A Secret Sharing Scheme Based on Group Presentations and the Word Problem

TL;DR

This paper introduces two novel secret sharing schemes based on non-abelian groups, leveraging group presentations and the word problem to enhance security and flexibility in secret distribution.

Contribution

It proposes new secret sharing methods using non-abelian groups, including a threshold scheme combining Shamir's scheme with group-theoretic techniques.

Findings

One scheme requires all participants to recover the secret.

The other scheme is a (t,n)-threshold scheme combining Shamir's and group-based methods.

The schemes utilize the word problem in non-abelian groups for security.

Abstract

A (t,n)-threshold secret sharing scheme is a method to distribute a secret among n participants in such a way that any t participants can recover the secret, but no t-1 participants can. In this paper, we propose two secret sharing schemes using non-abelian groups. One scheme is the special case where all the participants must get together to recover the secret. The other one is a (t,n)-threshold scheme that is a combination of Shamir's scheme and the group-theoretic scheme proposed in this paper.