Advertisement

Direct Construction of Recursive MDS Diffusion Layers Using Shortened BCH Codes

  • Daniel Augot
  • Matthieu Finiasz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8540)

Abstract

MDS matrices allow to build optimal linear diffusion layers in block ciphers. However, MDS matrices cannot be sparse and usually have a large description, inducing costly software/hardware implementations. Recursive MDS matrices allow to solve this problem by focusing on MDS matrices that can be computed as a power of a simple companion matrix, thus having a compact description suitable even for constrained environments. However, up to now, finding recursive MDS matrices required to perform an exhaustive search on families of companion matrices, thus limiting the size of MDS matrices one could look for. In this article we propose a new direct construction based on shortened BCH codes, allowing to efficiently construct such matrices for whatever parameters. Unfortunately, not all recursive MDS matrices can be obtained from BCH codes, and our algorithm is not always guaranteed to find the best matrices for a given set of parameters.

Keywords

Linear diffusion Recursive MDS matrices BCH codes 

Supplementary material

References

  1. 1.
    Augot, D., Finiasz, M.: Exhaustive search for small dimension recursive MDS diffusion layers for block ciphers and hash functions. In: 2013 IEEE International Symposium on Information Theory Proceedings (ISIT), pp. 1551–1555. IEEE (2013)Google Scholar
  2. 2.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Daemen, J.: Cipher and hash function design, strategies based on linear and differential cryptanalysis, Ph.D. thesis. K.U. Leuven (1995)Google Scholar
  4. 4.
    Daemen, J., Rijmen, V.: The Design of Rijndael. Information Security and Cryptography. Springer, Heidelberg (2002)CrossRefzbMATHGoogle Scholar
  5. 5.
    Guo, J., Peyrin, T., Poschmann, A.: The PHOTON family of lightweight hash functions. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 222–239. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Guo, J., Peyrin, T., Poschmann, A., Robshaw, M.: The LED block cipher. In: Preneel, B., Takagi, T. (eds.) CHES 2011. LNCS, vol. 6917, pp. 326–341. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  7. 7.
    Junod, P., Vaudenay, S.: FOX: a new family of block ciphers. In: Handschuh, H., Hasan, M.A. (eds.) SAC 2004. LNCS, vol. 3357, pp. 114–129. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  8. 8.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error Correcting Codes. North-Holland Mathematical Library. North-Holland, Amsterdam (1978) Google Scholar
  9. 9.
    Sajadieh, M., Dakhilalian, M., Mala, H., Sepehrdad, P.: Recursive diffusion layers for block ciphers and hash functions. In: Canteaut, A. (ed.) FSE 2012. LNCS, vol. 7549, pp. 385–401. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  10. 10.
    Singleton, R.: Maximum distance \(q\)-nary codes. IEEE Trans. Inf. Theor. 10(2), 116–118 (1964)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Wu, S., Wang, M., Wu, W.: Recursive diffusion layers for (lightweight) block ciphers and hash functions. In: Knudsen, L.R., Wu, H. (eds.) SAC 2012. LNCS, vol. 7707, pp. 355–371. Springer, Heidelberg (2013) CrossRefGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2015

Authors and Affiliations

  1. 1.INRIA - LIX UMR 7161 X-CNRSParisFrance
  2. 2.CryptoExpertsParisFrance

Personalised recommendations