Cryptanalysis of the CLR-cryptosystem

  • Giacomo Micheli
  • Violetta Weger


In this paper we break a variant of the El-Gamal cryptosystem for a ring action of the matrix space \(E_p^{(m)}\) on \(\mathbb {Z}/p\mathbb {Z}\times \mathbb {Z}/p^2\mathbb {Z}\times \dots \times \mathbb {Z}/p^m\mathbb {Z}\). Also, we describe a general vulnerability of the protocol using tools from p-adic analysis.


Finite fields Cryptography p-Adic numbers 

Mathematics Subject Classification

11T71 94A60 11S99 11C20 



The first author is thankful to the Swiss National Science Foundation under grant number 171248. The second author has been supported by the Swiss National Science Foundation under grant number 169510 We would also like to thank Karan Khathuria for his valuable inputs for the implementation of the attack. The authors are very grateful to the three anonymous referees whose suggestions greatly improved both the mathematics and the readability of the paper.


  1. 1.
    Bergman G.M.: Some examples in PI ring theory. Israel J. Math. 18(3), 257–277 (1974).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bernstein D.J., Buchmann J., Dahmen E.: Post-Quantum Cryptography. Springer, Heidelberg (2009).CrossRefzbMATHGoogle Scholar
  3. 3.
    Chevallier-Mames B., Naccache D., Stern J.: Linear bandwidth naccache-stern encryption. In: International Conference on Security and Cryptography for Networks, pp. 327–339. Springer (2008).Google Scholar
  4. 4.
    Climent J.-J., Navarro P.R., Tortosa L.: On the arithmetic of the endomorphisms ring end \((\mathbb{Z}\_p \times \mathbb{Z}\_p^2)\). Appl. Algebra Eng. Commun. Comput. 22(2), 91–108 (2011).CrossRefzbMATHGoogle Scholar
  5. 5.
    Climent J.-J., Navarro P.R., Tortosa L.: An extension of the noncommutative Bergmans ring with a large number of noninvertible elements. Appl. Algebr. Eng. Commun. Comput. 25(5), 347–361 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Climent J.-J., Ramos J.A.L.: Public key protocols over the ring \(E\_p^{(m)}\). Adv. Math. Commun. 10(4), 861–870 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ding J., Yang B.-Y.: Multivariate public key cryptography. In: Post-Quantum Cryptography, pp. 193–241. Springer (2009).Google Scholar
  8. 8.
    Feng C., Nóbrega R.W., Kschischang F.R., Silva D.: Communication over finite-chain-ring matrix channels. IEEE Trans. Inf. Theory 60(10), 5899–5917 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fine B., Habeeb M., Kahrobaei D., Rosenberger G.: Aspects of nonabelian group based cryptography: a survey and open problems. JP J. Algebra Number Theory Appl. 21(1), 1–40 (2011).MathSciNetzbMATHGoogle Scholar
  10. 10.
    Jao D., De Feo L.: Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. In: International Workshop on Post-Quantum Cryptography, pp. 19–34. Springer (2011).Google Scholar
  11. 11.
    Kamal A.A., Youssef M.: Cryptanalysis of a key exchange protocol based on the endomorphisms ring end \((\mathbb{Z}\_p \times \mathbb{Z}\_p^2)\). Appl. Algebr. Eng. Commun. Comput. 23(3), 143–149 (2012).CrossRefzbMATHGoogle Scholar
  12. 12.
    López-Ramos J.A., Rosenthal J., Schipani D., Schnyder R.: Group key management based on semigroup actions. J. Algebra Appl. 16(8), 1750148 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Maze G., Monico C., Rosenthal J.: Public key cryptography based on semigroup actions. Adv. Math. Commun. 1(4), 489–507 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    McDonald B.R.: Enumeration of classes of row equivalent matrices over a principal ideal domain modulo \(p^n\). Duke Math. J. 37(1), 163–169 (1970).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    McEliece R.J.: A public-key cryptosystem based on algebraic coding theory. Deep Space Network Progress Report 44, 114–116 (1978).Google Scholar
  16. 16.
    Micciancio D., Regev O.: Lattice-based cryptography. In: Post-Quantum Cryptography, pp. 147–191. Springer (2009).Google Scholar
  17. 17.
    Micheli G.: Cryptanalysis of a non-commutative key exchange protocol. Adv. Math. Commun. 9(2), 247–253 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Micheli G., Rosenthal J., Schnyder R.: An information rate improvement for a polynomial variant of the Naccache-Stern knapsack cryptosystem. In: Physical and Data-Link Security Techniques for Future Communication Systems, pp. 173–180. Springer (2016).Google Scholar
  19. 19.
    Micheli G., Rosenthal J., Vettori P.: Linear spanning sets for matrix spaces. Linear Algebra Appl. 483, 309–322 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Micheli G., Schiavina M.: A general construction for monoid-based knapsack protocols. Adv. Math. Commun. 8(3), 343–358 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Myasnikov A., Shpilrain V., Ushakov A.: Group-Based Cryptography. Springer, Berlin (2008).zbMATHGoogle Scholar
  22. 22.
    Naccache D., Stern J.: A new public-key cryptosystem. In: International Conference on the Theory and Applications of Cryptographic Techniques, pp. 27–36. Springer (1997).Google Scholar
  23. 23.
    Nóbrega R.W., Feng C., Silva D., Uchôa-Filho B.F.: On multiplicative matrix channels over finite chain rings. In: 2013 International Symposium on Network Coding (NetCod), pp. 1–6. IEEE (2013).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.Institute of MathematicsUniversity of ZurichZurichSwitzerland

Personalised recommendations