Computationally secure group ring based multi-stage secret sharing schemes

Ronald Cramer, Ivan Damgård, Ueli M. Maurer

Amos Beimel

Zhichao Yang, Debiao He, Longjiang Qu +1 more

In response to the threat posed by quantum computers, Pilaram and Eghlidos proposed the first lattice-based multi-stage secret sharing scheme which is the only post-quantum multi-stage secret sharing scheme. In this paper, we introduce an efficient attack on it and show that any adversary can easily reconstruct unrecovered secrets as long as it collects enough pseudo-secret shares. For the sake of complete, we further list two countermeasures to protect the scheme from such attack.

Hossein Pilaram, Taraneh Eghlidos

In this paper, we construct a lattice based (t, n) 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.

R. Sreerama Kumar, Sahadeo Padhye

Jing Yang, Fang‐Wei Fu

Secret sharing was firstly proposed in 1979 by Shamir and Blakley respectively. To avoid deficiencies of original schemes, researchers presented improvement schemes, among which the multi-secret sharing scheme (MSS) is significant. There are three categories of MSSs, however, we focus on multi-stage secret sharing scheme (MSSS) recovering secrets with any order in this work. By observing inhomogeneous linear recursions (ILRs) in the literature, we conclude a general formula and divide ILRs into two types according to different variables in them. Utilizing these two kinds of ILRs, we propose four verifiable MSSSs with Ajtai's function, which is a lattice-based function. Our schemes have the following advantages. Firstly, our schemes can detect cheat of the dealer and participants, and are multi-use. Secondly, we have several ways to restore secrets. Thirdly, we can turn our schemes into other types of MSSs due to the universality of our method. Fourthly, since we utilize a lattice-based function to mask shares, our schemes can resist the attack from the quantum computer with computational security. Finally, although our schemes need more memory consumption than some known schemes, we need much less time consumption, which makes our schemes more suitable facing limited computing power.