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Automatic Search of Bit-Based Division Property for ARX Ciphers and Word-Based Division Property

  • Ling Sun
  • Wei Wang
  • Meiqin WangEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10624)

Abstract

Division property is a generalized integral property proposed by Todo at Eurocrypt 2015. Previous tools for automatic searching are mainly based on the Mixed Integer Linear Programming (MILP) method and trace the division property propagation at the bit level. In this paper, we propose automatic tools to detect ARX ciphers’ division property at the bit level and some specific ciphers’ division property at the word level.

For ARX ciphers, we construct the automatic searching tool relying on Boolean Satisfiability Problem (SAT) instead of MILP, since SAT method is more suitable in the search of ARX ciphers’ differential/linear characteristics. The propagation of division property is translated into a system of logical equations in Conjunctive Normal Form (CNF). Some logical equations can be dynamically adjusted according to different initial division properties and stopping rule, while the others corresponding to r-round propagations remain the same. Moreover, our approach can efficiently identify some optimized distinguishers with lower data complexity. As a result, we obtain a 17-round distinguisher for SHACAL-2, which gains four more rounds than previous work, and an 8-round distinguisher for LEA, which covers one more round than the former one.

For word-based division property, we develop the automatic search based on Satisfiability Modulo Theories (SMT), which is a generalization of SAT. We model division property propagations of basic operations and S-boxes by logical formulas, and turn the searching problem into an SMT problem. With some available solvers, we achieve some new distinguishers. For CLEFIA, 10-round distinguishers are obtained, which cover one more round than the previous work. For the internal block cipher of Whirlpool, the data complexities of 4/5-round distinguishers are improved. For Rijndael-192 and Rijndael-256, 6-round distinguishers are presented, which attain two more rounds than the published ones. Besides, the integral attacks for CLEFIA are improved by one round with the newly obtained distinguishers.

Keywords

Automatic search Division property ARX SAT/SMT 

Notes

Acknowledgements

The authors would like to thank the anonymous reviewers of Asiacrypt 2017 for their helpful comments. This work was supported by the 973 Program (No. 2013CB834205), NSFC Projects (No. 61572293), Science and Technology on Communication Security Laboratory of China (No. 9140c110207150c11050), as well as Chinese Major Program of National Cryptography Development Foundation (No. MMJJ20170102).

Supplementary material

References

  1. 1.
    Barreto, P.S., Rijmen, V.: The Whirlpool hashing function. In: First Open NESSIE Workshop, Leuven, Belgium, vol. 13, p. 14 (2000)Google Scholar
  2. 2.
    Barrett, C.W., Sebastiani, R., Seshia, S.A., Tinelli, C.: Satisfiability modulo theories. Handb. Satisf. 185, 825–885 (2009)Google Scholar
  3. 3.
    Beaulieu, R., Shors, D., Smith, J., Treatman-Clark, S., Weeks, B., Wingers, L.: The SIMON and SPECK lightweight block ciphers. In: Proceedings of the 52nd Annual Design Automation Conference, San Francisco, CA, USA, 7–11 June 2015, pp. 175:1–175:6 (2015)Google Scholar
  4. 4.
    Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. In: Menezes, A.J., Vanstone, S.A. (eds.) CRYPTO 1990. LNCS, vol. 537, pp. 2–21. Springer, Heidelberg (1991).  https://doi.org/10.1007/3-540-38424-3_1 CrossRefGoogle Scholar
  5. 5.
    Biryukov, A., Khovratovich, D., Perrin, L.: Multiset-algebraic cryptanalysis of reduced Kuznyechik, Khazad, and secret SPNs. IACR Trans. Symmetric Cryptol. 2016(2), 226–247 (2016)Google Scholar
  6. 6.
    Cook, S.A.: The complexity of theorem-proving procedures. In: Proceedings of the Third Annual ACM Symposium on Theory of Computing, pp. 151–158. ACM (1971)Google Scholar
  7. 7.
    Courtois, N.T., Bard, G.V.: Algebraic cryptanalysis of the data encryption standard. In: Galbraith, S.D. (ed.) Cryptography and Coding 2007. LNCS, vol. 4887, pp. 152–169. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-77272-9_10 CrossRefGoogle Scholar
  8. 8.
    Daemen, J., Rijmen, V.: The Design of Rijndael: AES - The Advanced Encryption Standard. Information Security and Cryptography. Springer, Heidelberg (2002)CrossRefzbMATHGoogle Scholar
  9. 9.
    Dinu, D., Perrin, L., Udovenko, A., Velichkov, V., Großschädl, J., Biryukov, A.: Design strategies for ARX with provable bounds: Sparx and LAX. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10031, pp. 484–513. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53887-6_18 CrossRefGoogle Scholar
  10. 10.
    Fu, K., Wang, M., Guo, Y., Sun, S., Hu, L.: MILP-based automatic search algorithms for differential and linear trails for speck. In: Peyrin, T. (ed.) FSE 2016. LNCS, vol. 9783, pp. 268–288. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-52993-5_14 CrossRefGoogle Scholar
  11. 11.
    Galice, S., Minier, M.: Improving integral attacks against Rijndael-256 up to 9 rounds. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 1–15. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-68164-9_1 CrossRefGoogle Scholar
  12. 12.
    Gerault, D., Minier, M., Solnon, C.: Constraint programming models for chosen key differential cryptanalysis. In: Rueher, M. (ed.) CP 2016. LNCS, vol. 9892, pp. 584–601. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-44953-1_37 CrossRefGoogle Scholar
  13. 13.
    Handschuh, H., Naccache, D.: SHACAL: a family of block ciphers. Submission to the NESSIE project (2002)Google Scholar
  14. 14.
    Hong, D., Lee, J.-K., Kim, D.-C., Kwon, D., Ryu, K.H., Lee, D.-G.: LEA: a 128-bit block cipher for fast encryption on common processors. In: Kim, Y., Lee, H., Perrig, A. (eds.) WISA 2013. LNCS, vol. 8267, pp. 3–27. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-05149-9_1 CrossRefGoogle Scholar
  15. 15.
    Hong, D., et al.: 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).  https://doi.org/10.1007/11894063_4 CrossRefGoogle Scholar
  16. 16.
    Kamal, A.A., Youssef, A.M.: Applications of SAT solvers to AES key recovery from decayed key schedule images. In: Fourth International Conference on Emerging Security Information Systems and Technologies, SECURWARE 2010, Venice, Italy, 18–25 July 2010, pp. 216–220 (2010)Google Scholar
  17. 17.
    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 CrossRefGoogle Scholar
  18. 18.
    Kölbl, S., Leander, G., Tiessen, T.: Observations on the SIMON block cipher family. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9215, pp. 161–185. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-47989-6_8 CrossRefGoogle Scholar
  19. 19.
    Li, Y., Wu, W., Zhang, L.: Improved integral attacks on reduced-round CLEFIA block cipher. In: Jung, S., Yung, M. (eds.) WISA 2011. LNCS, vol. 7115, pp. 28–39. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-27890-7_3 CrossRefGoogle Scholar
  20. 20.
    Matsui, M.: Linear cryptanalysis method for DES cipher. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 386–397. Springer, Heidelberg (1994).  https://doi.org/10.1007/3-540-48285-7_33 Google Scholar
  21. 21.
    Matsui, M.: New block encryption algorithm MISTY. In: Biham, E. (ed.) FSE 1997. LNCS, vol. 1267, pp. 54–68. Springer, Heidelberg (1997).  https://doi.org/10.1007/BFb0052334 CrossRefGoogle Scholar
  22. 22.
    Mendel, F., Rechberger, C., Schläffer, M., Thomsen, S.S.: The rebound attack: cryptanalysis of reduced whirlpool and Grøstl. In: Dunkelman, O. (ed.) FSE 2009. LNCS, vol. 5665, pp. 260–276. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-03317-9_16 CrossRefGoogle Scholar
  23. 23.
    Minier, M., Phan, R.C.-W., Pousse, B.: Distinguishers for ciphers and known key attack against Rijndael with large blocks. In: Preneel, B. (ed.) AFRICACRYPT 2009. LNCS, vol. 5580, pp. 60–76. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-02384-2_5 CrossRefGoogle Scholar
  24. 24.
    Mouha, N., Preneel, B.: Towards finding optimal differential characteristics for ARX: application to Salsa20. Technical report, Cryptology ePrint Archive, Report 2013/328 (2013)Google Scholar
  25. 25.
    Mouha, N., Wang, Q., Gu, D., Preneel, B.: Differential and linear cryptanalysis using mixed-integer linear programming. In: Wu, C.-K., Yung, M., Lin, D. (eds.) Inscrypt 2011. LNCS, vol. 7537, pp. 57–76. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-34704-7_5 CrossRefGoogle Scholar
  26. 26.
    Needham, R.M., Wheeler, D.J.: TEA extensions. Report, Cambridge University, Cambridge, UK, October 1997Google Scholar
  27. 27.
    PUB. FIPS 180–2: Secure hash standard (SHS). US Department of Commerce, National Institute of Standards and Technology (NIST) (2012)Google Scholar
  28. 28.
    Rijmen, V., Daemen, J.: Advanced encryption standard. In: Proceedings of Federal Information Processing Standards Publications, National Institute of Standards and Technology, pp. 19–22 (2001)Google Scholar
  29. 29.
    Shibayama, N., Kaneko, T.: A new higher order differential of CLEFIA. IEICE Trans. 97–A(1), 118–126 (2014)CrossRefGoogle Scholar
  30. 30.
    Shin, Y., Kim, J., Kim, G., Hong, S., Lee, S.: Differential-linear type attacks on reduced rounds of SHACAL-2. In: Wang, H., Pieprzyk, J., Varadharajan, V. (eds.) ACISP 2004. LNCS, vol. 3108, pp. 110–122. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-27800-9_10 CrossRefGoogle Scholar
  31. 31.
    Shirai, T., Shibutani, K., Akishita, T., Moriai, S., Iwata, T.: The 128-bit blockcipher CLEFIA (extended abstract). In: Biryukov, A. (ed.) FSE 2007. LNCS, vol. 4593, pp. 181–195. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-74619-5_12 CrossRefGoogle Scholar
  32. 32.
    Song, L., Huang, Z., Yang, Q.: Automatic differential analysis of ARX block ciphers with application to SPECK and LEA. In: Liu, J.K., Steinfeld, R. (eds.) ACISP 2016. LNCS, vol. 9723, pp. 379–394. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-40367-0_24 CrossRefGoogle Scholar
  33. 33.
    KASUMI Specification: Specification of the 3GPP confidentiality and integrity algorithms. Version, vol. 1, pp. 8–17Google Scholar
  34. 34.
    Sugio, N., Igarashi, Y., Kaneko, T.: Integral characteristics of MISTY2 derived by division property. In: 2016 International Symposium on Information Theory and Its Applications, ISITA 2016, Monterey, CA, USA, 30 October–2 November 2016, pp. 151–155 (2016)Google Scholar
  35. 35.
    Sugio, N., Igarashi, Y., Kaneko, T., Higuchi, K.: New integral characteristics of KASUMI derived by division property. In: Choi, D., Guilley, S. (eds.) WISA 2016. LNCS, vol. 10144, pp. 267–279. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56549-1_23 CrossRefGoogle Scholar
  36. 36.
    Sun, L., Wang, W., Liu, R., Wang, M.: MILP-aided bit-based division property for ARX-based block cipher. IACR Cryptology ePrint Archive, 2016:1101 (2016)Google Scholar
  37. 37.
    Sun, L., Wang, W., Wang, M.: MILP-aided bit-based division property for primitives with non-bit-permutation linear layers. IACR Cryptology ePrint Archive, 2016:811 (2016)Google Scholar
  38. 38.
    Sun, S., Gerault, D., Lafourcade, P., Yang, Q., Todo, Y., Qiao, K., Hu, L.: Analysis of AES, SKINNY, and others with constraint programming. IACR Trans. Symmetric Cryptol. 2017(1), 281–306 (2017)Google Scholar
  39. 39.
    Sun, S., Hu, L., Wang, P., Qiao, K., Ma, X., Song, L.: Automatic security evaluation and (related-key) differential characteristic search: application to SIMON, PRESENT, LBlock, DES(L) and other bit-oriented block ciphers. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014. LNCS, vol. 8873, pp. 158–178. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-45611-8_9 Google Scholar
  40. 40.
    Todo, Y.: Integral cryptanalysis on full MISTY1. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9215, pp. 413–432. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-47989-6_20 CrossRefGoogle Scholar
  41. 41.
    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 Google Scholar
  42. 42.
    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 CrossRefGoogle Scholar
  43. 43.
    Wen, L., Wang, M.: Integral zero-correlation distinguisher for ARX block cipher, with application to SHACAL-2. In: Susilo, W., Mu, Y. (eds.) ACISP 2014. LNCS, vol. 8544, pp. 454–461. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-08344-5_32 Google Scholar
  44. 44.
    Wheeler, D.J., Needham, R.M.: TEA, a tiny encryption algorithm. In: Preneel, B. (ed.) FSE 1994. LNCS, vol. 1008, pp. 363–366. Springer, Heidelberg (1995).  https://doi.org/10.1007/3-540-60590-8_29 CrossRefGoogle Scholar
  45. 45.
    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 CrossRefGoogle Scholar
  46. 46.
    Zhang, H., Wu, W.: Structural evaluation for generalized feistel structures and applications to LBlock and TWINE. In: Biryukov, A., Goyal, V. (eds.) INDOCRYPT 2015. LNCS, vol. 9462, pp. 218–237. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-26617-6_12 CrossRefGoogle Scholar
  47. 47.
    Zhang, W., Rijmen, V.: Division cryptanalysis of block ciphers with a binary diffusion layer. IACR Cryptology ePrint Archive, 2017:188 (2017)Google Scholar
  48. 48.
    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, New York (1990).  https://doi.org/10.1007/0-387-34805-0_42 CrossRefGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2017

Authors and Affiliations

  1. 1.Key Laboratory of Cryptologic Technology and Information Security, Ministry of EducationShandong UniversityJinanChina
  2. 2.Science and Technology on Communication Security LaboratoryChengduChina
  3. 3.State Key Laboratory of CryptologyBeijingChina

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