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Secret Sharing on Superconcentrator

Yuan Li

TL;DR

It is proved that any unrestricted arithmetic circuit computing the shares of a threshold secret sharing scheme must satisfy superconcentrator-like connectivity properties, and that any graph satisfying these properties can be transformed into a linear arithmetic circuit computing the shares of a threshold secret sharing scheme, assuming a sufficiently large field.

Abstract

We study the arithmetic circuit complexity of threshold secret sharing schemes by characterizing the graph-theoretic properties of arithmetic circuits that compute the shares. Using information inequalities, we prove that any unrestricted arithmetic circuit (with arbitrary gates and unbounded fan-in) computing the shares must satisfy superconcentrator-like connectivity properties. Specifically, when the inputs consist of the secret and $t-1$ random elements, and the outputs are the $n$ shares of a $(t, n)$-threshold secret sharing scheme, the circuit graph must be a $(t, n)$-concentrator; moreover, after removing the secret input, the remaining graph is a $(t-1, n)$-concentrator. Conversely, we show that any graph satisfying these properties can be transformed into a linear arithmetic circuit computing the shares of a threshold secret sharing scheme, assuming a sufficiently large field. As a consequence, we derive upper and lower bounds on the arithmetic circuit complexity of computing the shares in threshold secret sharing schemes.

Secret Sharing on Superconcentrator

TL;DR

It is proved that any unrestricted arithmetic circuit computing the shares of a threshold secret sharing scheme must satisfy superconcentrator-like connectivity properties, and that any graph satisfying these properties can be transformed into a linear arithmetic circuit computing the shares of a threshold secret sharing scheme, assuming a sufficiently large field.

Abstract

We study the arithmetic circuit complexity of threshold secret sharing schemes by characterizing the graph-theoretic properties of arithmetic circuits that compute the shares. Using information inequalities, we prove that any unrestricted arithmetic circuit (with arbitrary gates and unbounded fan-in) computing the shares must satisfy superconcentrator-like connectivity properties. Specifically, when the inputs consist of the secret and random elements, and the outputs are the shares of a -threshold secret sharing scheme, the circuit graph must be a -concentrator; moreover, after removing the secret input, the remaining graph is a -concentrator. Conversely, we show that any graph satisfying these properties can be transformed into a linear arithmetic circuit computing the shares of a threshold secret sharing scheme, assuming a sufficiently large field. As a consequence, we derive upper and lower bounds on the arithmetic circuit complexity of computing the shares in threshold secret sharing schemes.
Paper Structure (19 sections, 27 theorems, 68 equations, 5 figures, 1 table)

This paper contains 19 sections, 27 theorems, 68 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our results
  3. Preliminaries
  4. Entropy and information inequalities
  5. Secret sharing scheme
  6. Arithmetic circuit model
  7. Inverse Ackermann-type function
  8. Superconcentrators and concentrators
  9. Characterization via graph-theoretic properties
  10. Necessity
  11. Sufficiency
  12. Consequences
  13. Size lower bounds
  14. Size upper bounds
  15. Depth 2
  16. ...and 4 more sections

Key Result

Theorem 1

An $(t, n)$-concentrator, where $t \le n$, is a directed acyclic graph with $t$ inputs and $n$ outputs in which every set of $t$ outputs is connected to distinct $t$ inputs by vertex-disjoint paths. Let $C$ be an unrestricted arithmetic circuit (arbitrary gates, unbounded fan-in) computing the $n$ s

Figures (5)

  • Figure 1: unrestricted arithmetic circuit computing $n$ shares
  • Figure 2: linear arithmetic circuit realizing SS distribution
  • Figure 3: Construction of depth-2 superconcentrator
  • Figure 4: Construction of linear-size depth-2 superconcentrator
  • Figure 5: Construction of linear-size depth-3 superconcentrator

Theorems & Definitions (54)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1
  • Proposition 1
  • proof
  • Definition 2
  • Definition 3
  • Proposition 2
  • ...and 44 more