Bounds on the Statistical Leakage-Resilience of Shamir's Secret Sharing

TL;DR

This paper introduces a statistical leakage model for secret sharing, analyzes the security resilience of Shamir's scheme under noisy leakage, and provides bounds showing exponential security improvements in certain cases.

Contribution

It is the first to analyze Shamir's secret sharing scheme under a general statistical noisy leakage model and derive security bounds.

Findings

Security of each secret bit improves exponentially with the number of users in certain scenarios.

Provides bounds on semantic and mutual information security under leakage.

First analysis of secret sharing security with noisy leakage.

Abstract

Secret sharing is an instrumental tool for sharing secret keys in distributed systems. In a classical threshold setting, this involves a dealer who has a secret/key, a set of parties/users to which shares of the secret are sent, and a threshold on the number of users whose presence is needed in order to recover the secret. In secret sharing, secure links with no leakage are often assumed between the involved parties. However, when the users are nodes in a communication network and all the links are physical links, e.g., wireless, such assumptions are not valid anymore. In order to study this critical problem, we propose a statistical leakage model of secret sharing, where some noisy versions of all the secret shares might be independently leaked to an adversary. We then study the resilience of the seminal Shamir's secret sharing scheme with statistical leakage, and bound certain measures of security (i.e., semantic security, mutual information security), given other parameters of the system including the amount of leakage from each secret share. We show that for an extreme scenario of Shamir's scheme, in particular when the underlying field characteristic is $2$, the security of each bit of the secret against leakage improves exponentially with the number of users. To the best of our knowledge, this is the first attempt towards understanding secret sharing under general statistical noisy leakage.