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New linear approximation of modular addition and improved differential-linear cryptanalysis of SPARX-64/128

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Abstract

Differential-linear cryptanalysis is an efficient cryptanalysis method to attack ARX ciphers, which have been used to present the best attacks on many ARX primitives such as Chaskey and Chacha. In this paper, we present the differential-linear cryptanalysis of another ARX-based block cipher SPARX-64/128. We first construct multiple 6-round differential-linear distinguishers based on the structure of SPARX-64/128, and then extend them into 14-round differential-linear distinguishers by adding a 7-round differential characteristic before and a one-round linear approximation after the distinguishers. Then we introduce a new linear approximation of modular addition, and use it to extend one more round after the 14-round differential-linear distinguishers. With the 15-round differential-linear distinguishers, we present a differential-linear attack on 18-round SPARX-64/128.

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References

  1. Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. In: Menezes, A., Vanstone, S.A. (eds.) Advances in Cryptology - CRYPTO ’90, 10th Annual International Cryptology Conference, Santa Barbara, California, USA, August 11–15, 1990, Proceedings. Lecture Notes in Computer Science, vol. 537, pp. 2–21. Springer, Berlin, Heidelberg (1990). https://doi.org/10.1007/3-540-38424-3_1

  2. Matsui, M.: Linear cryptanalysis method for DES cipher. In: Helleseth, T. (ed.) Advances in Cryptology - EUROCRYPT ’93, Workshop on the Theory and Application of of Cryptographic Techniques, Lofthus, Norway, May 23-27, 1993, Proceedings. Lecture Notes in Computer Science, vol. 765, pp. 386–397. Springer, Berlin, Heidelberg (1993). https://doi.org/10.1007/3-540-48285-7_33

  3. Langford, S.K., Hellman, M.E.: Differential-linear cryptanalysis. In: Desmedt, Y. (ed.) Advances in Cryptology - CRYPTO ’94, 14th Annual International Cryptology Conference, Santa Barbara, California, USA, August 21–25, 1994, Proceedings. Lecture Notes in Computer Science, vol. 839, pp. 17–25. Springer, Berlin, Heidelberg (1994). https://doi.org/10.1007/3-540-48658-5_3

  4. Biham, E., Dunkelman, O., Keller, N.: Enhancing differential-linear cryptanalysis. In: Zheng, Y. (ed.) Advances in Cryptology - ASIACRYPT 2002, 8th International Conference on the Theory and Application of Cryptology and Information Security, Queenstown, New Zealand, December 1-5, 2002, Proceedings. Lecture Notes in Computer Science, vol. 2501, pp. 254–266. Springer, Berlin, Heidelberg (2002). https://doi.org/10.1007/3-540-36178-2_16

  5. Huang, T., Tjuawinata, I., Wu, H.: Differential-linear cryptanalysis of ICEPOLE. In: Leander, G. (ed.) Fast Software Encryption - 22nd International Workshop, FSE 2015, Istanbul, Turkey, March 8-11, 2015, Revised Selected Papers. Lecture Notes in Computer Science, vol. 9054, pp. 243–263. Springer, Berlin, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48116-5_12

  6. Lu, J.: A methodology for differential-linear cryptanalysis and its applications - (extended abstract). In: Canteaut, A. (ed.) Fast Software Encryption - 19th International Workshop, FSE 2012, Washington, DC, USA, March 19-21, 2012. Revised Selected Papers. Lecture Notes in Computer Science, vol. 7549, pp. 69–89. Springer, Berlin, Heidelberg (2012). https://doi.org/10.1007/978-3-642-34047-5_5

  7. Bar-On, A., Dunkelman, O., Keller, N., Weizman, A.: DLCT: A new tool for differential-linear cryptanalysis. In: Ishai, Y., Rijmen, V. (eds.) Advances in Cryptology – EUROCRYPT 2019, pp. 313–342. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17653-2_11

  8. Liu, M., Lu, X., Lin, D.: Differential-linear cryptanalysis from an algebraic perspective. In: Malkin, T., Peikert, C. (eds.) Advances in Cryptology – CRYPTO 2021, pp. 247–277. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-84252-9_9

  9. Lipmaa, H., Moriai, S.: Efficient algorithms for computing differential properties of addition. In: Matsui, M. (ed.) Fast Software Encryption, 8th International Workshop, FSE 2001, Yokohama, Japan, April 2-4, 2001, Revised Papers. Lecture Notes in Computer Science, vol. 2355, pp. 336–350. Springer, Berlin, Heidelberg (2001). https://doi.org/10.1007/3-540-45473-X_28

  10. Wallén, J.: Linear approximations of addition modulo 2\(^{n}\). In: Johansson, T. (ed.) Fast Software Encryption, 10th International Workshop, FSE 2003, Lund, Sweden, February 24-26, 2003, Revised Papers. Lecture Notes in Computer Science, vol. 2887, pp. 261–273. Springer, Berlin, Heidelberg (2003). https://doi.org/10.1007/978-3-540-39887-5_20

  11. Schulte-Geers, E.: On ccz-equivalence of addition mod 2\(^{n}\). Des. Codes Cryptogr. 66(1–3), 111–127 (2013). https://doi.org/10.1007/s10623-012-9668-4

    Article  MathSciNet  Google Scholar 

  12. 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.) Fast Software Encryption - 23rd International Conference, FSE 2016, Bochum, Germany, March 20-23, 2016, Revised Selected Papers. Lecture Notes in Computer Science, vol. 9783, pp. 268–288. Springer, Berlin, Heidelberg (2016). https://doi.org/10.1007/978-3-662-52993-5_14

  13. Liu, Y., Wang, Q., Rijmen, V.: Automatic search of linear trails in ARX with applications to SPECK and chaskey. In: Manulis, M., Sadeghi, A., Schneider, S.A. (eds.) Applied Cryptography and Network Security - 14th International Conference, ACNS 2016, Guildford, UK, June 19-22, 2016. Proceedings. Lecture Notes in Computer Science, vol. 9696, pp. 485–499. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-39555-5_26

  14. Leurent, G.: Improved differential-linear cryptanalysis of 7-round chaskey with partitioning. In: Fischlin, M., Coron, J. (eds.) Advances in Cryptology - EUROCRYPT 2016 - 35th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Vienna, Austria, May 8-12, 2016, Proceedings, Part I. Lecture Notes in Computer Science, vol. 9665, pp. 344–371. Springer, Berlin, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49890-3_14

  15. Biham, E., Carmeli, Y.: An improvement of linear cryptanalysis with addition operations with applications to FEAL-8X. In: Joux, A., Youssef, A.M. (eds.) Selected Areas in Cryptography - SAC 2014 - 21st International Conference, Montreal, QC, Canada, August 14-15, 2014, Revised Selected Papers. Lecture Notes in Computer Science, vol. 8781, pp. 59–76. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13051-4_4

  16. Beierle, C., Leander, G., Todo, Y.: Improved differential-linear attacks with applications to ARX ciphers. In: Micciancio, D., Ristenpart, T. (eds.) Advances in Cryptology – CRYPTO 2020, pp. 329–358. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56877-1_12

  17. Beierle, C., Broll, M., Canale, F., David, N., Flórez-Gutiérrez, A., Leander, G., Naya-Plasencia, M., Todo, Y.: Improved differential-linear attacks with applications to ARX ciphers. J. Cryptol. 35(4), 29 (2022). https://doi.org/10.1007/s00145-022-09437-z

    Article  MathSciNet  Google Scholar 

  18. Dey, S., Garai, H.K., Sarkar, S., Sharma, N.K.: Revamped differential-linear cryptanalysis on reduced round ChaCha. In: Dunkelman, O., Dziembowski, S. (eds.) Advances in Cryptology - EUROCRYPT 2022 - 41st Annual International Conference on the Theory and Applications of Cryptographic Techniques, Trondheim, Norway, May 30 - June 3, 2022, Proceedings, Part III. Lecture Notes in Computer Science, vol. 13277, pp. 86–114. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-07082-2_4

  19. Wang, S., Liu, M., Hou, S., Lin, D.: Moving a step of ChaCha in syncopated rhythm. In: Handschuh, H., Lysyanskaya, A. (eds.) Advances in Cryptology - CRYPTO 2023 - 43rd Annual International Cryptology Conference, CRYPTO 2023, Santa Barbara, CA, USA, August 20-24, 2023, Proceedings, Part III. Lecture Notes in Computer Science, vol. 14083, pp. 273–304. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-38548-3_10

  20. Liu, Y., Sun, S., Li, C.: Rotational cryptanalysis from a differential-linear perspective - practical distinguishers for round-reduced friet, xoodoo, and alzette. In: Canteaut, A., Standaert, F. (eds.) Advances in Cryptology - EUROCRYPT 2021 - 40th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Zagreb, Croatia, October 17-21, 2021, Proceedings, Part I. Lecture Notes in Computer Science, vol. 12696, pp. 741–770. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77870-5_26

  21. Liu, Y., Niu, Z., Sun, S., Li, C., Hu, L.: Rotational differential-linear cryptanalysis revisited. J. Cryptol. 36(1), 3 (2023). https://doi.org/10.1007/s00145-022-09440-4

    Article  MathSciNet  Google Scholar 

  22. Niu, Z., Sun, S., Liu, Y., Li, C.: Rotational differential-linear distinguishers of ARX ciphers with arbitrary output linear masks. In: Dodis, Y., Shrimpton, T. (eds.) Advances in Cryptology - CRYPTO 2022 - 42nd Annual International Cryptology Conference, CRYPTO 2022, Santa Barbara, CA, USA, August 15-18, 2022, Proceedings, Part I. Lecture Notes in Computer Science, vol. 13507, pp. 3–32. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-15802-5_1

  23. 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.) Advances in Cryptology – ASIACRYPT 2016, pp. 484–513. Springer, Berlin, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53887-6_18

  24. Abdelkhalek, A., Tolba, M., Youssef, A.M.: Impossible differential attack on reduced round SPARX-64/128. In: Joye, M., Nitaj, A. (eds.) Progress in Cryptology - AFRICACRYPT 2017, pp. 135–146. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-57339-7_8

  25. Tolba, M., Abdelkhalek, A., Youssef, A.M.: Multidimensional zero-correlation linear cryptanalysis of reduced round SPARX-128. In: Adams, C., Camenisch, J. (eds.) Selected Areas in Cryptography - SAC 2017 - 24th International Conference, Ottawa, ON, Canada, August 16-18, 2017, Revised Selected Papers. Lecture Notes in Computer Science, vol. 10719, pp. 423–441. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-72565-9_22

  26. Ankele, R., List, E.: Differential cryptanalysis of round-reduced SPARX-64/128. In: Preneel, B., Vercauteren, F. (eds.) Applied Cryptography and Network Security - 16th International Conference, ACNS 2018, Leuven, Belgium, July 2-4, 2018, Proceedings. Lecture Notes in Computer Science, vol. 10892, pp. 459–475. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-93387-0_24

  27. Zhou, D., Chen, H., Zong, R., Song, N.: Zero-correlation linear cryptanalysis on SPARX-64. J. Sensors. 2021, 1–11 (2021). https://doi.org/10.1155/2021/2193543

    Article  Google Scholar 

  28. Langford, S.K., Hellman, M.E.: Differential-linear cryptanalysis. In: Desmedt, Y.G. (ed.) Advances in Cryptology — CRYPTO ’94, pp. 17–25. Springer, Berlin, Heidelberg (1994). https://doi.org/10.1007/3-540-48658-5_3

  29. Selçuk, A.A.: On probability of success in linear and differential cryptanalysis. J. Cryptol. 21(1), 131–147 (2008). https://doi.org/10.1007/s00145-007-9013-7

    Article  MathSciNet  Google Scholar 

  30. Sun, L., Wang, W., Wang, M.: Accelerating the search of differential and linear characteristics with the SAT method. IACR Trans. Symmetric Cryptol. 2021(1), 269–315 (2021). https://doi.org/10.46586/tosc.v2021.i1.269-315

    Article  Google Scholar 

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Authors and Affiliations

  1. Information Engineering University, Kexue Road, Zheng Zhou, 450001, HeNan, China

    Zhichao Xu, Hong Xu, Lin Tan & Wenfeng Qi

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  2. Hong Xu
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Z.X. and H.X. wrote the main manuscript text, while L.T. and W.Q. finished all the experiments and prepared all the figures. All authors reviewed the manuscript.

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Correspondence to Hong Xu.

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Xu, Z., Xu, H., Tan, L. et al. New linear approximation of modular addition and improved differential-linear cryptanalysis of SPARX-64/128. Cryptogr. Commun. (2024). https://doi.org/10.1007/s12095-024-00708-z

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