Secret Sharing with Binary Shares

TL;DR

This paper introduces new secret sharing schemes with binary shares that achieve near-optimal secret length and security against adaptive and non-adaptive adversaries, even in the challenging regime of small gap ratios and minimal share length.

Contribution

It provides explicit constructions of secret sharing schemes with binary shares that are secure under semantic security, overcoming limitations of traditional ramp schemes in extreme parameters.

Findings

Achieves secret length close to optimal for non-adaptive adversaries.

Constructs schemes secure against adaptive adversaries with linear secret length.

Eliminates impossibility of ramp schemes in the small gap ratio regime.

Abstract

Shamir's celebrated secret sharing scheme provides an efficient method for encoding a secret of arbitrary length $\ell$ among any $N \leq 2^\ell$ players such that for a threshold parameter $t$, (i) the knowledge of any $t$ shares does not reveal any information about the secret and, (ii) any choice of $t+1$ shares fully reveals the secret. It is known that any such threshold secret sharing scheme necessarily requires shares of length $\ell$, and in this sense Shamir's scheme is optimal. The more general notion of ramp schemes requires the reconstruction of secret from any $t+g$ shares, for a positive integer gap parameter $g$. Ramp secret sharing scheme necessarily requires shares of length $\ell/g$. Other than the bound related to secret length $\ell$, the share lengths of ramp schemes can not go below a quantity that depends only on the gap ratio $g/N$. In this work, we study secret sharing in the extremal case of bit-long shares and arbitrarily small gap ratio $g/N$, where standard ramp secret sharing becomes impossible. We show, however, that a slightly relaxed but equally effective notion of semantic security for the secret, and negligible reconstruction error probability, eliminate the impossibility. Moreover, we provide explicit constructions of such schemes. One of the consequences of our relaxation is that, unlike standard ramp schemes with perfect secrecy, adaptive and non-adaptive adversaries need different analysis and construction. For non-adaptive adversaries, we explicitly construct secret sharing schemes that provide secrecy against any $\tau$ fraction of observed shares, and reconstruction from any $\rho$ fraction of shares, for any choices of $0 \leq \tau < \rho \leq 1$. Our construction achieves secret length $N(\rho-\tau-o(1))$, which we show to be optimal. For adaptive adversaries, we construct explicit schemes attaining a secret length $\Omega(N(\rho-\tau))$.