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 $m$ ( $m\ge t$ ) 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.