1 Introduction

Part I: Synchronous Communication Setting

2 Preliminaries and Definitions

2.1 Definitions

2.2 Properties of Polynomials Over a Finite Field

3 Lower Bounds

4 Upper Bounds

4.1 VSS Schemes with \( n>3t \)

4.2 VSS Scheme with \( n>4t \)

4.3 VSS Scheme with a Single Round

Part III: Asynchronous Communication Setting

5 Preliminaries and Definitions

5.1 Asynchronous Tools

6 Perfectly-Secure AVSS Schemes

6.1 BCG - AVSS Scheme

6.2 PCR - AVSS Scheme

6.3 CHP - AVSS Scheme

Part III: Hybrid Communication Setting

7 Preliminaries and Definitions for Hybrid Communication Setting

8 Hybrid AVSS Protocol with \( t<n/3 \)

9 Open Problems