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One-Shot Secret Sharing with Monotone Access Structures over Classical-Quantum Broadcast Channels

Truman Welling, Rémi A. Chou, Aylin Yener

Abstract

We consider a secret sharing setting with a monotone access structure involving a control node and $L$ users, connected via a classical-quantum broadcast channel whose input is controlled by the control node, referred to as the dealer. Unlike traditional secret sharing settings, where the dealer fully controls the shares given to each user, in our model, the dealer encodes the secret for transmission over the broadcast channel. This means that the shares received by users are perturbed by the channel and are not fully controlled by the dealer. Our main results are achievable one-shot secret sharing rates, as well as converse bounds for arbitrary monotone access structures. We further derive second-order and asymptotic achievable rates for arbitrary monotone access structures. In the special case where all shares are required to recover the secret, we show that our result coincides with the existing secret sharing capacity over classical channels.

One-Shot Secret Sharing with Monotone Access Structures over Classical-Quantum Broadcast Channels

Abstract

We consider a secret sharing setting with a monotone access structure involving a control node and users, connected via a classical-quantum broadcast channel whose input is controlled by the control node, referred to as the dealer. Unlike traditional secret sharing settings, where the dealer fully controls the shares given to each user, in our model, the dealer encodes the secret for transmission over the broadcast channel. This means that the shares received by users are perturbed by the channel and are not fully controlled by the dealer. Our main results are achievable one-shot secret sharing rates, as well as converse bounds for arbitrary monotone access structures. We further derive second-order and asymptotic achievable rates for arbitrary monotone access structures. In the special case where all shares are required to recover the secret, we show that our result coincides with the existing secret sharing capacity over classical channels.

Paper Structure

This paper contains 27 sections, 28 theorems, 96 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Problem Statement
  4. One-shot Regime
  5. Asymptotic Regime
  6. Results
  7. Supporting Results
  8. Hash Functions
  9. Source Coding with Compound Quantum Side Information
  10. Channel Coding from Source Coding with Compound Quantum Side Information
  11. Proof of Theorem \ref{['thm:general_one_shot_secret_sharing_ach']}
  12. Overview
  13. Coding Scheme
  14. Analysis
  15. Reliability
  16. ...and 12 more sections

Key Result

Theorem 1

For secret sharing over a classical-quantum broadcast channel $W$ among $L$ users with monotone access structure $\mathbb{A}$, there exists an $\epsilon$-good $2^R$-code satisfying where $\epsilon_1,\epsilon_2,\delta>0$, $\epsilon'=\epsilon_1(|\mathbb{A}|+1)+\epsilon_2$, and $\epsilon = |\mathbb{A}|(3\epsilon'+2^{-\delta/2-1}) + |\mathbb{B}|(4\epsilon' + 2^{-\delta/2})$, and $\blacktriangleleft$

Figures (1)

  • Figure 1: A secret $S$ is shared among three users over a classical-quantum broadcast channel $W$ with access structure $\mathbb{A}=\{\{1,2,3\},\{1,2\},\{2,3\}\}$. Only sets of users in the authorized set can recover the secret.

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Corollary 4: Classical Capacity
  • ...and 24 more