Skip to main content
SpringerLink
Log in
Book cover

Australasian Conference on Information Security and Privacy

ACISP 2022: Information Security and Privacy pp 106–124Cite as

Improved Division Property for Ciphers with Complex Linear Layers

  • Conference paper
  • First Online:

  • 587 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13494))

Abstract

The division property proposed by Todo at EUROCRYPT 2015 as a generalized integral property has been applied to many symmetric ciphers. Automatic search methods of the division property assisted by modeling technique, such as Mixed Integer Linear Programming (MILP) and Boolean Satisfiability Problem (SAT), have become the most popular approach to searching integral distinguishers. The accuracy of the model in searching algorithms has an effect on the search results of integral distinguishers. For the block cipher, constructing an accurate and efficient model of the division property propagation on complex linear layers remains hard. This paper observes that the non-independent propagations of the bit-based division property (BDP) on complex linear layers can generate redundant division trails, which will affect the accuracy of the model if it is not taken into account in modeling. Based on this, we propose a method that can build a more accurate model by handling matrices containing non-independent propagations in the linear layer. To verify the effectiveness of our method, we apply the method to two block ciphers uBlock-128 and MIBS. For uBlock-128, our results improve the previous 8-round integral distinguisher by more balanced bits. For MIBS, a 9-round integral distinguisher is given for the first time, which is 4 rounds longer than the previous best.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Knudsen, L., Wagner, D.: Integral cryptanalysis. In: Daemen, J., Rijmen, V. (eds.) FSE 2002. LNCS, vol. 2365, pp. 112–127. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45661-9_9

    Chapter  Google Scholar 

  2. Daemen, J., Knudsen, L., Rijmen, V.: The block cipher Square. In: Biham, E. (ed.) FSE 1997. LNCS, vol. 1267, pp. 149–165. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0052343

    Chapter  Google Scholar 

  3. Todo, Y.: Structural evaluation by generalized integral property. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 287–314. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46800-5_12

    Chapter  Google Scholar 

  4. Xiang, Z., Zhang, W., Bao, Z., Lin, D.: Applying MILP method to searching integral distinguishers based on division property for 6 lightweight block ciphers. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10031, pp. 648–678. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53887-6_24

    Chapter  Google Scholar 

  5. Wang, Q., Hao, Y., Todo, Y., Li, C., Isobe, T., Meier, W.: Improved division property based cube attacks exploiting algebraic properties of superpoly. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10991, pp. 275–305. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96884-1_10

    Chapter  Google Scholar 

  6. Liu, M., Yang, J., Wang, W., Lin, D.: Correlation cube attacks: from weak-key distinguisher to key recovery. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10821, pp. 715–744. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78375-8_23

    Chapter  Google Scholar 

  7. Wang, S., Hu, B., Guan, J., Zhang, K., Shi, T.: MILP-aided method of searching division property using three subsets and applications. In: Galbraith, S.D., Moriai, S. (eds.) ASIACRYPT 2019. LNCS, vol. 11923, pp. 398–427. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-34618-8_14

    Chapter  Google Scholar 

  8. Hao, Y., Leander, G., Meier, W., Todo, Y., Wang, Q.: Modeling for three-subset division property without unknown subset. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020. LNCS, vol. 12105, pp. 466–495. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45721-1_17

    Chapter  Google Scholar 

  9. Hebborn, P., Lambin, B., Leander, G., Todo, Y.: Lower bounds on the degree of block ciphers. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020. LNCS, vol. 12491, pp. 537–566. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64837-4_18

    Chapter  Google Scholar 

  10. Hu, K., Sun, S., Wang, M., Wang, Q.: An algebraic formulation of the division property: revisiting degree evaluations, cube attacks, and key-independent sums. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020. LNCS, vol. 12491, pp. 446–476. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64837-4_15

    Chapter  Google Scholar 

  11. Hebborn, P., Lambin, B., Leander, G., Todo, Y.: Strong and tight security guarantees against integral distinguishers. In: Tibouchi, M., Wang, H. (eds.) ASIACRYPT 2021. LNCS, vol. 13090, pp. 362–391. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-92062-3_13

    Chapter  Google Scholar 

  12. Sun, L., Wang, W., Wang, M.Q.: MILP-aided bit-based division property for primitives with non-bit-permutation linear layers. IET Inf. Secur. 14, 12–20 (2020)

    Article  Google Scholar 

  13. Zhang, W.Y., Rijmen, V.: Division cryptanalysis of block ciphers with a binary diffusion layer. IET Inf. Secur. 13, 87–95 (2019)

    Article  Google Scholar 

  14. Hu, K., Wang, Q.J., Wang, M.Q.: Finding bit-based division property for ciphers with complex linear layers. IACR Trans. Symmetric Cryptol. 2020, 396–424 (2020)

    Article  Google Scholar 

  15. Todo, Y., Morii, M.: Bit-based division property and application to Simon family. In: Peyrin, T. (ed.) FSE 2016. LNCS, vol. 9783, pp. 357–377. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-52993-5_18

    Chapter  Google Scholar 

  16. Wu, W.L., Zhang, L., Zheng, Y.F., Li, L.C.: The block cipher uBlock. J. Cryptol. Res. 6(6), 690–703 (2019). (in Chinese)

    Google Scholar 

  17. Tian, W., Hu, B.: Integral cryptanalysis on two block ciphers Pyjamask and uBlock. IET Inf. Secur. 14, 572–579 (2020)

    Article  Google Scholar 

  18. Hong, C., Zhang, S., Chen, S., Lin, D., Xiang, Z.: More accurate division property propagations based on optimized implementations of linear layers. In: Yu, Yu., Yung, M. (eds.) Inscrypt 2021. LNCS, vol. 13007, pp. 212–232. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-88323-2_11

    Chapter  Google Scholar 

  19. Izadi, M., Sadeghiyan, B., Sadeghian, S.S., Khanooki, H.A.: MIBS: a new lightweight block cipher. In: Garay, J.A., Miyaji, A., Otsuka, A. (eds.) CANS 2009. LNCS, vol. 5888, pp. 334–348. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10433-6_22

    Chapter  Google Scholar 

  20. Bay, A., Nakahara, J., Vaudenay, S.: Cryptanalysis of reduced-round MIBS block cipher. In: Heng, S.-H., Wright, R.N., Goi, B.-M. (eds.) CANS 2010. LNCS, vol. 6467, pp. 1–19. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-17619-7_1

    Chapter  Google Scholar 

  21. Li, Y.J., Sun, Q., Ou, H.W., et al.: Improved integral attacks on MIBS-64 block cipher. J. Cryptol. Res. 8(4), 669–679 (2021). (in Chinese)

    Google Scholar 

Download references

Acknowledgement

The authors would like to thank Prof. Guomin Yang and the anonymous reviewers for their detailed and very helpful comments and suggestions to improve this article. This work is supported by the National Natural Science Foundation of China (No. 62072445).

Author information

Authors and Affiliations

  1. Trusted Computing and Information Assurance Laboratory, Institute of Software Chinese Academy of Sciences, Beijing, 100190, China

    Yongxia Mao, Wenling Wu, Bolin Wang & Li Zhang

  2. University of Chinese Academy of Sciences, Beijing, 100049, China

    Yongxia Mao, Wenling Wu, Bolin Wang & Li Zhang

Authors
  1. Yongxia Mao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenling Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bolin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Li Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenling Wu .

Editor information

Editors and Affiliations

  1. University of Wollongong, Wollongong, NSW, Australia

    Khoa Nguyen

  2. University of Wollongong, Wollongong, NSW, Australia

    Guomin Yang

  3. University of Wollongong, Wollongong, NSW, Australia

    Fuchun Guo

  4. University of Wollongong, Wollongong, NSW, Australia

    Willy Susilo

Appendices

A Linear Inequalities for S-Boxes in uBlock-128

The following inequalities are the 12 inequalities used to describe uBlock S-box in MILP model of BDP, and \((a_{3},a_{2},a_{1},a_{0})\rightarrow (b_{3},b_{2},b_{1},b_{0})\) denotes a division trail of S-box.

$$\begin{aligned} \left\{ \begin{aligned}&a_3+a_2+a_1+a_0-b_3-b_2-b_1-b_0\ge 0\\&-3a_3-a_2-2a_1-4a_0+3b_3+b_2+2b_1-b_0\ge -5\\&2a_3-a_0-2b_3-b_2-b_1+b_0\ge -2\\&-4a_3-3a_2-2a_1-2a_0-b_3+3b_2+b_1+2b_0\ge -6\\&-a_1+2a_0-b_3-b_2+b_1-2b_0\ge -2\\&-a_3-a_2-2a_0+b_3+2b_2+3b_1+2b_0\ge 0\\&a_3+a_0+b_3-2b_2-2b_1-b_0\ge -2\\&a_1+2a_0-b_3-b_2-b_1-b_0\ge -1\\&a_3-b_1-b_0\ge -1\\&-a_3-a_1+b_3+2b_2+b_1+b_0\ge 0\\&a_1-b_3-b_1\ge -1\\&a_2-b_2-b_1\ge -1\\ \end{aligned} \right. \end{aligned}$$

B Linear Inequalities for S-Boxes in MIBS

The following inequalities are the 12 inequalities used to describe MIBS S-box in MILP model of BDP, and \((d_{3},d_{2},d_{1},d_{0})\rightarrow (b_{3},b_{2},b_{1},b_{0})\) denotes a division trail of S-box.

$$\begin{aligned} \left\{ \begin{aligned}&d_3 + d_2 + 4 d_1 + d_0 - 2 b_3 - 2 b_2 - 2 b_1 - 2 b_0 \ge -1\\&3 d_2 - b_3 - b_2 - b_1 - b_0 \ge -1\\&-d_3 - 2 d_2 - 2 d_1 - d_0 - b_3 - 2 b_2 + 4 b_1 - b_0 \ge -6\\&-d_3 - 2 d_2 - 2 d_1 - d_0 + 5 b_3 + 4 b_2 + 5 b_1 + 5 b_0 \ge 0\\&-d_3 - d_2 - d_1- b_3 + 3 b_2 - 2 b_1 - b_0 \ge -4\\&-d_3 - d_0 - 2 b_3 - b_2 - b_1 + 3 b_0 \ge -3\\&d_3 + b_3 - b_2 - b_1 - b_0 \ge -1\\&-d_3 - d_2 - d_0 + b_3 + 2 b_2 + 2 b_1 + b_0 \ge -1\\&- d_1 - b_3 - b_2 + 2 b_1 - b_0 \ge -2\\ \end{aligned} \right. \end{aligned}$$

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Mao, Y., Wu, W., Wang, B., Zhang, L. (2022). Improved Division Property for Ciphers with Complex Linear Layers. In: Nguyen, K., Yang, G., Guo, F., Susilo, W. (eds) Information Security and Privacy. ACISP 2022. Lecture Notes in Computer Science, vol 13494. Springer, Cham. https://doi.org/10.1007/978-3-031-22301-3_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-22301-3_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-22300-6

  • Online ISBN: 978-3-031-22301-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation