MDS Ideal Secret Sharing Scheme from AG-codes on Elliptic Curves

TL;DR

This paper explores the construction of MDS linear secret sharing schemes using algebraic-geometric codes on elliptic curves, demonstrating many are optimal in error correction and secret recovery.

Contribution

It introduces a new class of MDS secret sharing schemes based on AG-codes on elliptic curves, expanding the theoretical framework and practical options.

Findings

Many AG-code based schemes are MDS and optimal for secret sharing.

The schemes extend the known constructions from Shamir's scheme.

The paper provides conditions for MDS properties in elliptic curve codes.

Abstract

For a secret sharing scheme, two parameters $d_{min}$ and $d_{cheat}$ are defined in [12] and [13]. These two parameters measure the error-correcting capability and the secret-recovering capability of the secret sharing scheme against cheaters. Some general properties of the parameters have been studied in [12,[9] and [13]. The MDS secret-sharing scheme was defined in [12] and it is proved that MDS perfect secret sharing scheme can be constructed for any monotone access structure. The famous Shamir $(k,n)$ threshold secret sharing scheme is the MDS with $d_{min}=d_{cheat}=n-k+1$. In [3] we proposed the linear secret sharing scheme from algebraic-geometric codes. In this paper the linear secret sharing scheme from AG-codes on elliptic curves is studied and it is shown that many of them are MDS linear secret sharing scheme.