Skip to main content
SpringerLink
Log in

A graph theoretic method for securing key fobs

  • Original Research
  • Published:
Mathematical Sciences Aims and scope Submit manuscript

Abstract

Key fobs are small security hardware devices that are used for controlling access to doors, cars and etc. There are many types of these devices, more secure types of them are rolling code. They often use a light weight cryptographic schemas to protect them against the replay attack. In this paper, by the mean of constructing a Hamiltonian graph, we propose a simple to implement and secure method which overrides some drawbacks of traditional ones. Let \(m,n>1\) be two integers, and \({\mathbb {Z}}_n\) be a \({\mathbb {Z}}_m\)-module. Here, we determine the values of m and n for which the \({\mathbb {Z}}_n\)-intersection graph of ideals of \({\mathbb {Z}}_m\) is Hamiltonian. Then a suitable sequence will be produced which by some criteria can be used as the authenticator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Benadjila, R., Renard, M., Lopes-Esteves, J., Kasmi, C.: One car, two frames: attacks on hitag-2 remote keyless entry systems revisited. In: Enck, W., Mulliner, C. (eds.) Proceedings of the 11th USENIX Workshop on Offensive Technologies, WOOT 2017, Vancouver, BC, Canada, August 14–15, 2017. USENIX Association (2017)

  2. Verstegen, A., Verdult, R., Bokslag, W.: Hitag 2 hell-brutally optimizing guess-and-determine attacks. In: Rossow, C., Younan, Y. (eds.) Proceedings of the 12th USENIX Workshop on Offensive Technologies, WOOT 2018. USENIX Association, Baltimore, MD, USA, 13–14 Aug 2018.

  3. Indesteege, S., Keller, N., Dunkelman, O., Biham, E., Preneel, B.: A practical attack on keeloq. In: Smart, N.P. (ed.) Proceedings of the Advances in Cryptology: EUROCRYPT 2008, 27th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Istanbul, Turkey, April 13–17, 2008., vol. 4965 of Lecture Notes in Computer Science, pp. 1–18, Springer, New York (2008)

  4. Courtois, N., O’Neil, S., Quisquater, J.: Practical algebraic attacks on the hitag2 stream cipher. In: Samarati, P., Yung, M., Martinelli, F., Ardagna, C.A. (eds) Proceedings of the Information Security, 12th International Conference, ISC 2009, Pisa, Italy, September 7–9, 2009, vol. 5735 of Lecture Notes in Computer Science, pp. 167–176. Springer, New York (2009)

  5. Lemke, K., Sadeghi, A.R., Stüble, C.: Anti-theft Protection: Electronic Immobilizers, pp. 51–67. Springer, Berlin (2006)

  6. Verdult, R., Garcia, F.D., Ege, B.: Dismantling megamos crypto: Wirelessly lockpicking a vehicle immobilizer. In: King, S. T. (ed.) Proceedings of the 22th USENIX Security Symposium, Washington, DC, USA, August 14–16, 2013, pp. 703–718, USENIX Association (2013)

  7. Akbari, S., Heydari, F.: The regular graph of a commutative ring. Period. Math. Hung. 67(2), 211–220 (2013)

    Article  MathSciNet  Google Scholar 

  8. Nikandish, R., Maimani, H.R., Izanloo, H.: The annihilating-ideal graph of \({\mathbb{Z}}_n\) is weakly perfect. Contrib. Discrete Math. 11(1), 16–21 (2016)

    MathSciNet  MATH  Google Scholar 

  9. Nikmehr, M.J., Heydari, F.: The \(M\)-regular graph of a commutative ring. Math. Slovaca 65(1), 1–12 (2015)

    Article  MathSciNet  Google Scholar 

  10. Nikmehr, M.J., Heydari, F.: The \(M\)-principal graph of a commutative ring. Period. Math. Hung. 68(2), 185–192 (2014)

    Article  MathSciNet  Google Scholar 

  11. Akbari, S., Tavallaee, H.A., Ghezelahmad, S.K.: On the complement of the intersection graph of submodules of a module. J. Algebra Appl. 14(08), 1550116 (2015)

    Article  MathSciNet  Google Scholar 

  12. Chakrabarty, I., Ghosh, S., Mukherjee, T.K., Sen, M.K.: Intersection graphs of ideals of rings. Discrete Math. 309(17), 5381–5392 (2009)

    Article  MathSciNet  Google Scholar 

  13. Csákány, B., Pollák, G.: The graph of subgroups of a finite group. Czechoslovak Math. J. 19, 241–247 (1969)

    Article  MathSciNet  Google Scholar 

  14. Heydari, F.: The \(M\)-intersection graph of ideals of a commutative ring. Discrete Math. Algorithms Appl. 10(03), 1850038 (2018)

    Article  MathSciNet  Google Scholar 

  15. Khojasteh, S.: The intersection graph of ideals of \({\mathbb{Z}_m}\). Discrete Math. Alg. Appl. 11(4), 1950037 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors highly appreciate the support provided by the Karaj Branch, Islamic Azad University.

Author information

Authors and Affiliations

  1. Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

    Farideh Heydari

  2. School of Electrical and Computer Engineering, Malek Ashtar University of Technology, Tehran, Iran

    Alireza Ghahremanian

Authors
  1. Farideh Heydari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alireza Ghahremanian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Farideh Heydari.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Heydari, F., Ghahremanian, A. A graph theoretic method for securing key fobs. Math Sci 15, 41–46 (2021). https://doi.org/10.1007/s40096-020-00363-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40096-020-00363-4

Keywords

Mathematics Subject Classification

Search

Navigation