Advertisement

Lightweight MDS Involution Matrices

  • Siang Meng Sim
  • Khoongming Khoo
  • Frédérique Oggier
  • Thomas Peyrin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9054)

Abstract

In this article, we provide new methods to look for lightweight MDS matrices, and in particular involutory ones. By proving many new properties and equivalence classes for various MDS matrices constructions such as circulant, Hadamard, Cauchy and Hadamard-Cauchy, we exhibit new search algorithms that greatly reduce the search space and make lightweight MDS matrices of rather high dimension possible to find. We also explain why the choice of the irreducible polynomial might have a significant impact on the lightweightness, and in contrary to the classical belief, we show that the Hamming weight has no direct impact. Even though we focused our studies on involutory MDS matrices, we also obtained results for non-involutory MDS matrices. Overall, using Hadamard or Hadamard-Cauchy constructions, we provide the (involutory or non-involutory) MDS matrices with the least possible XOR gates for the classical dimensions \(4 \times 4\), \(8 \times 8\), \(16 \times 16\) and \(32 \times 32\) in \(\mathrm {GF}(2^4)\) and \(\mathrm {GF}(2^8)\). Compared to the best known matrices, some of our new candidates save up to 50 % on the amount of XOR gates required for an hardware implementation. Finally, our work indicates that involutory MDS matrices are really interesting building blocks for designers as they can be implemented with almost the same number of XOR gates as non-involutory MDS matrices, the latter being usually non-lightweight when the inverse matrix is required.

Keywords

Lightweight cryptography Hadamard matrix Cauchy matrix Involution MDS 

Notes

Acknowledgments

The authors would like to thank the anonymous referees for their helpful comments. We also wish to thank Wang HuaXiong for providing useful and valuable suggestions.

Supplementary material

References

  1. 1.
    Andreeva, E., Bilgin, B., Bogdanov, A., Luykx, A., Mendel, F., Mennink, B., Mouha, N., Wang, Q., Yasuda, K.: PRIMATEs v1. Submission to the CAESAR Competition (2014). http://competitions.cr.yp.to/round1/primatesv1.pdf
  2. 2.
    Augot, D., Finiasz, M.: Direct construction of recursive MDS diffusion layers using shortened BCH codes. In: Cid, C., Rechberger, C. (eds.) FSE 2014. LNCS, vol. 8540, pp. 3–17. Springer, Heidelberg (2015) Google Scholar
  3. 3.
    Augot, D., Finiasz, M.: Exhaustive search for small dimension recursive MDS diffusion layers for block ciphers and hash functions. In: ISIT, pp. 1551–1555 (2013)Google Scholar
  4. 4.
    Aumasson, J.-P., Henzen, L., Meier, W., Naya-Plasencia, M.: Quark: a lightweight hash. In: Mangard, S., Standaert, F.-X. (eds.) CHES 2010. LNCS, vol. 6225, pp. 1–15. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  5. 5.
    Barreto, P., Rijmen, V.: The Anubis Block Cipher. Submission to the NESSIE Project (2000)Google Scholar
  6. 6.
    Barreto, P., Rijmen, V.: The Khazad legacy-level block cipher. In: First Open NESSIE Workshop (2000)Google Scholar
  7. 7.
    Barreto, P., Nikov, V., Nikova, S., Rijmen, V., Tischhauser, E.: Whirlwind a new cryptographic hash function. Des. Codes Crypt. 56(2–3), 141–162 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Barreto, P., Rijmen, V.: Whirlpool. In: Encyclopedia of Cryptography and Security, 2nd edn. Springer, Heidelberg (2011) Google Scholar
  9. 9.
    Beaulieu, R., Shors, D., Smith, J., Treatman-Clark, S., Weeks, B., Wingers, L.: The SIMON and SPECK Families of Lightweight Block Ciphers. Cryptology ePrint Archive, Report 2013/404 (2013)Google Scholar
  10. 10.
    Berger, T.P.: Construction of recursive MDS diffusion layers from Gabidulin codes. In: Paul, G., Vaudenay, S. (eds.) INDOCRYPT 2013. LNCS, vol. 8250, pp. 274–285. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  11. 11.
    Bogdanov, A., Knežević, M., Leander, G., Toz, D., Varıcı, K., Verbauwhede, I.: spongent: a lightweight hash function. In: Preneel, B., Takagi, T. (eds.) CHES 2011. LNCS, vol. 6917, pp. 312–325. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  12. 12.
    Bogdanov, A.A., Knudsen, L.R., Leander, G., Paar, C., Poschmann, A., Robshaw, M., Seurin, Y., Vikkelsoe, C.: PRESENT: an ultra-lightweight block cipher. In: Paillier, P., Verbauwhede, I. (eds.) CHES 2007. LNCS, vol. 4727, pp. 450–466. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  13. 13.
    Borghoff, J., Canteaut, A., Güneysu, T., Kavun, E.B., Knezevic, M., Knudsen, L.R., Leander, G., Nikov, V., Paar, C., Rechberger, C., Rombouts, P., Thomsen, S.S., Yalçın, T.: PRINCE – a low-latency block cipher for pervasive computing applications. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 208–225. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  14. 14.
    De Cannière, C., Dunkelman, O., Knežević, M.: KATAN and KTANTAN — a family of small and efficient hardware-oriented block ciphers. In: Clavier, C., Gaj, K. (eds.) CHES 2009. LNCS, vol. 5747, pp. 272–288. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  15. 15.
    Cui, T., Jin, C.I, Kong, Z.: On compact cauchy matrices for substitution-permutation networks. IEEE Trans. Comput. 99(PrePrints), 1 (2014)Google Scholar
  16. 16.
    Daemen, J., Knudsen, L.R., Rijmen, V.: The block cipher SQUARE. In: Biham, E. (ed.) FSE 1997. LNCS, vol. 1267, pp. 149–165. Springer, Heidelberg (1997) CrossRefGoogle Scholar
  17. 17.
    Daemen, J., Rijmen, V.: The Design of Rijndael: AES - The Advanced Encryption Standard. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  18. 18.
    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
  19. 19.
    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
  20. 20.
    Chand Gupta, K., Ghosh Ray, I.: On constructions of involutory MDS matrices. In: Youssef, A., Nitaj, A., Hassanien, A.E. (eds.) AFRICACRYPT 2013. LNCS, vol. 7918, pp. 43–60. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  21. 21.
    Chand Gupta, K., Ghosh Ray, I.: On constructions of circulant MDS matrices for lightweight cryptography. In: Huang, X., Zhou, J. (eds.) ISPEC 2014. LNCS, vol. 8434, pp. 564–576. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  22. 22.
    Jean, J., Nikolić, I., Peyrin, T.: Joltik v1.1, 2014. Submission to the CAESAR competition. http://www1.spms.ntu.edu.sg/~syllab/Joltik
  23. 23.
    Nakahara Jr., J., Abraho, I.: A new involutory mds matrix for the aes. I. J Netw. Secur. 9(2), 109–116 (2009)Google Scholar
  24. 24.
    Junod, P., Vaudenay, S.: Perfect diffusion primitives for block ciphers. In: Handschuh, H., Hasan, M.A. (eds.) SAC 2004. LNCS, vol. 3357, pp. 84–99. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  25. 25.
    Kavun, E.B., Lauridsen, M.M., Leander, G., Rechberger, C., Schwabe, P., Yalçın, T.: Prøst v1.1, 2014. Submission to the CAESAR competition. http://competitions.cr.yp.to/round1/proestv11.pdf
  26. 26.
    Khoo, K., Peyrin, T., Poschmann, A.Y., Yap, H.: FOAM: searching for hardware-optimal SPN structures and components with a fair comparison. In: Batina, L., Robshaw, M. (eds.) CHES 2014. LNCS, vol. 8731, pp. 433–450. Springer, Heidelberg (2014) Google Scholar
  27. 27.
    Lacan, J., Fimes, J.: Systematic MDS erasure codes based on Vandermonde matrices. IEEE Commun. Lett. 8(9), 570–572 (2004)CrossRefGoogle Scholar
  28. 28.
    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
  29. 29.
    Sajadieh, M.I., Dakhilalian, M., Mala, H., Omoomi, B.: On construction of involutory MDS matrices from Vandermonde matrices in GF(2 q ). Des. Codes Crypt. 64(3), 287–308 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Shirai, T., Shibutani, K.: On the diffusion matrix employed in the Whirlpool hashing function. NESSIE Phase 2 Report NES/DOC/EXT/WP5/002/1Google Scholar
  31. 31.
    Sim, S.M., Khoo, K., Oggier, F., Peyrin, T.: Lightweight mds involution matrices. Cryptology ePrint Archive, Report 2015/258 (2015). http://eprint.iacr.org/
  32. 32.
    Standaert, F.-X., Piret, G., Rouvroy, G., Quisquater, J.-J., Legat, J.-D.: ICEBERG : an involutional cipher efficient for block encryption in reconfigurable hardware. In: Roy, B., Meier, W. (eds.) FSE 2004. LNCS, vol. 3017, pp. 279–299. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  33. 33.
    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
  34. 34.
    Youssef, A.M., Mister, S., Tavares, S.E.: On the design of linear transformations for substitution permutation encryption networks. In: Workshop On Selected Areas in Cryptography, pp. 40–48 (1997)Google Scholar

Copyright information

© International Association for Cryptologic Research 2015

Authors and Affiliations

  • Siang Meng Sim
    • 1
  • Khoongming Khoo
    • 2
  • Frédérique Oggier
    • 1
  • Thomas Peyrin
    • 1
  1. 1.Nanyang Technological UniversitySingaporeSingapore
  2. 2.DSO National LaboratoriesSingaporeSingapore

Personalised recommendations