On asymptotically good ramp secret sharing schemes

TL;DR

This paper extends the analysis of asymptotically good linear ramp secret sharing schemes by examining partial information leakage and reconstruction, using generalized Hamming weights of algebraic geometric codes.

Contribution

It provides a comprehensive analysis of partial privacy and reconstruction parameters, enriching the understanding of access structures in ramp secret sharing schemes.

Findings

Characterization of partial information leakage

Analysis of partial reconstruction capabilities

Use of generalized Hamming weights for asymptotic behavior

Abstract

Asymptotically good sequences of linear ramp secret sharing schemes have been intensively studied by Cramer et al. in terms of sequences of pairs of nested algebraic geometric codes. In those works the focus is on full privacy and full reconstruction. In this paper we analyze additional parameters describing the asymptotic behavior of partial information leakage and possibly also partial reconstruction giving a more complete picture of the access structure for sequences of linear ramp secret sharing schemes. Our study involves a detailed treatment of the (relative) generalized Hamming weights of the considered codes.