[Paper Review] Probabilistic Infinite Secret Sharing

László Csirmaz|arXiv (Cornell University)|Oct 1, 2013

Cryptography and Data Security6 references3 citations

TL;DR

This paper introduces a framework for probabilistic infinite secret sharing using arbitrary probability spaces and infinitely many participants, characterizing which access structures can be realized via measurable schemes. It shows that every access structure admits a non-measurable perfect scheme, while measurable perfect/almost perfect schemes exist iff the access structure is open in the Sierpiński topology—equivalently, realizable by a span program—while measurable ramp/almost ramp schemes exist iff the structure is a Gδ set, equivalent to realization via a Hilbert-space program.

ABSTRACT

The study of probabilistic secret sharing schemes using arbitrary probability spaces and possibly infinite number of participants lets us investigate abstract properties of such schemes. It highlights important properties, explains why certain definitions work better than others, connects this topic to other branches of mathematics, and might yield new design paradigms. A probabilistic secret sharing scheme is a joint probability distribution of the shares and the secret together with a collection of secret recovery functions for qualified subsets. The scheme is measurable if the recovery functions are measurable. Depending on how much information an unqualified subset might have, we define four scheme types: perfect, almost perfect, ramp, and almost ramp. Our main results characterize the access structures which can be realized by schemes of these types. We show that every access structure can be realized by a non-measurable perfect probabilistic scheme. The construction is based on a paradoxical pair of independent random variables which determine each other. For measurable schemes we have the following complete characterization. An access structure can be realized by a (measurable) perfect, or almost perfect scheme if and only if the access structure, as a subset of the Sierpiński space {0, 1} P, is open, if and only if it can be realized by a span program. The access structure can be realized by a (measurable) ramp or almost ramp scheme if and only if the access structure is a Gδ set (intersection of countably many open sets) in the Sierpiński topology, if and only if it can be realized by a Hilbert-space program.

Motivation & Objective

• To investigate the abstract properties of probabilistic secret sharing schemes with arbitrary probability spaces and infinitely many participants.
• To clarify why certain definitions of secret sharing (e.g., perfect, almost perfect) are more suitable than others in infinite settings.
• To establish connections between secret sharing and broader mathematical structures such as topology, measure theory, and functional analysis.
• To provide a complete characterization of which access structures can be realized by different types of measurable probabilistic secret sharing schemes.

Proposed method

• Models secret sharing as a joint probability distribution over shares and the secret, with recovery functions for qualified subsets.
• Defines four scheme types—perfect, almost perfect, ramp, almost ramp—based on the information leakage to unqualified subsets.
• Uses the Sierpiński space {0,1}^P to topologize access structures and analyze their properties via open sets and Gδ sets.
• Applies measure-theoretic concepts to define measurable schemes, ensuring recovery functions are measurable.
• Constructs a paradoxical pair of independent random variables that determine each other to realize non-measurable perfect schemes.
• Establishes equivalence between access structure types and program-theoretic models: span programs for perfect schemes and Hilbert-space programs for ramp schemes.

Experimental results

Research questions

• RQ1 Which access structures can be realized by non-measurable perfect probabilistic secret sharing schemes in infinite settings?
• RQ2 What topological conditions on access structures ensure realizability by measurable perfect or almost perfect schemes?
• RQ3 How do ramp and almost ramp schemes relate to the Borel hierarchy, particularly Gδ sets, in the Sierpiński topology?
• RQ4 Can measurable secret sharing schemes be characterized via algebraic or functional-analytic models such as span programs or Hilbert-space programs?
• RQ5 What role does the paradoxical independence of random variables play in constructing non-measurable secret sharing schemes?

Key findings

• Every access structure can be realized by a non-measurable perfect probabilistic secret sharing scheme, leveraging a paradoxical pair of independent random variables that mutually determine each other.
• A measurable perfect or almost perfect secret sharing scheme exists if and only if the access structure is an open set in the Sierpiński topology on {0,1}^P.
• An access structure admits a measurable perfect or almost perfect scheme if and only if it can be realized by a span program.
• A measurable ramp or almost ramp secret sharing scheme exists if and only if the access structure is a Gδ set (countable intersection of open sets) in the Sierpiński topology.
• Such schemes are equivalent to realization via a Hilbert-space program, establishing a deep link between topological structure and functional-analytic representation.
• The paper establishes a complete topological and program-theoretic classification of access structures across all four scheme types in the infinite, probabilistic setting.