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.

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.  Google Scholar

[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.  Google Scholar

[3]

I. CascudoR. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. CleveD. Gottesman and H.-K. Lo, How to share a quantum secret, Phys. Rev. Lett., 83 (1999), 648-651.  doi: 10.1103/PhysRevLett.83.648.  Google Scholar

[8]

D. Gottesman, Theory of quantum secret sharing, Phys. Rev. A, 61 (2000), 042311.  doi: 10.1103/PhysRevA.61.042311.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

T. OgawaA. SasakiM. 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.  Google Scholar

[14]

A. Shamir, How to share a secret, Comm. ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.  Google Scholar

[15]

A. D. Smith, Quantum secret sharing for general access structures, 2000, arXiv:quant-ph/0001087, Google Scholar

[16]

H. Stichtenoth, Algebraic Function Fields and Codes, vol. 254 of Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, Berlin Heidelberg, 2009.  Google Scholar

[17] D. R. Stinson, Cryptography Theory and Practice, 3rd edition, Chapman & Hall/CRC, 2006.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

I. CascudoR. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. CleveD. Gottesman and H.-K. Lo, How to share a quantum secret, Phys. Rev. Lett., 83 (1999), 648-651.  doi: 10.1103/PhysRevLett.83.648.  Google Scholar

[8]

D. Gottesman, Theory of quantum secret sharing, Phys. Rev. A, 61 (2000), 042311.  doi: 10.1103/PhysRevA.61.042311.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

T. OgawaA. SasakiM. 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.  Google Scholar

[14]

A. Shamir, How to share a secret, Comm. ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.  Google Scholar

[15]

A. D. Smith, Quantum secret sharing for general access structures, 2000, arXiv:quant-ph/0001087, Google Scholar

[16]

H. Stichtenoth, Algebraic Function Fields and Codes, vol. 254 of Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, Berlin Heidelberg, 2009.  Google Scholar

[17] D. R. Stinson, Cryptography Theory and Practice, 3rd edition, Chapman & Hall/CRC, 2006.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[1]

Sushil Kumar Dey, Bibhas C. Giri. Coordination of a sustainable reverse supply chain with revenue sharing contract. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020165

[2]

Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307

[3]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[4]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[5]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[6]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020449

[7]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[8]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018

[9]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[10]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[11]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[12]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[13]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[14]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[15]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[16]

Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (251)
  • HTML views (453)
  • Cited by (1)

Other articles
by authors

[Back to Top]