Classification of Ideal Secret Sharing Schemes

Secret sharing schemes were introduced independently by Shamir and Blakley in 1979 and are now widely used in many cryptographic protocols. They are ideal for storing information that is highly sensitive and highly important. In a secret sharing scheme, each user is given a piece of information called a “share” so that only authorized coalitions can recover the secret when combining their shares. The collection of authorized coalitions is called an access structure. One of the central open problems in secret sharing is finding a characterization of ideal secret sharing schemes. Brickell and Davenport revealed an important connection between ideal secret sharing schemes and matroids. Therefore matroid theory is a natural tool to understand the structure of ideal secret sharing schemes. This thesis contributes to the characterization of ideal secret sharing schemes and the forensic aspects of secret sharing. We focus on ideal hierarchical secret sharing schemes and find a matroidal characterization of ideal hierarchical access structures. In particular, we show that every ideal hierarchical access structure corresponds to a lattice path matroid. We also give a matroidal characterization for different classes of ideal access structures, including conjunctive hierarchical access structures, disjunctive access structures, ideal weighted access structures, and ideal hierarchical roughly weighted access structures. We explore how the seniority of users influences their ability to frame other users. The possibility that the users can be framed is a price we have to pay for using ideal secret sharing schemes, and we find a (non-ideal) frameproof secret sharing scheme for every access structure.

Name thesis:degree_name

PhD

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor dc:publisher

ResearchSpace@Auckland

Year dc:date.issued

2023

Author dc:creator

• Mo, Songbao

Advisors dc:contributor.advisor

• Slinko, Arkadii
• Verret, Gabriel

Statement dc:rights

• Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated.

Licence dc:rights.uri

• https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm

Handle dc:identifier.uri

https://hdl.handle.net/2292/65526

OAI identifier oai:identifier

oai:researchspace.auckland.ac.nz:2292/65526