{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/124114"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/124114","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Breaking barriers in secret sharing","abstract":"In this thesis, we study secret sharing schemes for general (non-threshold) access functions. In a secret sharing scheme for n parties associated to a monotone function [mathematical formula], a dealer distributes shares of a secret among n parties. Any subset of parties [mathematical formula] can jointly reconstruct the secret if F(T) = 1, and should have no information about the secret if F(T) = 0. One of the major long-standing questions in information-theoretic cryptography is to determine the minimum size of the shares in a secret-sharing scheme for an access function F. There is an exponential gap between lower and upper bounds for share size: the best known scheme for general monotone functions has shares of size 2[superscript n-o(n)], while the best lower bound is n² / log(n). In this thesis, we improve this more-than-30-year-old upper bound by construct- ing a secret sharing scheme for any access function with shares of size 2[superscript 0.994n] and a linear secret sharing scheme for any access function with shares of size 2[superscript 0.994n]. As a contribution of independent interest, we also construct a secret sharing scheme with shares of size [mathematical formula] for a family of [mathematical formula] monotone access functions, out of a total of [mathematical formula] of them. As an intermediate result, we construct the first conditional disclosure of secrets (CDS) with sub-exponential communication complexity. CDS is a variant of secret sharing, in which a group of parties want to disclose a secret to a referee the parties' respective inputs satisfy some predicate. The key conceptual contribution of this thesis is a novel connection between secret sharing and CDS, and the notion of (2-server, information-theoretic) private information retrieval.","abstract_html":"In this thesis, we study secret sharing schemes for general (non-threshold) access functions. In a secret sharing scheme for n parties associated to a monotone function [mathematical formula], a dealer distributes shares of a secret among n parties. Any subset of parties [mathematical formula] can jointly reconstruct the secret if F(T) = 1, and should have no information about the secret if F(T) = 0. One of the major long-standing questions in information-theoretic cryptography is to determine the minimum size of the shares in a secret-sharing scheme for an access function F. There is an exponential gap between lower and upper bounds for share size: the best known scheme for general monotone functions has shares of size 2[superscript n-o(n)], while the best lower bound is n² / log(n). In this thesis, we improve this more-than-30-year-old upper bound by construct- ing a secret sharing scheme for any access function with shares of size 2[superscript 0.994n] and a linear secret sharing scheme for any access function with shares of size 2[superscript 0.994n]. As a contribution of independent interest, we also construct a secret sharing scheme with shares of size [mathematical formula] for a family of [mathematical formula] monotone access functions, out of a total of [mathematical formula] of them. As an intermediate result, we construct the first conditional disclosure of secrets (CDS) with sub-exponential communication complexity. CDS is a variant of secret sharing, in which a group of parties want to disclose a secret to a referee the parties' respective inputs satisfy some predicate. The key conceptual contribution of this thesis is a novel connection between secret sharing and CDS, and the notion of (2-server, information-theoretic) private information retrieval.","abstract_has_math":false,"creators":["Liu, Tianren,Ph. D.Massachusetts Institute of Technology."],"institution":"Massachusetts Institute of Technology","degree_name":"Doctoral","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. 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They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1721.1/124114","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Vinod Vaikuntanathan."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science","EECS"]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science."]},{"key":"dc:creator","label":"Author","values":["Liu, Tianren,Ph. 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D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2019","Cataloged from student-submitted PDF version of thesis.","Includes bibliographical references (pages 50-53)."]},{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis, we study secret sharing schemes for general (non-threshold) access functions. In a secret sharing scheme for n parties associated to a monotone function [mathematical formula], a dealer distributes shares of a secret among n parties. Any subset of parties [mathematical formula] can jointly reconstruct the secret if F(T) = 1, and should have no information about the secret if F(T) = 0. One of the major long-standing questions in information-theoretic cryptography is to determine the minimum size of the shares in a secret-sharing scheme for an access function F. There is an exponential gap between lower and upper bounds for share size: the best known scheme for general monotone functions has shares of size 2[superscript n-o(n)], while the best lower bound is n² / log(n). In this thesis, we improve this more-than-30-year-old upper bound by construct- ing a secret sharing scheme for any access function with shares of size 2[superscript 0.994n] and a linear secret sharing scheme for any access function with shares of size 2[superscript 0.994n]. As a contribution of independent interest, we also construct a secret sharing scheme with shares of size [mathematical formula] for a family of [mathematical formula] monotone access functions, out of a total of [mathematical formula] of them. As an intermediate result, we construct the first conditional disclosure of secrets (CDS) with sub-exponential communication complexity. CDS is a variant of secret sharing, in which a group of parties want to disclose a secret to a referee the parties' respective inputs satisfy some predicate. The key conceptual contribution of this thesis is a novel connection between secret sharing and CDS, and the notion of (2-server, information-theoretic) private information retrieval."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:title","label":"Title","values":["Breaking barriers in secret sharing"]}]}],"canonical_facts":{"dc:contributor.advisor":["Vinod Vaikuntanathan."],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science","EECS"],"dc:contributor.other":["Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science."],"dc:creator":["Liu, Tianren,Ph. D.Massachusetts Institute of Technology."],"dc:date.accessioned":["2020-03-09T18:58:37Z"],"dc:date.available":["2020-03-09T18:58:37Z"],"dc:date.issued":["2019"],"dc:description":["This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.","Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2019","Cataloged from student-submitted PDF version of thesis.","Includes bibliographical references (pages 50-53)."],"dc:description.abstract":["In this thesis, we study secret sharing schemes for general (non-threshold) access functions. In a secret sharing scheme for n parties associated to a monotone function [mathematical formula], a dealer distributes shares of a secret among n parties. Any subset of parties [mathematical formula] can jointly reconstruct the secret if F(T) = 1, and should have no information about the secret if F(T) = 0. One of the major long-standing questions in information-theoretic cryptography is to determine the minimum size of the shares in a secret-sharing scheme for an access function F. There is an exponential gap between lower and upper bounds for share size: the best known scheme for general monotone functions has shares of size 2[superscript n-o(n)], while the best lower bound is n² / log(n). In this thesis, we improve this more-than-30-year-old upper bound by construct- ing a secret sharing scheme for any access function with shares of size 2[superscript 0.994n] and a linear secret sharing scheme for any access function with shares of size 2[superscript 0.994n]. As a contribution of independent interest, we also construct a secret sharing scheme with shares of size [mathematical formula] for a family of [mathematical formula] monotone access functions, out of a total of [mathematical formula] of them. As an intermediate result, we construct the first conditional disclosure of secrets (CDS) with sub-exponential communication complexity. CDS is a variant of secret sharing, in which a group of parties want to disclose a secret to a referee the parties' respective inputs satisfy some predicate. The key conceptual contribution of this thesis is a novel connection between secret sharing and CDS, and the notion of (2-server, information-theoretic) private information retrieval."],"dc:description.degree":["Ph. D."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/124114"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Electrical Engineering and Computer Science."],"dc:title":["Breaking barriers in secret sharing"],"dc:type":["Thesis"],"thesis:degree_name":["Doctoral"]},"updated_at":"2026-07-22T22:21:45Z"}