A Modern Day Approach to Combinatorial Secret Sharing/ Anandarup Roy

Thesis (Ph.D.) - Indian Statistical Institute, 2024

Includes bibliography

Introduction -- Mathematical Preliminaries -- IoT-Applicable Generalized Frameproof Combinatorial Designs -- Applications to IoT and Verifiability -- Access Structure Hiding Verifiable Tensor Designs -- A Secret Sharing Application on a Public Transport
Model

Guided by Prof. Bimal Kumar Roy & Prof. Mridul Nandi

In this thesis, we aim to develop generalised secret sharing protocols to enhance privacy, security and robustness in various applications. We begin by introducing various existing concepts related to secret sharing, including combinatorial repairable threshold schemes (RTSs), ramp schemes, balanced incomplete block designs (BIBDs), frameproofness, verifiability and hierarchy in the access structure. Our first work, motivated by the concepts of reparable threshold schemes by Stinson et al. develops extendable tensor designs built on balanced incomplete block designs. It then combines this construc- tion with the concepts of frameproofness by Desmedt et al. and consequently presents a frameproof version (which by definition, loses the property of share repairability). This results in a method of generalizing multiple BIBDs into a single, multi-level, ramp-type extendable secret sharing scheme, along with a discussion focusing on improvement of security, and reduction of share size as well as computation, particularly for application in IoT environments. A new graphical approach can be found in our paper that deals with the problem of secret and share reconstruction in the frameproof setup. Furthermore, a generalised combinatorial design resistant to framing has interesting implications in many areas of interest in distributed IoT devices. Vulnerabilities may arise in communication networks at various stages. For example, at the share distribution stage, anomalies may be introduced during data transfer from the dealer to some players. It is also possible that some (malicious) players try to frame others. Furthermore, there may occur false share contributions by some (malicious) players during the secret reconstruction stage. We present a novel approach to verify correct submission of shares by each participant during secret reconstruction through a lightweight cheater identification algorithm, which significantly improves the computational complexity of verification compared to existing algorithms. We move on to exploring ramp-type verifiable secret sharing schemes, and the application of hidden access structures in such cryptographic protocols. Inspired by Sehrawat et al.’s access structure hiding scheme, we develop an ϵ-almost access structure hiding scheme, which is verifiable as well as frameproof. We detail how the concept of ϵ-almost hiding is important for incorporating ramp schemes, thus making a fundamental generalisation of this concept. In particular, this proves that tensor designs are verifiable ramp-type secret sharing schemes. Finally, we explore hierarchy in access structures and formalize our ϵ-almost access structure hiding framework in the context of zero-knowledge proofs. We aim to achieve this by modelling a smart transportation system implemented through a new Hierarchical Secret Sharing (HSS) ramp scheme within this framework and instantiated with ASCON, a good lightweight verification authenticated encryption scheme.