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

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

Ryutaroh Matsumoto 1,2,,

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.

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