In this paper, we construct a lattice based threshold multi-stage secret sharing (MSSS) scheme according to Ajtai’s construction for one-way functions. In an MSSS scheme, the authorized subsets of participants can recover a subset of secrets at each stage while other secrets remain undisclosed. In this paper, each secret is a vector from a   $t$  -dimensional lattice and the basis of each lattice is kept private. A   $t$      -subset of    $n$       participants can recover the secret(s) using their assigned shares. Using a lattice based one-way function, even after some secrets are revealed, the computational security of the unrecovered secrets is provided against quantum computers. The scheme is multi-use in the sense that to share a new set of secrets, it is sufficient to renew some public information such that a new share distribution is no longer required. Furthermore, the scheme is verifiable meaning that the participants can verify the shares received from the dealer and the recovered secrets from the combiner, using public information.
['Hossein Pilaram', 'Taraneh Eghlidos']
An Efficient Lattice Based Multi-Stage Secret Sharing Scheme