Secret Sharing for mNP: Completeness Results

Abstract

We show completeness results for secret sharing schemes realizing \(\mathsf {mNP}\) access structures. We begin by proposing a new, Euclidean-type, division technique for access structures. Using this new technique we obtain several results in characterizing access structures for efficient (unconditionally secure) secret sharing schemes: We show a useful transformation that achieves efficient schemes for complex access structures using schemes realizing simple access structures. We show that, assuming every access structure in \(\mathsf {P} \cap \mathsf {mono}\) admits efficient secret sharing, the existence of an efficient secret sharing for an access structure in \(\mathsf {mNP}\) that is also complete for \(\mathsf {mNP}\) under Karp/Levin monotone-reductions implies secret sharing schemes for all of \(\mathsf {mNP}\). We finally improve upon the above completeness result by obtaining the same under ordinary Karp/Levin reductions.