Many researchers studied computationally-secure (verifiable) secret sharing schemes which distribute multiple secrets with a bulletin board. However, the security definition is ambiguous in many of the past articles. In this paper, we first review existing schemes based on formal definitions of indistinguishability of secrets, verifiability of consistency, and cheater-detectability. And then, we propose a new secret sharing scheme which is the first scheme with indistinguishability of secrets, verifiability, and cheater-detectability, and allows to share secrets with arbitrary access structures. Further, our scheme is provably secure under well known computational assumptions.

On-line secret sharing scheme, introduced by Cachin, is a computational variation of secret sharing scheme. It supports dynamic changing of access structures and reusable shares, by grace of public bulletin board. In this paper, first we introduce a formal model of on-line secret sharing scheme, and analyze existing on-line secret sharing schemes. As a result, it is shown that they are all vulnerable by giving concrete attacks. Next, we propose a novel on-line secret sharing scheme which is provably secure.