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

TL;DR

This paper introduces four quantum-resistant, multi-stage secret sharing schemes based on inhomogeneous linear recursions and Ajtai's lattice function, offering cheat detection, multiple secret recovery methods, and efficiency advantages.

Contribution

It proposes novel, verifiable multi-stage secret sharing schemes utilizing inhomogeneous linear recursions and Ajtai's lattice function, enhancing security and flexibility.

Findings

Schemes can detect cheating by dealer and participants.

Multiple methods for secret recovery are provided.

Schemes are quantum-resistant with reduced time consumption.

Abstract

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.