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On relations between CCZ- and EA-equivalences

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In the present paper we introduce some sufficient conditions and a procedure for checking whether, for a given function, CCZ-equivalence is more general than EA-equivalence together with taking inverses of permutations. It is known from Budaghyan et al. (IEEE Trans. Inf. Theory 52.3, 1141–1152 2006; Finite Fields Appl. 15(2), 150–159 2009) that for quadratic APN functions (both monomial and polynomial cases) CCZ-equivalence is more general. We prove hereby that for non-quadratic APN functions CCZ-equivalence can be more general (by studying the only known APN function which is CCZ-inequivalent to both power functions and quadratics). On the contrary, we prove that for power non-Gold APN functions, CCZ equivalence coincides with EA-equivalence and inverse transformation for n ≤ 8. We conjecture that this is true for any n.

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  1. 1.

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

  2. 2.

    Beth, T., Ding, C.: On almost perfect nonlinear permutations, Advances in cryptology-EUROCRYPT’93, Lecture Notes in Computer science, vol. 765, pp 65–76. Springer, New York (1993)

  3. 3.

    Brinkmann, M., Leander, G.: On the classification of APN functions up to dimension five. Des. Codes Cryptogr. 49.1-3, 273–288 (2008)

  4. 4.

    Browning, K. A., Dillon, J. F., McQuistan, M. T., Wolfe, A. J.: An APN permutation in dimension six. Finite Fields: Theory Appl. 518, 33–42 (2010)

  5. 5.

    Budaghyan, L.: The Simplest Method for Constructing APN Polynomials EA-Inequivalent to Power Functions. In: Carlet, C., Sunar, B. (eds.) WAIFI 2007. LNCS, vol. 4547, pp 177–188. Springer, Heidelberg (2007)

  6. 6.

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

  7. 7.

    Budaghyan, L., Carlet, C., Leander, G.: On inequivalence between known power APN functions. In: Masnyk-Hansen, O., Michon, J.-F., Valarcher, P., Yunes, J.-B. (eds.) Proceedings of the conference BFCA’08. Copenhagen, Denmark (2008)

  8. 8.

    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)

  9. 9.

    Calderini, M., Sala, M., Villa, I.: A note on APN permutations in even dimension. Finite Fields Appl. 46, 1–16 (2017)

  10. 10.

    Canteaut, A., Perrin, L.: On CCZ-equivalence, Extended-Affine Equivalence, and Function Twisting. Finite Fields Appl. 56, 209–246 (2019)

  11. 11.

    Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15.2, 125–156 (1998)

  12. 12.

    Charpin, P., Kyureghyan, G.: On a Class of Permutation Polynomials over F2n, In: SETA 2008, Lecture Notes in Comput. Sci., vol. 5203, pp 368–376. Springer, Berlin (2008)

  13. 13.

    Dempwolff, U.: CCZ Equivalence of power functions, submitted to Designs, Codes and Cryptography (2017)

  14. 14.

    Dobbertin, H.: Almost perfect nonlinear power functions over GF(2n): the Welch case. IEEE Trans. Inform. Theory 45, 1271–1275 (1999)

  15. 15.

    Dobbertin, H.: Almost perfect nonlinear power functions over GF(2n): the Niho case. Inform. Comput. 151, 57–72 (1999)

  16. 16.

    Dobbertin, H.: Almost perfect nonlinear power functions over GF(2n): a new case for n divisible by 5. In: Proceedings of Finite Fields and Applications FQ5, pp. 113–121 (2000)

  17. 17.

    Edel, Y., Pott, A.: A new almost perfect nonlinear function which is not quadratic. Adv. Math. Comm. 3.1, 59–81 (2009)

  18. 18.

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

  19. 19.

    Janwa, H., Wilson, R.: hyperplane sections of Fermat varieties in p3 in char. 2 and some applications to cycle codes, Proceedings of AAECC-10, LNCS, vol. 673, pp 180–194. Springer, Berlin (1993)

  20. 20.

    Kasami, T.: The weight enumerators for several classes of subcodes of the second order binary Reed-Muller codes. Inform. Control 18, 369–394 (1971)

  21. 21.

    Ling, S., Qu, L.J.: A note on linearized polynomials and the dimension of their kernels. Finite Fields Appl. 18, 56–62 (2012)

  22. 22.

    Leander, G., Poschmann, A.: On the classification of 4 bit s-boxes. International Workshop on the Arithmetic of Finite Fields, pp 159–176. Springer, Berlin (2007)

  23. 23.

    Nyberg, K.: Differentially uniform mappings for cryptography, Advances in Cryptography, EUROCRYPT’93, Lecture Notes in Computer Science 765, pp. 55–64 (1994)

  24. 24.

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

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Correspondence to Marco Calderini.

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Budaghyan, L., Calderini, M. & Villa, I. On relations between CCZ- and EA-equivalences. Cryptogr. Commun. 12, 85–100 (2020). https://doi.org/10.1007/s12095-019-00367-5

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  • CCZ-equivalence
  • EA-equivalence
  • APN
  • Boolean functions

Mathematics Subject Classification (2010)

  • 94A60
  • 06E30
  • 11T71