Skip to main content
SpringerLink
Log in

On construction of involutory MDS matrices from Vandermonde Matrices in GF(2q)

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

Due to their remarkable application in many branches of applied mathematics such as combinatorics, coding theory, and cryptography, Vandermonde matrices have received a great amount of attention. Maximum distance separable (MDS) codes introduce MDS matrices which not only have applications in coding theory but also are of great importance in the design of block ciphers. Lacan and Fimes introduce a method for the construction of an MDS matrix from two Vandermonde matrices in the finite field. In this paper, we first suggest a method that makes an involutory MDS matrix from the Vandermonde matrices. Then we propose another method for the construction of 2n × 2n Hadamard MDS matrices in the finite field GF(2q). In addition to introducing this method, we present a direct method for the inversion of a special class of 2n × 2n Vandermonde matrices.

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

Similar content being viewed by others

References

  1. Althaus H.L., Leake R.J.: Inverse of a finite-field Vandermonde matrix. IEEE Trans. Inform. Theory 15, 173 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  2. Biham E., Shamir A.: Differential Cryptanalysis of the Data Encryption Standard. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  3. Barreto P., Rijmen V.: The Anubis Block Cipher. Submission to the NESSIE Project (2000). Available at http://cryptonessie.org.

  4. Barreto P., Rijmen V.: The Khazad Legacy-Level Block Cipher. Submission to the NESSIE Project (2000). Available at http://cryptonessie.org.

  5. Daemen J., Rijmen V.: The Design of Rijndael: AES—The Advanced Encryption Standard. Springer, Berlin (2002)

    MATH  Google Scholar 

  6. Filho G.D., Barreto P., Rijmen V.: The Maelstrom-0 hash function. In: Proceedings of the 6th Brazilian Symposium on Information and Computer Systems Security (2006).

  7. Gauravaram P., Knudsen L.R., Matusiewicz K., Mendel F., Rechberger C., Schlaffer M., Thomsen S.: Grøstl a SHA-3 Candidate. Submission to NIST (2008). Available at http://www.groestl.info.

  8. Junod P., Vaudenay S.: Perfect Diffusion primitives for block ciphers building efficient MDS matrices. In: SAC’04, pp. 84–99. Springer, Heidelberg (2004).

  9. Lacan J., Fimes J.: Systematic MDS erasure codes based on vandermonde matrices. IEEE Trans. Commun. Lett. 8(9), 570–572 (2004)

    Article  Google Scholar 

  10. Lin S., Costello D.: Error Control Coding: Fundamentals and Applications, 2nd edn. Prentice Hall, Englewood Cliffs (2004)

    Google Scholar 

  11. MacWilliams F.J., Sloane N.J.A.: The theory of error correcting codes. North-Holland (1977).

  12. Matsui M.: Linear cryptanalysis method for DES cipher. In: EUROCRYPT’93, pp. 386–397. Springer, Heidelberg (1993).

  13. Nakahara J. Jr., Abrahao E.: A new involutory MDS matrix for the AES. IJNS 9(2), 109–116 (2009)

    Google Scholar 

  14. Rijmen V.: Cryptanalysis and Design of Iterated Block Ciphers. Ph.D. thesis, Dept. Elektrotechniek Katholieke Universiteit Leuven, pp. 228–238 (1998).

  15. Sony Corporation: The 128-bit Block cipher CLEFIA: Algorithm Specification (2007). Available at http://www.sony.co.jp/Products/cryptography/clefia/download/data/clefia-spec-1.0.pdf.

  16. Yan S., Yang A.: Explicit algorithm to the inverse of Vandermonde matrix. In: ICTM 2009, pp. 176–179 (2009).

  17. Youssef A.M., Mister S., Tavares S.E.: On the design of linear transformations for substitution permutation encryption networks. In: SAC’97, pp. 1–9 (1997).

Download references

Author information

Authors and Affiliations

  1. Cryptography & System Security Research Laboratory, Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, Iran

    Mahdi Sajadieh & Mohammad Dakhilalian

  2. Department of Information Technology Engineering, University of Isfahan, Isfahan, Iran

    Hamid Mala

  3. Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran

    Behnaz Omoomi

Authors
  1. Mahdi Sajadieh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohammad Dakhilalian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hamid Mala
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Behnaz Omoomi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mahdi Sajadieh.

Additional information

Communicated by J. Jedwab.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sajadieh, M., Dakhilalian, M., Mala, H. et al. On construction of involutory MDS matrices from Vandermonde Matrices in GF(2q). Des. Codes Cryptogr. 64, 287–308 (2012). https://doi.org/10.1007/s10623-011-9578-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-011-9578-x

Keywords

Mathematics Subject Classification (2000)

Search

Navigation