On the Security of Secret Sharing Over a Ring and the Fast Implementation

Secret sharing is the method to share secrets among a group of shares, and the secret can be reconstructed if one obtains a predefined number of shares. The polynomial secret sharing is usually constructed over a field. In this letter, a novel polynomial secret sharing over a ring is proposed. In particular, by choosing a certain ring, the fast Fourier transform can be applied on the encoding of secret sharing. The analysis shows that the proposed secret sharing scheme requires <inline-formula><tex-math notation="LaTeX">$O(N\log _2 N)$</tex-math></inline-formula> Boolean operations per secret bit, which improves the prior result <inline-formula><tex-math notation="LaTeX">$O(8^{\log ^* N}N\log _2 N)$</tex-math></inline-formula> Boolean operations per secret bit. The simulation shows that the proposed scheme is in average four times faster than the conventional approach.