Homomorphic extensions of CRT-based secret sharing

Abstract

In this work, we explore the homomorphic aspect of CRT-based secret sharing schemes. Secret sharing homomorphism is the notion of operating on multiple secrets by direct computation on shares. There are schemes based on polynomial interpolation which have partially homomorphic properties, whereas CRT-based secret sharing homomorphism is an open area.Because of the disparate structure of the CRT-based schemes, we introduce advanced security notion and analyze the existing schemes. We formulate homomorphic inadequacy caused by the overflow problem and present sufficient and necessary homomorphism conditions. Then, we show the impossibility of homomorphic and secure threshold Asmuth–Bloom scheme while keeping the original structure. Accordingly, we propose possible homomorphic extensions to the Asmuth–Bloom SSS.Our first extension, additively homomorphic ramp scheme, can attain arbitrarily large information rate. Besides, it is the only CRT-based scheme possessing perfect secrecy. The second scheme allows homomorphic addition and also multiplication to some point. The bound on addition operations is correlated with the share size of the scheme, whereas multiplication bound is inversely proportional to the secrecy threshold.We give detailed analyses of the extensions and their security proofs as well as their properties like information rate, security characteristic, and homomorphic capabilities.