On Maximum Differential Probability of Generalized Feistel

  • Kazuhiko Minematsu
  • Tomoyasu Suzaki
  • Maki Shigeri
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6812)


The maximum differential probability (MDP) is an important security measure for blockciphers. We investigate MDP of Type-2 generalized Feistel structure (Type-2 GFS), one of the most popular cipher architectures. Previously MDP of Type-2 GFS has been studied for partition number (number of sub-blocks) k = 2 by Aoki and Ohta, and k = 4 by Kim et al. These studies are based on ad-hoc case analysis and it seems rather difficult to analyze larger k by hand. In this paper, we abstract the idea of previous studies and generalize it for any k, and implement it using computers. We investigate Type-2 GFS of k = 4,6,8 and 10 with k + 1 rounds, and obtain O(pk) bound for all cases, when the round function is invertible and its MDP is p. The bound for k = 4 is improved from Kim et al. and those for larger k are new. We also investigate an improvement of Type-2 GFS proposed by Suzaki and Minematsu, and obtain similar bounds as Type-2.


blockcipher generalized Feistel differential probability 


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  1. 1.
    Massey, J.: On the Optimality of SAFER+ Diffusion. In: Second AES Candidate Conference. National Institute of Standards and Technology (1999)Google Scholar
  2. 2.
    Zheng, Y., Matsumoto, T., Imai, H.: On the construction of block ciphers provably secure and not relying on any unproved hypotheses. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 461–480. Springer, Heidelberg (1990)Google Scholar
  3. 3.
    Shirai, T., Shibutani, K., Akishita, T., Moriai, S., Iwata, T.: The 128-bit Blockcipher CLEFIA. In: Biryukov, A. (ed.) FSE 2007. LNCS, vol. 4593, pp. 181–195. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Shibutani, K.: On the Diffusion Properties of Generalized Feistel Structures. In: Biryukov, A., Gong, G., Stinson, D.R. (eds.) SAC 2010. LNCS, vol. 6544, pp. 211–228. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. 5.
    Park, S., Sung, S., Lee, S., Lim, J.: Improving the upper bound on the maximum differential and the maximum linear hull probability for SPN structures and AES. In: Johansson, T. (ed.) FSE 2003. LNCS, vol. 2887, pp. 247–260. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Hong, S., Lee, S., Lim, J., Sung, J., Cheon, D., Cho, I.: Provable security against differential and linear cryptanalysis for the SPN structure. In: Schneier, B. (ed.) FSE 2000. LNCS, vol. 1978, p. 273. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Hong, D., Sung, J., Hong, S., Lim, J., Lee, S., Koo, B., Lee, C., Chang, D., Lee, J., Jeong, K., Kim, H., Kim, J., Chee, S.: HIGHT: A new block cipher suitable for low-resource device. In: Goubin, L., Matsui, M. (eds.) CHES 2006. LNCS, vol. 4249, pp. 46–59. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Nyberg, K.: Generalized Feistel Networks. In: Kim, K.-c., Matsumoto, T. (eds.) ASIACRYPT 1996. LNCS, vol. 1163, pp. 90–104. Springer, Heidelberg (1996)Google Scholar
  9. 9.
    Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. In: Menezes, A., Vanstone, S.A. (eds.) CRYPTO 1990. LNCS, vol. 537, pp. 2–21. Springer, Heidelberg (1991)Google Scholar
  10. 10.
    Kim, J., Lee, C., Sung, J., Hong, S., Lee, S., Lim, J.: Seven New Block Cipher Structures with Provable Security against Differential Cryptanalysis. IEICE Trans. Fundamentals E91-A(10) (2008)Google Scholar
  11. 11.
    Corporation, S.: The 128-bit Blockcipher CLEFIA Security and Performance Evaluations. Revision 1.0 (June 1, 2007)Google Scholar
  12. 12.
    Nyberg, K., Knudsen, L.R.: Provable security against differential cryptanalysis. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 566–574. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  13. 13.
    Aoki, K., Ohta, K.: Strict Evaluation of the Maximum Average of Differential Probability and the Maximum Average of Linear Probability. IEICE Trans. Fundamentals E80-A(1), 2–8 (1997)Google Scholar
  14. 14.
    Suzaki, T., Minematsu, K.: Improving the generalized feistel. In: Hong, S., Iwata, T. (eds.) FSE 2010. LNCS, vol. 6147, pp. 19–39. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  15. 15.
    Matsui, M.: New Structure of Block Ciphers With Provable Security against Differential and Linear Cryptanalysis. In: Gollmann, D. (ed.) FSE 1996. LNCS, vol. 1039, Springer, Heidelberg (1996)CrossRefGoogle Scholar
  16. 16.
    Lai, X.: On the Design and Security of Block Ciphers. Hartung-Gorre (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Kazuhiko Minematsu
    • 1
  • Tomoyasu Suzaki
    • 1
  • Maki Shigeri
    • 2
  1. 1.NEC Corporation.KawasakiJapan
  2. 2.NEC Software Hokuriku, Ltd.IshikawaJapan

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