Designs, Codes and Cryptography

, Volume 64, Issue 3, pp 287–308 | Cite as

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

  • Mahdi Sajadieh
  • Mohammad Dakhilalian
  • Hamid Mala
  • Behnaz Omoomi


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 2 n × 2 n Hadamard MDS matrices in the finite field GF(2 q ). In addition to introducing this method, we present a direct method for the inversion of a special class of 2 n  × 2 n Vandermonde matrices.


MDS matrix Vandermonde matrix Hadamard matrix Blockcipher 

Mathematics Subject Classification (2000)

11T71 14G50 51E22 94B05 20H30 15A09 


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  1. 1.
    Althaus H.L., Leake R.J.: Inverse of a finite-field Vandermonde matrix. IEEE Trans. Inform. Theory 15, 173 (1969)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Biham E., Shamir A.: Differential Cryptanalysis of the Data Encryption Standard. Springer, Berlin (1993)MATHCrossRefGoogle Scholar
  3. 3.
    Barreto P., Rijmen V.: The Anubis Block Cipher. Submission to the NESSIE Project (2000). Available at
  4. 4.
    Barreto P., Rijmen V.: The Khazad Legacy-Level Block Cipher. Submission to the NESSIE Project (2000). Available at
  5. 5.
    Daemen J., Rijmen V.: The Design of Rijndael: AES—The Advanced Encryption Standard. Springer, Berlin (2002)MATHGoogle Scholar
  6. 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).Google Scholar
  7. 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
  8. 8.
    Junod P., Vaudenay S.: Perfect Diffusion primitives for block ciphers building efficient MDS matrices. In: SAC’04, pp. 84–99. Springer, Heidelberg (2004).Google Scholar
  9. 9.
    Lacan J., Fimes J.: Systematic MDS erasure codes based on vandermonde matrices. IEEE Trans. Commun. Lett. 8(9), 570–572 (2004)CrossRefGoogle Scholar
  10. 10.
    Lin S., Costello D.: Error Control Coding: Fundamentals and Applications, 2nd edn. Prentice Hall, Englewood Cliffs (2004)Google Scholar
  11. 11.
    MacWilliams F.J., Sloane N.J.A.: The theory of error correcting codes. North-Holland (1977).Google Scholar
  12. 12.
    Matsui M.: Linear cryptanalysis method for DES cipher. In: EUROCRYPT’93, pp. 386–397. Springer, Heidelberg (1993).Google Scholar
  13. 13.
    Nakahara J. Jr., Abrahao E.: A new involutory MDS matrix for the AES. IJNS 9(2), 109–116 (2009)Google Scholar
  14. 14.
    Rijmen V.: Cryptanalysis and Design of Iterated Block Ciphers. Ph.D. thesis, Dept. Elektrotechniek Katholieke Universiteit Leuven, pp. 228–238 (1998).Google Scholar
  15. 15.
    Sony Corporation: The 128-bit Block cipher CLEFIA: Algorithm Specification (2007). Available at
  16. 16.
    Yan S., Yang A.: Explicit algorithm to the inverse of Vandermonde matrix. In: ICTM 2009, pp. 176–179 (2009).Google Scholar
  17. 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).Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Mahdi Sajadieh
    • 1
  • Mohammad Dakhilalian
    • 1
  • Hamid Mala
    • 2
  • Behnaz Omoomi
    • 3
  1. 1.Cryptography & System Security Research Laboratory, Department of Electrical and Computer EngineeringIsfahan University of TechnologyIsfahanIran
  2. 2.Department of Information Technology EngineeringUniversity of IsfahanIsfahanIran
  3. 3.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran

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