Randomized Component and Its Application to ( $t$ , $m$ , $n$ )-Group Oriented Secret Sharing

A basic (t,n)-secret sharing (SS) scheme allows a secret s to be divided into n shares and shared among n shareholders. In the scheme, any t or more than t shareholders can recover the secret while fewer than t shareholders cannot obtain the secret s. But an adversary without any valid share may obtain the secret if there are over t participants in the secret reconstruction. To address this type of attack, we first introduce the notion of randomized component (RC), which binds a share with all participants and protects the share from being exposed to outside without any computational assumption; at the same time, RCs can be used to reconstruct the secret. As one of the applications of RCs, a (t,m,n)-group oriented SS scheme is proposed to cope with the attack in basic (t,n)-SSs, in which once m (m ≥ t) participants form a tightly couple group by generating RCs, the secret can be recovered only if all m RCs are correct, which requires each participant to have a valid share in advance. Moreover, the scheme can secure the secret without any user authentication or share verification. Analyses show the proposed (t,m,n)-group oriented SS is asymptotically perfect and unconditionally secure. RCs can also be applied to build other schemes in a simple way, such as multi-SS, group authentication, and so on.