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On the equivalence between a new family of APN quadrinomials and the power APN functions

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Abstract

Finding new (up to CCZ-equivalence) constructions of APN functions is one of the important but difficult topics in the study of cryptographic functions. Up to now, only 6 infinite families of power APN functions and 14 infinite families of APN polynomials are known. Although the CCZ-equivalence between power APN functions has been completely characterized, a similar theoretical analysis between polynomial APN functions and power APN functions is still missing. In this paper we prove that, in dimension not multiple of 3, a recently introduced family of APN quadrinomials is CCZ-inequivalent to any power APN function. Moreover, we prove that a more general family of APN functions is EA-equivalent to this family.

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References

  1. Beth, T., Ding, C.: On almost perfect nonlinear permutations, in Workshop on the Theory and Application of Cryptographic Techniques. Springer, pp. 65-76 (1993)

  2. Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bracken, C., Byrne, E., Markin, N., McGuire, G.: New families of quadratic almost perfect nonlinear trinomials and multinomias. Finite Fields Appl. 14(3), 703–714 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bracken, C., Byrne, E., Markin, N., McGuire, G.: A few more quadratic APN functions. Cryptogr. Commun. 3(1), 43–53 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brinkmann, M., Leander, G.: On the classification of APN functions up to dimension five. Des. Codes Cryptogr. 49, 273–288 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Budaghyan, L.: On the equivalence between known power APN functions. Proc. Conference BFCA 2008, Copenhagen (2008)

  7. Budaghyan, L., Carlet, C.: Classes of quadratic APN trinomials and hexanomials and related structures. IEEE Trans. Inf. Theory 54(5), 2354–2357 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Budaghyan, L., Calderini, M., Carlet, C., Coulter, R.S., Villa, I.: Constructing APN functions through isotopic shifts. IEEE Trans. Inf. Theory 66(8), 5299–5309 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Budaghyan, L., Calderini, M., Villa, I.: On equivalence between known families of quadratic APN functions. Finite Fields Appl. 66, (2020). https://doi.org/10.1016/j.ffa.2020.101704

  10. Budaghyan, L., Carlet, C., Leander, G.: Two classes of quadratic APN binomials inequivalent to power functions. IEEE Trans. Inf. Theory 54(9), 4218–4229 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Budaghyan, L., Carlet, C., Leander, G.: Constructing new APN functions from known ones. Finite Fields Appl. 15(2), 150–159 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Budaghyan, L., Carlet, C., Leander, G.: On a construction of quadratic APN functions. In: Proceedings of IEEE Information Theory Workshop, pp. 374–378. (2009)

  13. Budaghyan, L., Carlet, C., Pott, A.: New classes of almost bent and almost perfect nonlinear polynomials. IEEE Trans. Inf. Theory 52(3), 1141–1152 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Budaghyan, L., Helleseth, T., Kaleyski, N.: A new family of APN quadrinomials. IEEE Trans. Inf. Theory (2020). https://doi.org/10.1109/tit.2020.3007513

    Article  MathSciNet  MATH  Google Scholar 

  15. Canteaut, A., Charpin, P., Dobbertin, H.: Weight divisibility of cyclic codes, highly nonlinear functions on \(\mathbb{F}_{2}^{m}\), and crosscorrelation of maximum-length sequences. SIAM J. Discrete Math. 13(1), 105–138 (2000)

  16. Carlet, C.: Boolean functions for cryptography and coding theory, p. 562. Cambridge Univ. Press, Cambridge, U.K. (2021)

  17. Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptsystems. Des. Codes Cryptogr. 15(2), 125–156 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dempwolff, U.: CCZ equivalence of power functions. Des. Codes Cryptogr. 86(3), 665–692 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dobbertin, H.: Almost perfect nonlinear power functions on GF(\(2^{n}\)): the Welch case. IEEE Trans. Inf. Theory 45(4), 1271–1275 (1999)

  20. Dobbertin, H.: Almost perfect nonlinear power functions on GF(\(2^{n}\)): the Niho case. Inform. Comput. 151(1), 57–72 (1999)

  21. Dobbertin, H.: Almost perfect nonlinear power functions on GF(\(2^{n}\)): a new case for \(n\) divisible by 5. In: Proceedings of Finite Fields and Applications, pp. 113-121. Springer, Berlin, Heidelberg (2001)

  22. Gold, R.: Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory 14(1), 154–156 (1968)

    Article  MATH  Google Scholar 

  23. Janwa, H., Wiison, R.M.: Hyperplane sections of Fermat varieties in \(P^{3}\) in char. 2 and some applications to cyclic codes. International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes., pp. 180–194. Springer (1993)

  24. Kasami, T.: The weight enumerators for several classes of subcodes of the 2nd order binary reed-muller codes. Inf. Control 18(4), 369–394 (1971)

    Article  MATH  Google Scholar 

  25. Kaspers, C., Zhou, Y.: The number of almost perfect nonlinear functions grows exponentially, J. Cryptol. 34(4), (2021). https://doi.org/10.1007/s00145-020-09373-w

  26. Lachaud, G., Wolfmann, J.: The weights of the orthogonals of the extended quadraic binary Goppa codes. IEEE Trans. Inf. Theory 36(3), 686–692 (1990)

    Article  MATH  Google Scholar 

  27. Nyberg, K.: Differential uniform mappings for cryptography. In: Helleseth, T. (eds.) Advances in Cryptography. EUROCRYPT’93 (Lecture Notes in Computer Science), vol. 765, pp. 55–64. New York: Springer-Verlag (1993)

  28. Taniguchi, H.: On some quadratic APN functions. Des. Codes Cryptogr. 87(9), 1973–1983 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wan, Q., Qu, L., Li, C.: On equivalence between known polynomial APN functions and power APN functions. Finite Fields Appl. 71, (2021). https://doi.org/10.1016/j.ffa.2020.101762

  30. Yoshiara, S.: Equivalences of quadratic APN functions. J. Algebraic Comb. 35(3), 461–475 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yoshiara, S.: Equivalences of power APN functions with power or quadratic APN functions. J. Algebraic Comb. 44(3), 561–585 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yu, Y., Kaleyski, N., Budaghyan, L., Li, Y.: Classification of quadratic APN functions with coefficients in \(\mathbb{F} _{2}\) for dimensions up to 9. Finite Fields Appl. 68, (2020). https://doi.org/10.1016/j.ffa.2020.101733

  33. Zheng, L., Kan, H., Li, Y., Peng, J., Tang, D.: Constructing new APN functions through relative trace functions. arXiv: 210111535v1 [IEEE Trans. Inf. Theory] (2021)

  34. Zhou, Y., Pott, A.: A new family of semifields with 2 parameters. Adv. Math. 234, 43–60 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the anonymous reviewers for their valuable suggestions which improved both the quality and the presentation of this paper.

Funding

This work was supported in part by the National Natural Science Foundation of China under Grant 61972258 and by the Scientific Research Fund of Hunan Provincial Education Department under Grant 19B485.

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Authors and Affiliations

  1. Mathematics and Science College of Shanghai Normal University, Shanghai, 200234, China

    Chenmiao Shi, Jie Peng & Shihao Lu

  2. School of Mathematics and Physics, University of South China, Hengyang, Hunan, 421001, China

    Lijing Zheng

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  1. Chenmiao Shi
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  2. Jie Peng
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  3. Lijing Zheng
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  4. Shihao Lu
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Correspondence to Jie Peng.

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Shi, C., Peng, J., Zheng, L. et al. On the equivalence between a new family of APN quadrinomials and the power APN functions. Cryptogr. Commun. 15, 351–363 (2023). https://doi.org/10.1007/s12095-022-00606-2

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  • DOI: https://doi.org/10.1007/s12095-022-00606-2

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