Cryptanalysis of an asymmetric cipher protocol using a matrix decomposition problem

一个基于矩阵分解的非对称密码协议的分析

Abstract

Advances in quantum computation threaten to break public key cryptosystems such as RSA, ECC, and ElGamal that are based on the difficulty of factorization or taking a discrete logarithm, although up to now, no quantum algorithms have been found that are able to solve certain mathematical problems on noncommutative algebraic structures. Against this background, Raulynaitis et al. have proposed a novel asymmetric cipher protocol using a matrix decomposition problem. Their proposed scheme is vulnerable to a linear algebra attack based on the probable occurrence of weak keys in the generation process. In this paper, we show that the asymmetric cipher of the non-commutative cryptography scheme is vulnerable to a linear algebra attack and that it only requires polynomial time to obtain the equivalent keys for some given public keys. We also propose an improvement to enhance the scheme of Raulynaitis et al.

摘要

创新点

量子计算技术的发展对基于大整数因子分解,离散对数等问题具有交换代数结构的密码体制 (如 RSA,ECC 和 ElGamal 密码)构成威胁, 因此研究具有非交换代数结构的密码体制是一项富有挑战性的课题.针对该课题, Raulynaitis 等人基于矩阵分解构造了一个非对称密码协议. 本文对基于有限域上的非对称密码协议,提出了一种结构攻击方法并且给出了对应的算法描述和有效性分析.通过分析可知, 该结构攻击算法能够在多项式计算复杂度内从相关的公钥获得等价密钥. 最后,本文在给出攻击算法的基础上对该非对称密码协议给出一个修正方案.

This is a preview of subscription content, access via your institution.

References

  1. 1

    Cao Z. New Directions of Modern Cryptography. Boca Raton: CRC Press, 2012. 10–255

    Google Scholar 

  2. 2

    Peikert C. Lattice cryptography for the internet. In: Mosca M, ed. Post-Quantum Cryptography. Waterloo: Springer, 2014. 197–219

    Google Scholar 

  3. 3

    Shi J J, Shi R H, Guo Y, et al. Batch proxy quantum blind signature scheme. Sci China Inf Sci, 2013, 56: 052115

    MathSciNet  Google Scholar 

  4. 4

    Song F. A note on quantum security for post-quantum cryptography. In: Mosca M, ed. Post-Quantum Cryptography. Waterloo: Springer, 2014. 246–265

    Google Scholar 

  5. 5

    Tsaban B. Polynomial-time solutions of computational problems in noncommutative-algebraic cryptography. J Cryptol, 2013, 28: 601–622

    MathSciNet  Article  MATH  Google Scholar 

  6. 6

    Zhang H G, Liu J H, Jia J W, et al. A survey on applications of matrix decomposition in cryptography. J Cryptol Res, 2014, 1: 341–357

    Google Scholar 

  7. 7

    Mao S W, Zhang H G, Wu W Q, et al. A resistant quantum key exchange protocol and its corresponding encryption scheme. China Commun, 2014, 11: 131–141

    Article  Google Scholar 

  8. 8

    Wang H Z, Zhang H G, Wang Z Y, et al. Extended multivariate public key cryptosystems with secure encryption function. Sci China Inf Sci, 2011, 54: 1161–1171

    MathSciNet  Article  MATH  Google Scholar 

  9. 9

    Ling S, Phan D H, Stehle D, et al. Hardness of k-LWE and applications in traitor tracing. In: Proceedings of Advances in Cryptology-CRYPTO. Berlin: Springer, 2014. 315–334

    Google Scholar 

  10. 10

    Regev O. On lattices, learning with errors, random linear codes, and cryptography. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing. New York: ACM Press, 2005. 84–93

    Google Scholar 

  11. 11

    Braun J, Buchmann J, Mullan C, et al. Long term confidentiality: a survey. Design Code Cryptogr, 2014, 71: 459–478

    MathSciNet  Article  MATH  Google Scholar 

  12. 12

    Wang S B, Zhu Y, Ma D, et al. Lattice-based key exchange on small integer solution problem. Sci China Inf Sci, 2014, 57: 112111

  13. 13

    Albrecht M R, Faugere J C, Fitzpatrick R, et al. Practical cryptanalysis of a public-key encryption scheme based on new multivariate quadratic assumptions. In: Proceedings of Public Key Cryptography-PKC. Berlin: Springer, 2014. 446–464

    Google Scholar 

  14. 14

    Raulynaitis A, Sakalauskas E, Japertas S. Security analysis of asymmetric cipher protocol based on matrix decomposition problem. Informatica, 2010, 21: 215–228

    MathSciNet  MATH  Google Scholar 

  15. 15

    Raulynaitis A, Japertas S. Asymmetric cipher protocol using decomposition problem. In: Proceedings of Information Research and Applications, Varna, 2008. 107–111

    Google Scholar 

  16. 16

    Gashkov S B, Sergeev I S. Complexity of computation in finite fields. J Math Sci, 2013, 191: 661–685

    MathSciNet  Article  MATH  Google Scholar 

  17. 17

    Gu L, Zheng S. Conjugacy systems based on nonabelian factorization problems and their applications in cryptography. J Appl Math, 2014, 52: 1–9

    MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Huanguo Zhang.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, J., Zhang, H., Jia, J. et al. Cryptanalysis of an asymmetric cipher protocol using a matrix decomposition problem. Sci. China Inf. Sci. 59, 052109 (2016). https://doi.org/10.1007/s11432-015-5443-2

Download citation

Keywords

  • cryptography
  • post-quantum computational cryptography
  • asymmetric cipher protocol
  • cryptanalysis
  • matrix decomposition

关键词

  • 密码学
  • 抗量子计算密码学
  • 非对称密码协议
  • 密码分析
  • 矩阵分解