Extending Brickell–Davenport theorem to non-perfect secret sharing schemes
One important result in secret sharing is the Brickell–Davenport theorem: every ideal perfect secret sharing scheme defines a matroid that is uniquely determined by the access structure. We present a generalization of the Brickell–Davenport theorem to the general case, in which non-perfect schemes a...
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| Vydané v: | Designs, codes, and cryptography Ročník 74; číslo 2; s. 495 - 510 |
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| Jazyk: | English |
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01.02.2015
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| Abstract | One important result in secret sharing is the Brickell–Davenport theorem: every ideal perfect secret sharing scheme defines a matroid that is uniquely determined by the access structure. We present a generalization of the Brickell–Davenport theorem to the general case, in which non-perfect schemes are also considered. After analyzing that result under a new point of view and identifying its combinatorial nature, we present a characterization of the (not necessarily perfect) secret sharing schemes that are associated with matroids. Some optimality properties of such schemes are discussed. |
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| AbstractList | One important result in secret sharing is the Brickell–Davenport theorem: every ideal perfect secret sharing scheme defines a matroid that is uniquely determined by the access structure. We present a generalization of the Brickell–Davenport theorem to the general case, in which non-perfect schemes are also considered. After analyzing that result under a new point of view and identifying its combinatorial nature, we present a characterization of the (not necessarily perfect) secret sharing schemes that are associated with matroids. Some optimality properties of such schemes are discussed. |
| Author | Farràs, Oriol Padró, Carles |
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| References | Massey J.L.: Minimal codewords and secret sharing. In: Proceedings of the 6th joint Swedish–Russian workshop on information theory, Molle, Sweden, August 1993, pp. 269–279 (1993). SeymourP.D.On secret-sharing matroidsJ. Comb. Theory B1992566973 Cover T.M., Thomas J.A.: Elements of information theory. Wiley, New York (1991). Ogata W., Kurosawa K., Tsujii S.: Nonperfect secret sharing schemes. In: Advances in Cryptology, Auscrypt 92. Lecture Notes in Computer Science, vol. 718, pp. 56–66 (1993). BrickellE.F.Some ideal secret sharing schemesJ. Comb. Math. Comb. Comput.19899105113 MatúšF.Two constructions on limits of entropy functionsIEEE Trans. Inf. Theory200753320330 StinsonD.R.An explication of secret sharing schemesDes. Codes Cryptogr.19922357390 CsirmazL.The size of a share must be largeJ. Cryptol.199710223231 Martí-FarréJ.PadróC.On secret sharing schemes, matroids and polymatroidsJ. Math. Cryptol.2010495120 SeymourP.D.A forbidden minor characterization of matroid portsQ. J. Math. Oxf. Ser.197627407413 SimonisJ.AshikhminA.Almost affine codesDes. Codes Cryptogr.199814179197 Blakley G.R., Meadows C.: Security of ramp schemes. In: Advances in Cryptology, Crypto 84. Lecture Notes in Computer Science, vol. 196, pp. 242–268 (1985). BeimelA.OrlovI.Secret sharing and non-Shannon information inequalitiesIEEE Trans. Inf. Theory20115756345649 Martin K.M.: Discrete structures in the theory of secret sharing. Ph.D. Thesis, University of London (1991). Farràs O., Padró C., Xing C., Yang A.: Natural generalizations of threshold secret sharing. In: Advances in Cryptology, Asiacrypt 2011. Lecture Notes in Computer Science, vol. 7073, pp. 610–627 (2011). ShamirA.How to share a secretCommun. ACM197922612613 FarràsO.PadróC.Ideal hierarchical secret sharing schemesIEEE Trans. Inf. Theory20125832733286 Welsh D.J.A.: Matroid theory. Academic Press, London (1976). BrickellE.F.DavenportD.M.On the classification of ideal secret sharing schemesJ. Cryptol.19914123134 Oxley J.G.: Matroid theory. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1992). FarràsO.Martí-FarréJ.PadróC.Ideal multipartite secret sharing schemesJ. Cryptol.201225434463 MatúšF.Matroid representations by partitionsDiscret. Math.1999203169194 Paillier P.: On ideal non-perfect secret sharing schemes. In: Security protocols, 5th international workshop. Lecture Notes in Computer Science, vol. 1361, pp. 207–216 (1998). Kurosawa K., Okada K., Sakano K., Ogata W., Tsujii S.: Nonperfect secret sharing schemes and matroids. In: Advances in Cryptology, EUROCRYPT 1993. Lecture Notes in Computer Science, vol. 765, pp. 126–141 (1993). FujishigeS.Polymatroidal dependence structure of a set of random variablesInf. Control1978395572 KarninE.D.GreeneJ.W.HellmanM.E.On secret sharing systemsIEEE Trans. Inf. Theory1983293541 PadróC.VázquezL.YangA.Finding lower bounds on the complexity of secret sharing schemes by linear programmingDiscret. Appl. Math.201316110721084 BeimelA.WeinrebE.Separating the power of monotone span programs over different fieldsSIAM J. Comput.20053411961215 LehmanA.A solution of the Shannon switching gameJ. Soc. Ind. Appl. Math.196412687725 Beimel A.: Secret-sharing schemes: a survey. In: Coding and Cryptology. Third International Workshop, IWCC. Lecture Notes in Computer Science, vol. 6639, pp. 11–46 (2011). Schrijver A.: Combinatorial optimization. Polyhedra and efficiency. Springer, Berlin (2003). LehmanA.Matroids and portsNotices Am. Math. Soc.197612356360 Beimel A., Livne N., Padró C.: Matroids can be far from ideal secret sharing. In: Fifth theory of cryptography conference, TCC 2008. Lecture Notes in Computer Science, vol. 4948, pp. 194–212 (2008). Cramer R., Daza V., Gracia I., Jiménez Urroz J., Leander G., Martí-Farré J., Padró C.: On codes, matroids and secure multi-party computation from linear secret sharing schemes. IEEE Trans. Inf. Theory 54, 2644–2657 (2008). Dougherty R., Freiling C., Zeger K.: Linear rank inequalities on five or more variables. SIAM J. Discret. Math. (2009). arXiv:0910.0284v3. 9858_CR28 9858_CR29 9858_CR26 9858_CR27 9858_CR2 9858_CR3 9858_CR1 9858_CR6 9858_CR20 9858_CR7 9858_CR21 9858_CR4 9858_CR5 9858_CR24 9858_CR25 9858_CR8 9858_CR22 9858_CR9 9858_CR23 9858_CR17 9858_CR18 9858_CR15 9858_CR16 9858_CR19 9858_CR31 9858_CR10 9858_CR32 9858_CR30 9858_CR13 9858_CR35 9858_CR14 9858_CR11 9858_CR33 9858_CR12 9858_CR34 |
| References_xml | – reference: SeymourP.D.A forbidden minor characterization of matroid portsQ. J. Math. Oxf. Ser.197627407413 – reference: BeimelA.WeinrebE.Separating the power of monotone span programs over different fieldsSIAM J. Comput.20053411961215 – reference: Martin K.M.: Discrete structures in the theory of secret sharing. Ph.D. Thesis, University of London (1991). – reference: FarràsO.Martí-FarréJ.PadróC.Ideal multipartite secret sharing schemesJ. Cryptol.201225434463 – reference: Cramer R., Daza V., Gracia I., Jiménez Urroz J., Leander G., Martí-Farré J., Padró C.: On codes, matroids and secure multi-party computation from linear secret sharing schemes. IEEE Trans. Inf. Theory 54, 2644–2657 (2008). – reference: Oxley J.G.: Matroid theory. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1992). – reference: Beimel A.: Secret-sharing schemes: a survey. In: Coding and Cryptology. Third International Workshop, IWCC. Lecture Notes in Computer Science, vol. 6639, pp. 11–46 (2011). – reference: SimonisJ.AshikhminA.Almost affine codesDes. Codes Cryptogr.199814179197 – reference: LehmanA.A solution of the Shannon switching gameJ. Soc. Ind. Appl. Math.196412687725 – reference: PadróC.VázquezL.YangA.Finding lower bounds on the complexity of secret sharing schemes by linear programmingDiscret. Appl. Math.201316110721084 – reference: Cover T.M., Thomas J.A.: Elements of information theory. Wiley, New York (1991). – reference: MatúšF.Two constructions on limits of entropy functionsIEEE Trans. Inf. Theory200753320330 – reference: StinsonD.R.An explication of secret sharing schemesDes. Codes Cryptogr.19922357390 – reference: SeymourP.D.On secret-sharing matroidsJ. Comb. Theory B1992566973 – reference: FarràsO.PadróC.Ideal hierarchical secret sharing schemesIEEE Trans. Inf. Theory20125832733286 – reference: BrickellE.F.Some ideal secret sharing schemesJ. Comb. Math. Comb. Comput.19899105113 – reference: Paillier P.: On ideal non-perfect secret sharing schemes. In: Security protocols, 5th international workshop. Lecture Notes in Computer Science, vol. 1361, pp. 207–216 (1998). – reference: Welsh D.J.A.: Matroid theory. Academic Press, London (1976). – reference: Beimel A., Livne N., Padró C.: Matroids can be far from ideal secret sharing. In: Fifth theory of cryptography conference, TCC 2008. Lecture Notes in Computer Science, vol. 4948, pp. 194–212 (2008). – reference: Farràs O., Padró C., Xing C., Yang A.: Natural generalizations of threshold secret sharing. In: Advances in Cryptology, Asiacrypt 2011. Lecture Notes in Computer Science, vol. 7073, pp. 610–627 (2011). – reference: LehmanA.Matroids and portsNotices Am. Math. Soc.197612356360 – reference: ShamirA.How to share a secretCommun. ACM197922612613 – reference: KarninE.D.GreeneJ.W.HellmanM.E.On secret sharing systemsIEEE Trans. Inf. Theory1983293541 – reference: Dougherty R., Freiling C., Zeger K.: Linear rank inequalities on five or more variables. SIAM J. Discret. Math. (2009). arXiv:0910.0284v3. – reference: FujishigeS.Polymatroidal dependence structure of a set of random variablesInf. Control1978395572 – reference: BeimelA.OrlovI.Secret sharing and non-Shannon information inequalitiesIEEE Trans. Inf. Theory20115756345649 – reference: Ogata W., Kurosawa K., Tsujii S.: Nonperfect secret sharing schemes. In: Advances in Cryptology, Auscrypt 92. Lecture Notes in Computer Science, vol. 718, pp. 56–66 (1993). – reference: Schrijver A.: Combinatorial optimization. Polyhedra and efficiency. Springer, Berlin (2003). – reference: Massey J.L.: Minimal codewords and secret sharing. In: Proceedings of the 6th joint Swedish–Russian workshop on information theory, Molle, Sweden, August 1993, pp. 269–279 (1993). – reference: Martí-FarréJ.PadróC.On secret sharing schemes, matroids and polymatroidsJ. Math. Cryptol.2010495120 – reference: Blakley G.R., Meadows C.: Security of ramp schemes. In: Advances in Cryptology, Crypto 84. Lecture Notes in Computer Science, vol. 196, pp. 242–268 (1985). – reference: Kurosawa K., Okada K., Sakano K., Ogata W., Tsujii S.: Nonperfect secret sharing schemes and matroids. In: Advances in Cryptology, EUROCRYPT 1993. Lecture Notes in Computer Science, vol. 765, pp. 126–141 (1993). – reference: BrickellE.F.DavenportD.M.On the classification of ideal secret sharing schemesJ. Cryptol.19914123134 – reference: CsirmazL.The size of a share must be largeJ. Cryptol.199710223231 – reference: MatúšF.Matroid representations by partitionsDiscret. Math.1999203169194 – ident: 9858_CR1 doi: 10.1007/978-3-642-20901-7_2 – ident: 9858_CR30 – ident: 9858_CR11 – ident: 9858_CR13 – ident: 9858_CR8 doi: 10.1002/0471200611 – ident: 9858_CR7 doi: 10.1007/BF00196772 – ident: 9858_CR4 – ident: 9858_CR17 – ident: 9858_CR19 – ident: 9858_CR2 – ident: 9858_CR6 – ident: 9858_CR21 – ident: 9858_CR15 doi: 10.1016/S0019-9958(78)91063-X – ident: 9858_CR33 doi: 10.1023/A:1008244215660 – ident: 9858_CR25 – ident: 9858_CR27 – ident: 9858_CR28 doi: 10.1007/BFb0028171 – ident: 9858_CR12 doi: 10.1007/s00145-011-9101-6 – ident: 9858_CR31 – ident: 9858_CR29 – ident: 9858_CR14 – ident: 9858_CR10 doi: 10.1007/s001459900029 – ident: 9858_CR35 – ident: 9858_CR18 doi: 10.1137/0112059 – ident: 9858_CR23 doi: 10.1016/S0012-365X(99)00004-7 – ident: 9858_CR16 – ident: 9858_CR22 – ident: 9858_CR3 – ident: 9858_CR32 doi: 10.1145/359168.359176 – ident: 9858_CR9 – ident: 9858_CR20 – ident: 9858_CR24 – ident: 9858_CR34 doi: 10.1007/BF00125203 – ident: 9858_CR5 doi: 10.1007/3-540-39568-7_20 – ident: 9858_CR26 |
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| SubjectTerms | Circuits Coding and Information Theory Computer Science Cryptology Data Structures and Information Theory Discrete Mathematics in Computer Science Information and Communication |
| Title | Extending Brickell–Davenport theorem to non-perfect secret sharing schemes |
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