Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Utkarsh Gupta, Hessam Mahdavifar

TL;DR

This work addresses leakage resilience in secret sharing by introducing a statistical leakage model where honest parties leak through independent wiretap channels. It analyzes Shamir's secret sharing under this model, deriving bounds on mutual-information security and related metrics. In the general-threshold case, the leakage is bounded by the sum of per-share leakages; in the extreme $(N,N)$ scenario over fields of characteristic $2$, per-bit leakage decays exponentially with the number of users, indicating strong information-theoretic resilience. The results provide a principled framework for understanding secret sharing in wireless and noisy environments and point to direction for generalizing the theory to other groups and leakage models with practical implications for secure distributed systems.

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.

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

TL;DR

This work addresses leakage resilience in secret sharing by introducing a statistical leakage model where honest parties leak through independent wiretap channels. It analyzes Shamir's secret sharing under this model, deriving bounds on mutual-information security and related metrics. In the general-threshold case, the leakage is bounded by the sum of per-share leakages; in the extreme scenario over fields of characteristic , per-bit leakage decays exponentially with the number of users, indicating strong information-theoretic resilience. The results provide a principled framework for understanding secret sharing in wireless and noisy environments and point to direction for generalizing the theory to other groups and leakage models with practical implications for secure distributed systems.

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 , 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.
Paper Structure (12 sections, 12 theorems, 22 equations, 1 figure)

This paper contains 12 sections, 12 theorems, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Shamir's Secret Sharing Scheme
  4. The Wiretap Channel and Security Metrics
  5. Statistical Leakage Model
  6. System Model
  7. Related Prior Work
  8. Bounds on Securi ty
  9. Shamir's Scheme with General Threshold
  10. The Extreme-Threshold Scenario
  11. Security Analysis
  12. Conclusion

Key Result

Theorem 1

For the wiretap channel, semantic security $\eta_{\text{SS}}$, distinguishing security $\eta_{\text{DS}}$, and mutual-information security $\eta_{\text{MIS}}$ satisfy the following inequality

Figures (1)

  • Figure 1: Secret sharing with statistical leakage

Theorems & Definitions (24)

  • Remark 1: guruswami2016repairing
  • Definition 1: Mutual information security, bellare2012semantic
  • Theorem 1: bellare2012semantic, Theorem 1,5
  • Lemma 2
  • proof
  • Corollary 3
  • Proposition 4
  • proof
  • Corollary 5
  • Remark 2
  • ...and 14 more