Nearly optimal robust secret sharing

We prove that a known general approach to improve Shamir’s celebrated secret sharing scheme; i.e., adding an information-theoretic authentication tag to the secret, can make it robust for n parties against any collusion of size δ n , for any constant δ ∈ ( 0 , 1 / 2 ) . Shamir’s original scheme is r...

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Vydáno v:	Designs, codes, and cryptography Ročník 87; číslo 8; s. 1777 - 1796
Hlavní autor:	Cheraghchi, Mahdi
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 15.08.2019 Springer Nature B.V
Témata:	Algebra Algorithms Circuits Codes Coding and Information Theory Computer Science Cryptology Data Structures and Information Theory Decoding Discrete Mathematics in Computer Science Information and Communication Information theory Mathematical models Parameter modification Reed-Solomon codes Robustness Coding and information theory 68P30 Algebraic coding theory 94A60 11T71 Cryptography
ISSN:	0925-1022, 1573-7586
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Abstract	We prove that a known general approach to improve Shamir’s celebrated secret sharing scheme; i.e., adding an information-theoretic authentication tag to the secret, can make it robust for n parties against any collusion of size δ n , for any constant δ ∈ ( 0 , 1 / 2 ) . Shamir’s original scheme is robust for all δ ∈ ( 0 , 1 / 3 ) . Beyond that, we employ the best known list decoding algorithms for Reed-Solomon codes and show that, with high probability, only the correct secret maintains the correct information-theoretic tag if an algebraic manipulation detection (AMD) code is used to tag secrets. This result holds in the so-called “non-rushing” model in which the n shares are submitted simultaneously for reconstruction. We thus obtain a fully explicit and robust secret sharing scheme in this model that is essentially optimal in all parameters including the share size which is k ( 1 + o ( 1 ) ) + O ( κ ) , where k is the secret length and κ is the security parameter. Like Shamir’s scheme, in this modified scheme any set of more than δ n honest parties can efficiently recover the secret. Using algebraic geometry codes instead of Reed-Solomon codes, the share length can be decreased to a constant (only depending on δ ) while the number of shares n can grow independently. In this case, when n is large enough, the scheme satisfies the “threshold” requirement in an approximate sense; i.e., any set of δ n ( 1 + ρ ) honest parties, for arbitrarily small ρ > 0 , can efficiently reconstruct the secret. From a practical perspective, the main importance of our result is in showing that existing systems employing Shamir-type secret sharing schemes can be made much more robust than previously thought with minimal change, essentially only involving the addition of a short and simple checksum to the original data.
AbstractList	We prove that a known general approach to improve Shamir’s celebrated secret sharing scheme; i.e., adding an information-theoretic authentication tag to the secret, can make it robust for n parties against any collusion of size δ n , for any constant δ ∈ ( 0 , 1 / 2 ) . Shamir’s original scheme is robust for all δ ∈ ( 0 , 1 / 3 ) . Beyond that, we employ the best known list decoding algorithms for Reed-Solomon codes and show that, with high probability, only the correct secret maintains the correct information-theoretic tag if an algebraic manipulation detection (AMD) code is used to tag secrets. This result holds in the so-called “non-rushing” model in which the n shares are submitted simultaneously for reconstruction. We thus obtain a fully explicit and robust secret sharing scheme in this model that is essentially optimal in all parameters including the share size which is k ( 1 + o ( 1 ) ) + O ( κ ) , where k is the secret length and κ is the security parameter. Like Shamir’s scheme, in this modified scheme any set of more than δ n honest parties can efficiently recover the secret. Using algebraic geometry codes instead of Reed-Solomon codes, the share length can be decreased to a constant (only depending on δ ) while the number of shares n can grow independently. In this case, when n is large enough, the scheme satisfies the “threshold” requirement in an approximate sense; i.e., any set of δ n ( 1 + ρ ) honest parties, for arbitrarily small ρ > 0 , can efficiently reconstruct the secret. From a practical perspective, the main importance of our result is in showing that existing systems employing Shamir-type secret sharing schemes can be made much more robust than previously thought with minimal change, essentially only involving the addition of a short and simple checksum to the original data. We prove that a known general approach to improve Shamir’s celebrated secret sharing scheme; i.e., adding an information-theoretic authentication tag to the secret, can make it robust for n parties against any collusion of size δn, for any constant δ∈(0,1/2). Shamir’s original scheme is robust for all δ∈(0,1/3). Beyond that, we employ the best known list decoding algorithms for Reed-Solomon codes and show that, with high probability, only the correct secret maintains the correct information-theoretic tag if an algebraic manipulation detection (AMD) code is used to tag secrets. This result holds in the so-called “non-rushing” model in which the n shares are submitted simultaneously for reconstruction. We thus obtain a fully explicit and robust secret sharing scheme in this model that is essentially optimal in all parameters including the share size which is k(1+o(1))+O(κ), where k is the secret length and κ is the security parameter. Like Shamir’s scheme, in this modified scheme any set of more than δn honest parties can efficiently recover the secret. Using algebraic geometry codes instead of Reed-Solomon codes, the share length can be decreased to a constant (only depending on δ) while the number of shares n can grow independently. In this case, when n is large enough, the scheme satisfies the “threshold” requirement in an approximate sense; i.e., any set of δn(1+ρ) honest parties, for arbitrarily small ρ>0, can efficiently reconstruct the secret. From a practical perspective, the main importance of our result is in showing that existing systems employing Shamir-type secret sharing schemes can be made much more robust than previously thought with minimal change, essentially only involving the addition of a short and simple checksum to the original data.
Author	Cheraghchi, Mahdi
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References	GuruswamiVAlgorithmic results in list decodingFound. Trends Theor. Comput. Sci.200722107195245314710.1561/04000000071203.94140 Bishop A., Pastro V., Rajaraman R., Wichs D.: Essentially optimal robust secret sharing with maximal corruptions. In: Proceedings of the 35th Annual International Conference on the Theory and Applications of Cryptographic Techniques (EUROCRYPT 2016), pp. 58–86 (2016). Guruswami, V., Xing, C.: Optimal rate list decoding of folded algebraic-geometric codes over constant-sized alphabets. In: SODA, pp. 1858–1866 (2014). ShamirAHow to share a secretCommun. ACM1979221161261354925210.1145/359168.3591760414.94021 Cramer R., Damgård I., Fehr S.: On the cost of reconstructing a secret, or VSS with optimal reconstruction phase. In: Proceedings of Advances in Cryptology CRYPTO 2001, Lecture Notes in Computer Science, vol. 2139, pp. 503–523. Springer (2001). GuruswamiVRudraAExplicit codes achieving list decoding capacity: error-correction with optimal redundancyIEEE Trans. Inf. Theory2008541135150244674510.1109/TIT.2007.9112221205.94125 StinsonDRAn explication of secret sharing schemesDes. Codes Cryptogr.199224357390119477610.1007/BF001252030793.68111 Rabin T., Ben-Or M.: Verifiable secret sharing and multiparty protocols with honest majority. In: Proceedings of the Twenty-first Annual ACM Symposium on Theory of Computing (STOC ’89), pp. 73–85 (1989). Chen H., Cramer R.: Algebraic geometric secret sharing schemes and secure multi-party computations over small fields. In: Advances in Cryptology—CRYPTO 2006, Lecture Notes in Computer Science, vol. 4117, pp. 521–536. Springer (2006). Cramer R., Damgård I., Döttling N., Fehr S., Spini G.: Linear secret sharing schemes from error correcting codes and universal hash functions. In: Advances in Cryptology—EUROCRYPT 2015, Lecture Notes in Computer Science, vol. 9057, pp. 313–336. Springer (2015). 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References_xml	– reference: CramerRPadróCXingCOptimal Algebraic Manipulation Detection Codes in the Constant-Error Model2015BerlinSpringer4815011354.94056 – reference: Bishop A., Pastro V., Rajaraman R., Wichs D.: Essentially optimal robust secret sharing with maximal corruptions. In: Proceedings of the 35th Annual International Conference on the Theory and Applications of Cryptographic Techniques (EUROCRYPT 2016), pp. 58–86 (2016). – reference: Guruswami, V., Xing, C.: Optimal rate list decoding of folded algebraic-geometric codes over constant-sized alphabets. In: SODA, pp. 1858–1866 (2014). – reference: Bishop A., Pastro V.: Robust secret sharing schemes against local adversaries. In: Proceedings of Public-Key Cryptography (PKC), pp. 327–356 (2016). – reference: CramerRDamgårdINielsenJBSecure Multiparty Computation and Secret Sharing2015CambridgeCambridge University Press10.1017/CBO97811073377561322.68003 – reference: StichtenothHAlgebraic Function Fields and Codes20092BerlinSpringer1155.14022 – reference: Blakley G.R.: Safeguarding cryptographic keys. In: National Computer Conference, vol. 48, pp. 313–317. Springer (1979). – reference: GuruswamiVRudraAExplicit codes achieving list decoding capacity: error-correction with optimal redundancyIEEE Trans. Inf. Theory2008541135150244674510.1109/TIT.2007.9112221205.94125 – reference: Cabello S., Padró C., Sáez G.: Secret sharing schemes with detection of cheaters for a general access structure. In: Proceedings of Fundamentals of Computation Theory, Lecture Notes in Computer Science, vol. 1684, pp. 185–194. Springer (1999). – reference: WynerADThe wire-tap channelBell Syst. Tech. J.1975541355138740897910.1002/j.1538-7305.1975.tb02040.x0316.94017 – reference: Bellare M., Tessaro S., Vardy A.: Semantic security for the wiretap channel. In: Proceedings of Advances in Cryptology CRYPTO 2012, Lecture Notes in Computer Science, vol. 7417, pp. 294–311. Springer (2012). – reference: Cevallos A., Fehr S., Ostrovsky R., Rabani Y.: Unconditionally-secure robust secret sharing with compact shares. In: Proceedings of Advances in Cryptology EUROCRYPT 2012, Lecture Notes in Computer Science, vol. 7237, pp. 195–208. Springer (2012). – reference: Safavi-NainiRWangPA model for adversarial wiretap channels and its applicationsJ. Inf. Process.2015235554561 – reference: CsiszárIKörnerJBroadcast channels with confidential messagesIEEE Trans. Inf. Theory197824333934849364610.1109/TIT.1978.10558920382.94017 – reference: Rabin T., Ben-Or M.: Verifiable secret sharing and multiparty protocols with honest majority. In: Proceedings of the Twenty-first Annual ACM Symposium on Theory of Computing (STOC ’89), pp. 73–85 (1989). – reference: Cramer R., Dodis Y., Fehr S., Padró C., Wichs D.: Detection of algebraic manipulation with applications to robust secret sharing and fuzzy extractors. In: Advances in Cryptology - EUROCRYPT 2008, Lecture Notes in Computer Science, vol. 4965, pp. 471–488. Springer (2008). – reference: GuruswamiVAlgorithmic results in list decodingFound. Trends Theor. Comput. Sci.200722107195245314710.1561/04000000071203.94140 – reference: RothRMIntroduction to Coding Theory2006CambridgeCambridge University Press10.1017/CBO97805118089681092.94001 – reference: Leung-Yan-CheongSOn a special class of wiretap channels (corresp.)IEEE Trans. Inf. Theory197723562562752961710.1109/TIT.1977.10557630373.94018 – reference: StinsonDRAn explication of secret sharing schemesDes. Codes Cryptogr.199224357390119477610.1007/BF001252030793.68111 – reference: Chen H., Cramer R.: Algebraic geometric secret sharing schemes and secure multi-party computations over small fields. In: Advances in Cryptology—CRYPTO 2006, Lecture Notes in Computer Science, vol. 4117, pp. 521–536. Springer (2006). – reference: Ishai Y., Ostrovsky R., Seyalioglu H.: Identifying cheaters without an honest majority. In: Proceedings of Theory of Cryptography (TCC 2012), Lecture Notes in Computer Science, vol. 7194, pp. 21–38. Springer (2012). – reference: Cramer R., Damgård I., Döttling N., Fehr S., Spini G.: Linear secret sharing schemes from error correcting codes and universal hash functions. In: Advances in Cryptology—EUROCRYPT 2015, Lecture Notes in Computer Science, vol. 9057, pp. 313–336. Springer (2015). – reference: ShamirAHow to share a secretCommun. ACM1979221161261354925210.1145/359168.3591760414.94021 – reference: EliasPError-correcting codes for list decodingIEEE Trans. Inf. Theory1991371512108788110.1109/18.611230712.94021 – reference: Guruswami V., Smith A.: Codes for computationally simple channels: explicit constructions with optimal rate. In: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pp. 723–732 (2010). – reference: Cramer R., Damgård I., Fehr S.: On the cost of reconstructing a secret, or VSS with optimal reconstruction phase. In: Proceedings of Advances in Cryptology CRYPTO 2001, Lecture Notes in Computer Science, vol. 2139, pp. 503–523. Springer (2001). – ident: 578_CR16 doi: 10.1109/FOCS.2010.74 – volume-title: Introduction to Coding Theory year: 2006 ident: 578_CR22 doi: 10.1017/CBO9780511808968 – ident: 578_CR8 – volume: 2 start-page: 107 issue: 2 year: 2007 ident: 578_CR17 publication-title: Found. Trends Theor. Comput. Sci. doi: 10.1561/0400000007 – ident: 578_CR6 – ident: 578_CR18 doi: 10.1137/1.9781611973402.134 – ident: 578_CR21 doi: 10.1145/73007.73014 – ident: 578_CR4 – volume-title: Secure Multiparty Computation and Secret Sharing year: 2015 ident: 578_CR9 doi: 10.1017/CBO9781107337756 – volume: 22 start-page: 612 issue: 11 year: 1979 ident: 578_CR23 publication-title: Commun. ACM doi: 10.1145/359168.359176 – volume: 2 start-page: 357 issue: 4 year: 1992 ident: 578_CR25 publication-title: Des. Codes Cryptogr. doi: 10.1007/BF00125203 – volume: 24 start-page: 339 issue: 3 year: 1978 ident: 578_CR11 publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.1978.1055892 – volume: 54 start-page: 135 issue: 1 year: 2008 ident: 578_CR15 publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.2007.911222 – volume: 54 start-page: 1355 year: 1975 ident: 578_CR27 publication-title: Bell Syst. Tech. J. doi: 10.1002/j.1538-7305.1975.tb02040.x – ident: 578_CR10 – start-page: 481 volume-title: Optimal Algebraic Manipulation Detection Codes in the Constant-Error Model year: 2015 ident: 578_CR13 – ident: 578_CR12 – ident: 578_CR19 – ident: 578_CR2 doi: 10.1007/978-3-662-49387-8_13 – volume: 23 start-page: 554 issue: 5 year: 2015 ident: 578_CR24 publication-title: J. Inf. Process. – ident: 578_CR7 – ident: 578_CR5 – volume: 23 start-page: 625 issue: 5 year: 1977 ident: 578_CR20 publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.1977.1055763 – volume: 37 start-page: 5 issue: 1 year: 1991 ident: 578_CR14 publication-title: IEEE Trans. Inf. Theory doi: 10.1109/18.61123 – ident: 578_CR1 – ident: 578_CR3 doi: 10.1007/978-3-662-49890-3_3 – volume-title: Algebraic Function Fields and Codes year: 2009 ident: 578_CR26 doi: 10.1007/978-3-540-76878-4
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Snippet	We prove that a known general approach to improve Shamir’s celebrated secret sharing scheme; i.e., adding an information-theoretic authentication tag to the...
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SubjectTerms	Algebra Algorithms Circuits Codes Coding and Information Theory Computer Science Cryptology Data Structures and Information Theory Decoding Discrete Mathematics in Computer Science Information and Communication Information theory Mathematical models Parameter modification Reed-Solomon codes Robustness
Title	Nearly optimal robust secret sharing
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