  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.
