Optimal Computational Secret Sharing

TL;DR

This paper introduces a secret sharing scheme that reduces share size to (|S| + |K|)/t and proves its optimality under certain cryptographic assumptions, improving efficiency over previous methods.

Contribution

It presents a new secret sharing construction achieving minimal share size and proves its optimality under specific cryptographic assumptions.

Findings

Share size of (|S| + |K|)/t achieved

Optimality proven under cryptographic assumptions

Improves efficiency over previous schemes

Abstract

In $(t, n)$-threshold secret sharing, a secret $S$ is distributed among $n$ participants such that any subset of size $t$ can recover $S$, while any subset of size $t-1$ or fewer learns nothing about it. For information-theoretic secret sharing, it is known that the share size must be at least as large as the secret, i.e., $|S|$. When computational security is employed using cryptographic encryption with a secret key $K$, previous work has shown that the share size can be reduced to $\tfrac{|S|}{t} + |K|$. In this paper, we present a construction achieving a share size of $\tfrac{|S| + |K|}{t}$. Furthermore, we prove that, under reasonable assumptions on the encryption scheme -- namely, the non-compressibility of pseudorandom encryption and the non-redundancy of the secret key -- this share size is optimal.