Communication efficient and strongly secure secret sharing schemes based on algebraic geometry codes

TL;DR

This paper introduces a flexible framework for constructing communication-efficient, strongly secure secret sharing schemes using algebraic geometry codes, achieving near-optimal overheads and security for large lengths over fixed fields.

Contribution

It presents a general framework based on nested linear codes, enabling the first constructions with universal optimal communication and security properties using algebraic geometry codes.

Findings

Schemes with near-optimal communication overheads for large n and fixed fields.

First constructions with universal optimal communication and strong security.

Schemes with near-optimal security and communication overheads for large n.

Abstract

Secret sharing schemes with optimal and universal communication overheads have been obtained independently by Bitar et al. and Huang et al. However, their constructions require a finite field of size q > n, where n is the number of shares, and do not provide strong security. In this work, we give a general framework to construct communication efficient secret sharing schemes based on sequences of nested linear codes, which allows to use in particular algebraic geometry codes and allows to obtain strongly secure and communication efficient schemes. Using this framework, we obtain: 1) schemes with universal and close to optimal communication overheads for arbitrarily large lengths n and a fixed finite field, 2) the first construction of schemes with universal and optimal communication overheads and optimal strong security (for restricted lengths), having in particular the component-wise security advantages of perfect schemes and the security and storage efficiency of ramp schemes, and 3) schemes with universal and close to optimal communication overheads and close to optimal strong security defined for arbitrarily large lengths n and a fixed finite field.