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Secret sharing was firstly proposed in 1979 by Shamir and Blakleyrespectively. To avoid deficiencies of original schemes, researchers presentedimprovement schemes, among which the multi-secret sharing scheme (MSS) issignificant. There are three categories of MSSs, however, we focus onmulti-stage secret sharing scheme (MSSS) recovering secrets with any order inthis work. By observing inhomogeneous linear recursions (ILRs) in theliterature, we conclude a general formula and divide ILRs into two typesaccording to different variables in them. Utilizing these two kinds of ILRs, wepropose four verifiable MSSSs with Ajtai's function, which is a lattice-basedfunction. Our schemes have the following advantages. Firstly, our schemes candetect cheat of the dealer and participants, and are multi-use. Secondly, wehave several ways to restore secrets. Thirdly, we can turn our schemes intoother types of MSSs due to the universality of our method. Fourthly, since weutilize a lattice-based function to mask shares, our schemes can resist theattack from the quantum computer with computational security. Finally, althoughour schemes need more memory consumption than some known schemes, we need muchless time consumption, which makes our schemes more suitable facing limitedcomputing power.

Post-quantum Multi-stage Secret Sharing Schemes using Inhomogeneous Linear Recursion and Ajtai's Function