Cryptanalysis of an ElGamal-Like Cryptosystem Based on Matrices Over Group Rings

  • Jianwei Jia
  • Houzhen Wang
  • Huanguo Zhang
  • Shijia Wang
  • Jinhui LiuEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 960)


ElGamal cryptography is one of the most important Public Key Cryptography (PKC) since Diffie-Hellman exchanges was proposed, however these PKCs which are based on the hard problems that discrete logarithm problem and integer factorization problem are weak with advances in quantum computers. So some alternatives should be proposed. Majid Khan et al. proposed two ElGamal-like public-key encryption schemes based on large abelian subgroup of general linear group over a residue ring, however the two schemes were not long before it was proved unsafe by us. Then, Saba Inam and Rashid (2016) proposed an improved cryptosystem which can resist my attack on “NEURAL COMPUTING & APPLICATIONS”. By analyzing the security of the public key cryptography, we propose an improved method of algebraic key-recovery attack in the polynomial computational complexity despiteing the designers’ claim the cryptosystem is optimal security. Besides, we provide corresponding practical attack example to illustrate the attack method in our cryptanalysis, which breaks instances claiming 80 bits of security less than one minute on a single desktop computer.


Cryptography Post-quantum computational cryptography Cryptanalysis Conjugator search problem Computational complexity 



The author would like to thank the anonymous reviewers for their constructive comments and suggestions. This work was supported by National Key R&D Program of China (2017YFB0802000), National Natural Science Foundation of China (61772326, 61572303, 61872229, 61802239), NSFC Research Fund for International Young Scientists (61750110528), National Cryptography Development Fund during the 13th Five-year Plan Period (MMJJ20170216, MMJJ201701304), Foundation of State Key Laboratory of Information Security (2017-MS-03), Fundamental Research Funds for the Central Universities (GK201702004, GK201803061) and China Postdoctoral Science Foundation (2018M631121).


  1. 1.
    Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Trans. Inf. Theor. 22(6), 644–654 (1976)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Annual ACM Symposium on Theory of Computing, pp. 212–219 (1996)Google Scholar
  3. 3.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM review 41(2), 303–332 (1999)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Zhang, H.G., Han, W.B., Lai, X.J., et al.: Survey on cyberspace security. Sci. China Inf. Sci. 58(11), 1–43 (2015)MathSciNetGoogle Scholar
  5. 5.
    Buchmann, J.A., Butin, D., Göpfert, F., Petzoldt, A.: Post-quantum cryptography: state of the art. IEEE Security & Privacy 15(4), 12–13 (2017)CrossRefGoogle Scholar
  6. 6.
    Bernstein, D.J., Lange, T.: Post-quantum cryptography. Nature 549(7671), 188 (2017)CrossRefGoogle Scholar
  7. 7.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. J. ACM (JACM) 56(6), 34 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ding, J., Petzoldt, A.: Current state of multivariate cryptography. IEEE Secur. Priv. 15(4), 28–36 (2017)CrossRefGoogle Scholar
  9. 9.
    Sendrier, N.: Code-based cryptography: state of the art and perspectives. IEEE Secur. Priv. 15(4), 44–50 (2017)CrossRefGoogle Scholar
  10. 10.
    Wu, W., Zhang, H.G., Wang, H.Z., et al.: A public key cryptosystem based on data complexity under quantum environment. Sci. China Inf. Sci. 58(11), 1–11 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Anshel, I., Anshel, M., Goldfeld, D.: An algebraic method for public-key cryptography. Math. Res. Lett. 6(3), 287–292 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dehornoy, P.: Braid-based cryptography. Contemp. Math. 7, 5–33 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Myasnikov, A.G., Shpilrain, V.: Group theory, statistics, and cryptography, vol. 360 (2004)Google Scholar
  14. 14.
    Hurley, B., Hurley, T.: Group ring cryptography. Mathematics 69(1), 67–86 (2012)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings of the 35th Annual Symposium on Foundations of Computer Science (FOCS 1994), pp. 124–134. IEEE (1994)Google Scholar
  16. 16.
    McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. Deep space network progress report 42-44 (1978).
  17. 17.
    Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. J. ACM 60, 43:1–43:35 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Patarin, J.: Hidden fields equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms. In: Maurer, U. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 33–48. Springer, Heidelberg (1996). Scholar
  19. 19.
    Petzoldt, A., Chen, M.S., Yang, B.Y., Tao, C., Ding, J.: Design principles for HFEv-based multivariate signature schemes. In: Iwata, T., Cheon, J. (eds.) ASIACRYPT 2015. LNCS, vol. 9452, pp. 311–334. Springer, Heidelberg (2015). Scholar
  20. 20.
    Bernstein, D.J., et al.: SPHINCS: practical stateless hash-based signatures. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 368–397. Springer, Heidelberg (2015). Scholar
  21. 21.
    Alkim, E., Ducas, L., Pöppelmann, T., Schwabe, P.: Post-quantum key exchange a new hope. In: Holz, T., Savage, S. (eds.) 25th USENIX Security Symposium, USENIX Security 2016, pp. 327–343. USENIX Association (2016)Google Scholar
  22. 22.
    PQCRYPTO Project: Initial recommendations of long-term secure post-quantum systems (2015).
  23. 23.
    Braithwaite, M.: Experimenting with post-quantum cryptography. Google Security Blog (2016).
  24. 24.
    NIST Information Technology Laboratory: Secure Hash Standard (SHS). Federal Information Processing Standards Publication 180-4. NIST (2012).
  25. 25.
    Bernstein, D.J., Lange, T.: Post-quantum cryptography. Nature 549(14), 188–195 (2018)Google Scholar
  26. 26.
    Jia, J., Liu, J., Wu, S., et al.: Break R.S. Bhalerao’s public key encryption scheme. J. Wuhan Univ. 62(5), 425–428 (2016)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Wu, W.Q., Zhang, H.G., Wang, H.Z., et al.: A public key cryptosystem based on data complexity under quantum environment. Sci. China Inf. Sci. 58(11), 110102 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Liu, J., Fan, A., Jia, J., et al.: Cryptanalysis of public key cryptosystems based on non-abelian factorization problems. Tsinghua Sci. Technol. 21(3), 344–351 (2016)CrossRefGoogle Scholar
  29. 29.
    Mao, S., Zhang, H., Wu, W., et al.: A resistant quantum key exchange protocol and its corresponding encryption scheme. China Commun. 11(9), 124–134 (2014)CrossRefGoogle Scholar
  30. 30.
    Liu, J., Zhang, H., Jia, J.: A linear algebra attack on the non-commuting cryptography class based on matrix power function. In: Chen, K., Lin, D., Yung, M. (eds.) Inscrypt 2016. LNCS, vol. 10143, pp. 343–354. Springer, Cham (2017). Scholar
  31. 31.
    Liu, J., Zhang, H., Jia, J.: Cryptanalysis of schemes based on polynomial symmetrical decomposition. Chin. J. Electron. 26(6), 1139–1146 (2017)CrossRefGoogle Scholar
  32. 32.
    Liu, J., Jia, J., Zhang, H., et al.: Cryptanalysis of a cryptosystem with non-commutative platform groups. China Commun. 15(2), 67–73 (2018)CrossRefGoogle Scholar
  33. 33.
    Jia, J., Liu, J., Zhang, H.: Cryptanalysis of a key exchange protocol based on commuting matrices. Chin. J. Electron. 26(5), 947–951 (2017)CrossRefGoogle Scholar
  34. 34.
    Liu, J., Zhang, H., Jia, J., et al.: Cryptanalysis of an asymmetric cipher protocol using a matrix decomposition problem. Sci. China Inf. Sci. 59(5), 1–11 (2016)MathSciNetGoogle Scholar
  35. 35.
    Mao, S., Zhang, H., Wanqing, W.U., et al.: Key exchange protocol based on tensor decomposition problem. China Commun. 13(3), 174–183 (2016)CrossRefGoogle Scholar
  36. 36.
    Habeeb, M., Kahrobaei, D., Koupparis, C., Shpilrain, V.: Public key exchange using semidirect product of (semi)groups. In: Jacobson, M., Locasto, M., Mohassel, P., Safavi-Naini, R. (eds.) ACNS 2013. LNCS, vol. 7954, pp. 475–486. Springer, Heidelberg (2013). Scholar
  37. 37.
    Kahrobaei, D., Koupparis, C., Shpilrain, V.: A CCA secure cryptosystem using matrices over group rings. (preprint)
  38. 38.
    Kahrobaei, D., Koupparis, C., Shpilrain, V.: Public key exchange using matrices over group rings. Groups Complex. Cryptol. 5, 97–115 (2013)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Miasnikov, A.G., Shpilrain, V., Ushakov, A.: Non-commutative cryptography and complexity of group-theoretic problems. In: Mathematical Surveys and Monographs. AMS (2011)Google Scholar
  40. 40.
    Myasnikov, A.D., Ushakov, A.: Quantum algorithm for discrete logarithm problem for matrices over finite group rings. (preprint)
  41. 41.
    Shpilrain, V., Zapata, G.: Combinatorial group theory and public key cryptography. Appl. Algebra Eng. Commun. Comput. 17(3–4), 291–302 (2006)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Jia, J., Liu, J., Zhang, H.: Cryptanalysis of cryptosystems based on general linear group. China Commun. 13(6), 217–224 (2016)CrossRefGoogle Scholar
  43. 43.
    Inam, S., Ali, R.: A new ElGamal-like cryptosystem based on matrices over group ring. Neural Comput. Appl. 29(11), 1279–1283 (2018)CrossRefGoogle Scholar
  44. 44.
    Roseblade, J.E.: The algebraic structure of group rings. Bull. Lond. Math. Soc. 11, 1–100 (2011)MathSciNetGoogle Scholar
  45. 45.
    Kusmus, O., Hanoymak, T.: On construction of cryptographic systems over units of group rings. Electron. J. Pure and Appl. Math. 9(1), 37–43 (2015)MathSciNetGoogle Scholar
  46. 46.
    Gu, L., Zheng, S.: Conjugacy systems based on nonabelian factorization problems and their applications in cryptography. J. Appl. Math. 2014(2), 1–10 (2014)MathSciNetGoogle Scholar
  47. 47.
    Khan, M., Shah, T.: A novel cryptosystem based on general linear group. 3D Res. 6(1), 1–8 (2015)CrossRefGoogle Scholar
  48. 48.
    Storjohann, A., Mulders, T.: Fast algorithms for linear algebra modulo N. In: Bilardi, G., Italiano, G.F., Pietracaprina, A., Pucci, G. (eds.) ESA 1998. LNCS, vol. 1461, pp. 139–150. Springer, Heidelberg (1998). Scholar
  49. 49.
    Gashkov, S.B., Sergeev, I.S.: Complexity of computation in finite fields. J. Math. Sci. 191(5), 661–685 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Jianwei Jia
    • 1
  • Houzhen Wang
    • 2
    • 3
  • Huanguo Zhang
    • 2
    • 3
  • Shijia Wang
    • 4
  • Jinhui Liu
    • 5
    Email author
  1. 1.Huawei Technologies Co., Ltd.Xi’anChina
  2. 2.School of Cyber Science and EngineeringWuhanChina
  3. 3.Key Laboratory of Aerospace Information Security and Trusted Computing Ministry of EducationWuhanChina
  4. 4.Department of Statistics and Actuarial ScienceSimon Fraser UniversityBurnabyCanada
  5. 5.School of Computer ScienceShaanxi Normal UniversityXi’anChina

Personalised recommendations