstract

Let Kt ( x ) denote the Levin-Kolmogorov Complexity of the string x , and let MKtP denote the language of pairs ( x , k ) having the property that Kt ( x ) ≤ k. We demonstrate that: • MKtP ∉ Heur neg BPP (i.e., MKtP is two-sided error mildly average-case hard) iff infinitely-often OWFs exist. • MKtP ∉ Avg neg BPP (i.e., MKtP is errorless mildly average-case hard) iff EXP ≠ BPP. Taken together, these results show that the only "gap" toward 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.