Strongly secure quantum ramp secret sharing constructed from algebraic curves over finite fields

Previous Article
AMC Home
This Issue
Next Article
The secrecy capacity of the arbitrarily varying wiretap channel under list decoding

February 2019, 13(1): 1-10. doi: 10.3934/amc.2019001

Strongly secure quantum ramp secret sharing constructed from algebraic curves over finite fields

1.	Department of Information and Communication Engineering, Nagoya University, Nagoya, Aichi 464-8603, Japan
2.	Department of Mathematical Sciences, Aalborg University, Denmark

* Corresponding author

Received March 2015 Published December 2018

Fund Project: This research is partly supported by the National Institute of Information and Communications Technology, Japan, and by the Japan Society for the Promotion of Science Grant Nos. 23246071 and 26289116, and the Villum Foundation through their VELUX Visiting Professor Programme 2013-2014

The first construction of strongly secure quantum ramp secret sharing by Zhang and Matsumoto had an undesirable feature that the dimension of quantum shares must be larger than the number of shares. By using algebraic curves over finite fields, we propose a new construction in which the number of shares can become arbitrarily large for fixed dimension of shares.

Keywords: Algebraic curve, quantum secret sharing, non-perfect secret sharing, ramp secret sharing.

Mathematics Subject Classification: Primary: 81P94; Secondary: 94A62, 94B27.

Citation: Ryutaroh Matsumoto. Strongly secure quantum ramp secret sharing constructed from algebraic curves over finite fields. Advances in Mathematics of Communications, 2019, 13 (1) : 1-10. doi: 10.3934/amc.2019001

References:

[1]	G. R. Blakley and C. Meadows, Security of ramp schemes, in Advances in Cryptology-CRYPTO'84, vol. 196 of Lecture Notes in Computer Science, Springer-Verlag, 1985, 242-269. doi: 10.1007/3-540-39568-7_20.
[2]	A. Bogdanov, S. Guo and I. Komargodski, Threshold secret sharing requires a linear size alphabet, in Theory of Cryptography (eds. M. Hirt and A. Smith), Springer Berlin Heidelberg, Berlin, Heidelberg, 9986 (2016), 471-484. doi: 10.1007/978-3-662-53644-5_18.
[3]	I. Cascudo, R. Cramer and C. Xing, Bounds on the threshold gap in secret sharing and its applications, IEEE Trans. Inform. Theory, 59 (2013), 5600-5612. doi: 10.1109/TIT.2013.2264504.
[4]	I. Cascudo, J. Skovsted Gundersen and D. Ruano, Improved bounds on the threashold gap in ramp secret sharing, 2018, Cryptology ePrint Archive 2018/099.
[5]	H. Chen and R. Cramer, Algebraic geometric secret sharing schemes and secure multi-party computations over small fields, in Advances in Cryptology - CRYPT 2006 (ed. C. Dwork), vol. 4117 of Lecture Notes in Computer Science, Springer-Verlag, 2006, 521-536. doi: 10.1007/11818175_31.
[6]	H. Chen, R. Cramer, R. de Haan and I. Cascudo Pueyo, Strongly multiplicative ramp schemes from high degree rational points on curves, in Advances in Cryptology - EUROCRYPT 2008 (ed. N. Smart), vol. 4965 of Lecture Notes in Computer Science, Springer-Verlag, 2008, 451-470 doi: 10.1007/978-3-540-78967-3_26.
[7]	R. Cleve, D. Gottesman and H.-K. Lo, How to share a quantum secret, Phys. Rev. Lett., 83 (1999), 648-651. doi: 10.1103/PhysRevLett.83.648.
[8]	D. Gottesman, Theory of quantum secret sharing, Phys. Rev. A, 61 (2000), 042311. doi: 10.1103/PhysRevA.61.042311.
[9]	M. Iwamoto and H. Yamamoto, Strongly secure ramp secret sharing schemes for general access structures, Inform. Process. Lett., 97 (2006), 52-57. doi: 10.1016/j.ipl.2005.09.012.
[10]	R. Matsumoto, Coding theoretic construction of quantum ramp secret sharing, IEICE Trans. Fundamentals, E101-A (2018), 1215-1222. doi: 10.1587/transfun.E101.A.1215.
[11]	R. Matsumoto, Strong security of the strongly multiplicative ramp secret sharing based on algebraic curves, IEICE Trans. Fundamentals, E98-A (2015), 1576-1578. doi: 10.1587/transfun.E98.A.1576.
[12]	R. J. McEliece and D. V. Sarwate, On sharing secrets and Reed-Solomon codes, Comm. ACM, 24 (1981), 583-584. doi: 10.1145/358746.358762.
[13]	T. Ogawa, A. Sasaki, M. Iwamoto and H Yamamoto, Quantum secret sharing schemes and reversibility of quantum operations, Phys. Rev. A, 72 (2005), 032318. doi: 10.1103/PhysRevA.72.032318.
[14]	A. Shamir, How to share a secret, Comm. ACM, 22 (1979), 612-613. doi: 10.1145/359168.359176.
[15]	A. D. Smith, Quantum secret sharing for general access structures, 2000, arXiv:quant-ph/0001087,
[16]	H. Stichtenoth, Algebraic Function Fields and Codes, vol. 254 of Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, Berlin Heidelberg, 2009.
[17]	D. R. Stinson, Cryptography Theory and Practice, 3rd edition, Chapman & Hall/CRC, 2006.
[18]	H. Yamamoto, Secret sharing system using (k; l; n) threshold scheme, Electronics and Communications in Japan (Part I: Communications), 69 (1986), 46-54, (the original Japanese version published in 1985) doi: 10.1002/ecja.4410690906.
[19]	P. Zhang and R. Matsumoto, Quantum strongly secure ramp secret sharing, Quantum Information Processing, 14 (2015), 715-729. doi: 10.1007/s11128-014-0863-2.

show all references

References:

[1]	G. R. Blakley and C. Meadows, Security of ramp schemes, in Advances in Cryptology-CRYPTO'84, vol. 196 of Lecture Notes in Computer Science, Springer-Verlag, 1985, 242-269. doi: 10.1007/3-540-39568-7_20.
[2]	A. Bogdanov, S. Guo and I. Komargodski, Threshold secret sharing requires a linear size alphabet, in Theory of Cryptography (eds. M. Hirt and A. Smith), Springer Berlin Heidelberg, Berlin, Heidelberg, 9986 (2016), 471-484. doi: 10.1007/978-3-662-53644-5_18.
[3]	I. Cascudo, R. Cramer and C. Xing, Bounds on the threshold gap in secret sharing and its applications, IEEE Trans. Inform. Theory, 59 (2013), 5600-5612. doi: 10.1109/TIT.2013.2264504.
[4]	I. Cascudo, J. Skovsted Gundersen and D. Ruano, Improved bounds on the threashold gap in ramp secret sharing, 2018, Cryptology ePrint Archive 2018/099.
[5]	H. Chen and R. Cramer, Algebraic geometric secret sharing schemes and secure multi-party computations over small fields, in Advances in Cryptology - CRYPT 2006 (ed. C. Dwork), vol. 4117 of Lecture Notes in Computer Science, Springer-Verlag, 2006, 521-536. doi: 10.1007/11818175_31.
[6]	H. Chen, R. Cramer, R. de Haan and I. Cascudo Pueyo, Strongly multiplicative ramp schemes from high degree rational points on curves, in Advances in Cryptology - EUROCRYPT 2008 (ed. N. Smart), vol. 4965 of Lecture Notes in Computer Science, Springer-Verlag, 2008, 451-470 doi: 10.1007/978-3-540-78967-3_26.
[7]	R. Cleve, D. Gottesman and H.-K. Lo, How to share a quantum secret, Phys. Rev. Lett., 83 (1999), 648-651. doi: 10.1103/PhysRevLett.83.648.
[8]	D. Gottesman, Theory of quantum secret sharing, Phys. Rev. A, 61 (2000), 042311. doi: 10.1103/PhysRevA.61.042311.
[9]	M. Iwamoto and H. Yamamoto, Strongly secure ramp secret sharing schemes for general access structures, Inform. Process. Lett., 97 (2006), 52-57. doi: 10.1016/j.ipl.2005.09.012.
[10]	R. Matsumoto, Coding theoretic construction of quantum ramp secret sharing, IEICE Trans. Fundamentals, E101-A (2018), 1215-1222. doi: 10.1587/transfun.E101.A.1215.
[11]	R. Matsumoto, Strong security of the strongly multiplicative ramp secret sharing based on algebraic curves, IEICE Trans. Fundamentals, E98-A (2015), 1576-1578. doi: 10.1587/transfun.E98.A.1576.
[12]	R. J. McEliece and D. V. Sarwate, On sharing secrets and Reed-Solomon codes, Comm. ACM, 24 (1981), 583-584. doi: 10.1145/358746.358762.
[13]	T. Ogawa, A. Sasaki, M. Iwamoto and H Yamamoto, Quantum secret sharing schemes and reversibility of quantum operations, Phys. Rev. A, 72 (2005), 032318. doi: 10.1103/PhysRevA.72.032318.
[14]	A. Shamir, How to share a secret, Comm. ACM, 22 (1979), 612-613. doi: 10.1145/359168.359176.
[15]	A. D. Smith, Quantum secret sharing for general access structures, 2000, arXiv:quant-ph/0001087,
[16]	H. Stichtenoth, Algebraic Function Fields and Codes, vol. 254 of Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, Berlin Heidelberg, 2009.
[17]	D. R. Stinson, Cryptography Theory and Practice, 3rd edition, Chapman & Hall/CRC, 2006.
[18]	H. Yamamoto, Secret sharing system using (k; l; n) threshold scheme, Electronics and Communications in Japan (Part I: Communications), 69 (1986), 46-54, (the original Japanese version published in 1985) doi: 10.1002/ecja.4410690906.
[19]	P. Zhang and R. Matsumoto, Quantum strongly secure ramp secret sharing, Quantum Information Processing, 14 (2015), 715-729. doi: 10.1007/s11128-014-0863-2.

[1]	Bagher Bagherpour, Shahrooz Janbaz, Ali Zaghian. Optimal information ratio of secret sharing schemes on Dutch windmill graphs. Advances in Mathematics of Communications, 2019, 13 (1) : 89-99. doi: 10.3934/amc.2019005
[2]	Stefka Bouyuklieva, Zlatko Varbanov. Some connections between self-dual codes, combinatorial designs and secret sharing schemes. Advances in Mathematics of Communications, 2011, 5 (2) : 191-198. doi: 10.3934/amc.2011.5.191
[3]	Adriana Navarro-Ramos, William Olvera-Lopez. A solution for discrete cost sharing problems with non rival consumption. Journal of Dynamics & Games, 2018, 5 (1) : 31-39. doi: 10.3934/jdg.2018004
[4]	Juliang Zhang, Jian Chen. Information sharing in a make-to-stock supply chain. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1169-1189. doi: 10.3934/jimo.2014.10.1169
[5]	Dan Mangoubi. A gradient estimate for harmonic functions sharing the same zeros. Electronic Research Announcements, 2014, 21: 62-71. doi: 10.3934/era.2014.21.62
[6]	Jong Soo Kim, Won Chan Jeong. A model for buyer and supplier coordination and information sharing in order-up-to systems. Journal of Industrial & Management Optimization, 2012, 8 (4) : 987-1015. doi: 10.3934/jimo.2012.8.987
[7]	João Correia-da-Silva, Joana Pinho. The profit-sharing rule that maximizes sustainability of cartel agreements. Journal of Dynamics & Games, 2016, 3 (2) : 143-151. doi: 10.3934/jdg.2016007
[8]	Daniele Bartoli, Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco. A 3-cycle construction of complete arcs sharing $(q+3)/2$ points with a conic. Advances in Mathematics of Communications, 2013, 7 (3) : 319-334. doi: 10.3934/amc.2013.7.319
[9]	Shiyong Li, Wei Sun, Quan-Lin Li. Utility maximization for bandwidth allocation in peer-to-peer file-sharing networks. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2018194
[10]	Xue-Yan Wu, Zhi-Ping Fan, Bing-Bing Cao. Cost-sharing strategy for carbon emission reduction and sales effort: A nash game with government subsidy. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-29. doi: 10.3934/jimo.2019040
[11]	Rich Stankewitz, Toshiyuki Sugawa, Hiroki Sumi. Hereditarily non uniformly perfect sets. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 2391-2402. doi: 10.3934/dcdss.2019150
[12]	Feng Rong. Non-algebraic attractors on $\mathbf{P}^k$. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 977-989. doi: 10.3934/dcds.2012.32.977
[13]	Jianjun Paul Tian. Algebraic model of non-Mendelian inheritance. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1577-1586. doi: 10.3934/dcdss.2011.4.1577
[14]	Olof Heden, Denis S. Krotov. On the structure of non-full-rank perfect $q$-ary codes. Advances in Mathematics of Communications, 2011, 5 (2) : 149-156. doi: 10.3934/amc.2011.5.149
[15]	Isaac A. García, Jaume Giné. Non-algebraic invariant curves for polynomial planar vector fields. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 755-768. doi: 10.3934/dcds.2004.10.755
[16]	Antonio Algaba, Natalia Fuentes, Cristóbal García, Manuel Reyes. Non-formally integrable centers admitting an algebraic inverse integrating factor. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 967-988. doi: 10.3934/dcds.2018041
[17]	Peter Markowich, Jesús Sierra. Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. Kinetic & Related Models, 2019, 12 (2) : 347-356. doi: 10.3934/krm.2019015
[18]	Denis S. Krotov, Patric R. J. Östergård, Olli Pottonen. Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code. Advances in Mathematics of Communications, 2016, 10 (2) : 393-399. doi: 10.3934/amc.2016013
[19]	Daniele Bartoli, Leo Storme. On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric. Advances in Mathematics of Communications, 2014, 8 (3) : 271-280. doi: 10.3934/amc.2014.8.271
[20]	Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu. On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3479-3520. doi: 10.3934/dcds.2019144

American Institute of Mathematical Sciences