## Doctoral Theses

### Secret Sharing and Its Variants, Matroids,Combinatorics.

12-20-2020

12-20-2021

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Applied Statistics Unit (ASU-Kolkata)

#### Supervisor

Roy, Bimal Kumar (ASU-Kolkata; ISI)

#### Abstract (Summary of the Work)

The main focus of this thesis is secret sharing. Secret Sharing is a very basic and fundamental cryptographic primitive. It is a method to share a secret by a dealer among different parties in such a way that only certain predetermined subsets of parties can together reconstruct the secret while some of the remaining subsets of parties can have no information about the secret. Secret sharing was introduced independently by Shamir [139] and Blakely [20]. What they introduced is called a threshold secret sharing scheme. In such a secret sharing scheme the subsets of parties that can reconstruct a secret are all those subsets whose cardinality is greater than a predetermined threshold. In a latter work by Ito, Saito and Nishizeki [93], secret sharing schemes were constructed where the subsets of parties who can reconstruct the secret did not have any concrete mathematical description.Illustrative ExampleSecret sharing schemes mirror a real life scenario. Consider the following situation :â€¢ A wealthy man (the dealer) keeps his money in a locker.â€¢ He has four children (parties/participants) and gives them keys such that : Atleast three of them has to co-operate (bring their keys together) to open the locker.â€¢ None of the children can open the locker on their own.â€¢ Even if two of them bring their keys together, still they cannot open the locker.â€¢ The above mentioned condition is a description of a (3-out-of-4) threshold secret sharing scheme.Formal DefinitionBefore going into the work that has been done in this thesis, we take a look at the formal definition of secret sharing schemes. The definitions have been taken from the survey of Amos Beimel [15]. Definition 1. Access structure : - For a set of parties P = {p1, . . . , pn}, a collection of subsets A âŠ† 2 P is said to be monotone if, B âˆˆ A and B âŠ† C =â‡’ C âˆˆ A. An access structure A is a monotone collection of non-empty subsets of P. A set A âˆˆ A, A âŠ† P is called an authorized set and a set A /âˆˆ A, A âŠ† P is called an unauthorized set .Definition 2. Distribution Scheme :- Given a domain of secrets K, a set of random strings R and domains of shares K1, . . . , Kn, a distribution scheme is a pair Î£ = hÎ , Âµi where Âµ is a probability distribution on R and Î is a mapping Î : K Ã— R âˆ’â†’ K1 Ã— . . . Ã— Kn.

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