Hierarchical Threshold Secret Sharing

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


We consider the problem of threshold secret sharing in groups with hierarchical structure. In such settings, the secret is shared among a group of participants that is partitioned into levels. The access structure is then determined by a sequence of threshold requirements: a subset of participants is authorized if it has at least fco members from the highest level, as well as at least k1> k0 members from the two highest levels and so forth. Such problems may occur in settings where the participants differ in their authority or level of confidence and the presence of higher level participants is imperative to allow the recovery of the common secret. Even though secret sharing in hierarchical groups has been studied extensively in the past, none of the existing solutions addresses the simple setting where, say, a bank transfer should be signed by three employees, at least one of whom mustbe a department manager. We present a perfect secret sharing scheme for this problem that, unlike most secret sharing schemes that are suitable for hierarchical structures, is ideal. As in Shamir's scheme, the secret is represented as the free coefficient of some polynomial. The novelty of our scheme is the usage of polynomial derivatives in order to generate lesser shares for participants of lower levels. Consequently, our scheme uses Birkhoff interpolation, i.e., the construction of a polynomial according to an unstructured set of point and derivative values. A substantial part of our discussion is dedicated to the question of how to assign identities to the participants from the underlying finite field so that the resulting. Birkhoff interpolation problem will be well posed. In the course of this discussion, we borrow some results from the theory of Birkhoff interpolation over R and import them to the context of finite fields.

Original languageEnglish
Title of host publicationTheory of Cryptography (TCC 2004)
EditorsMoni Naor
Place of PublicationBerlin, Heidelberg
Number of pages18
ISBN (Electronic)9783540246381
ISBN (Print)9783540210009
StatePublished - 2004

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


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