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Some Results on the Known Classes of Quadratic APN Functions

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Book cover Codes, Cryptology and Information Security (C2SI 2017)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10194))

Abstract

In this paper, we determine the Walsh spectra of three classes of quadratic APN functions and we prove that the class of quadratic trinomial APN functions constructed by Göloğlu is affine equivalent to Gold functions.

This work was supported by the Norwegian Research Council.

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References

  1. Beth, T., Ding, C.: On almost perfect nonlinear permutations. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 65–76. Springer, Heidelberg (1994). doi:10.1007/3-540-48285-7_7

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Bracken, C., Byrne, E., Markin, N., McGuire, G.: On the Walsh spectrum of a new APN function. In: Galbraith, S.D. (ed.) Cryptography and Coding 2007. LNCS, vol. 4887, pp. 92–98. Springer, Heidelberg (2007). doi:10.1007/978-3-540-77272-9_6

    Chapter  Google Scholar 

  6. Bracken, C., Byrne, E., Markin, N., McGuire, G.: On the Fourier spectrum of binomial APN functions. SIAM J. Discrete Math. 23(2), 596–608 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bracken, C., Byrne, E., Markin, N., McGuire, G.: Determining the nonlinearity of a new family of APN functions. In: Boztaş, S., Lu, H.-F.F. (eds.) AAECC 2007. LNCS, vol. 4851, pp. 72–79. Springer, Heidelberg (2007). doi:10.1007/978-3-540-77224-8_11

    Chapter  Google Scholar 

  8. Bracken, C., Zha, Z.: On the Fourier spectra of the infinite families of quadratic APN functions. Finite Fields Appl. 18(3), 537–546 (2012)

    Article  MathSciNet  Google Scholar 

  9. Brinkmann, M., Leander, G.: On the classification of APN functions up to dimension five. Des. Codes Crypt. 49(1–3), 273–288 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Browning, A.K., Dillon, F.J., Kibler, E.R., McQuistan, T.M.: APN polynomials and related codes. J. Comb. Inf. Syst. Sci. 34(1–4), 135–159 (2009). Special Issue in Honor of Prof. D.K Ray-Chaudhuri on the occasion of his 75th birthday

    Google Scholar 

  11. Browning, A.K., Dillon, F.J., McQuistan, T.M., Wolfe, J.A.: An APN permutation in dimension six. In: Post-proceedings of the 9-th International Conference on Finite Fields and Their Applications Fq 2009, Contemporary Math, AMS, vol. 518, pp. 33–42 (2010)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. 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 

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

    Google Scholar 

  16. Budaghyan, L., Carlet, C., Pott, A.: New classes of almost bent and almost perfect nonlinear functions. IEEE Trans. Inform. Theor. 52(3), 1141–1152 (2006)

    Article  MATH  Google Scholar 

  17. Canteaut, A., Charpin, P., Dobbertin, H.: Binary \(m\)-sequences with three-valued crosscorrelation: a proof of Welch’s conjecture. IEEE Trans. Inform. Theor. 46(1), 4–8 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Carlet, C.: Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions. Des. Codes Crypt. 59(1–3), 89–109 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Carlet, C.: Vectorial Boolean functions for cryptography. In: Crama, Y., Hammer, P. (eds.) Boolean Methods and Models. Cambridge University Press (2005–2006, to appear). Chapter of the Monography

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Chabaud, F., Vaudenay, S.: Links between differential and linear cryptanalysis. In: Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 356–365. Springer, Heidelberg (1995). doi:10.1007/BFb0053450

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  24. Dobbertin, H.: Almost perfect nonlinear power functions over \(GF(2^n)\): a new case for \(n\) divisible by 5. In: Proceedings of Finite Fields and Applications Fq5, pp. 113–121 (2000)

    Google Scholar 

  25. Dobbertin, H.: Another proof of Kasami’s theorem. Des. Codes Crypt. 17, 177–180 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Edel, Y.: Quadratic APN functions as subspaces of alternating bilinear forms. In: Contact Forum Coding Theory and Cryptography III 2009, Belgium, pp. 11–24 (2011)

    Google Scholar 

  27. Edel, Y., Pott, A.: A new almost perfect nonlinear function which is not quadratic. Adv. Math. Commun. 3(1), 59–81 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gold, R.: Maximal recursive sequences with 3-valued recursive crosscorrelation functions. IEEE Trans. Inform. Theor. 14, 154–156 (1968)

    Article  MATH  Google Scholar 

  29. Göloğlu, F.: Almost perfect nonlinear trinomials and hexanomials. Finite Fields Appl. 33, 258–282 (2015)

    Article  MathSciNet  Google Scholar 

  30. Hollmann, H., Xiang, Q.: A proof of the Welch and Niho conjectures on crosscorrelations of binary \(m\)-sequences. Finite Fields Appl. 7, 253–286 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Janwa, H., Wilson, R.M.: Hyperplane sections of fermat varieties in \(P^{3}\) in char. 2 and some applications to cyclic codes. In: Cohen, G., Mora, T., Moreno, O. (eds.) AAECC 1993. LNCS, vol. 673, pp. 180–194. Springer, Heidelberg (1993). doi:10.1007/3-540-56686-4_43

    Chapter  Google Scholar 

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

    Article  MATH  Google Scholar 

  33. Matsui, M.: Linear cryptanalysis method for DES cipher. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 386–397. Springer, Heidelberg (1994). doi:10.1007/3-540-48285-7_33

    Google Scholar 

  34. Nyberg, K.: Differentially uniform mappings for cryptography. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 55–64. Springer, Heidelberg (1994). doi:10.1007/3-540-48285-7_6

    Google Scholar 

  35. Nyberg, K.: Perfect nonlinear S-boxes. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 378–386. Springer, Heidelberg (1991). doi:10.1007/3-540-46416-6_32

    Google Scholar 

  36. Tan, Y., Qu, L., Ling, S., Tan, C.H.: On the Fourier spectra of new APN functions. SIAM J. Discrete Math. 27(2), 791–801 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Yu, Y., Wang, M., Li, Y.: A matrix approach for constructing quadratic APN functions. In: Pre-proceedings of the International Conference WCC 2013, Bergen, Norway (2013)

    Google Scholar 

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Budaghyan, L., Helleseth, T., Li, N., Sun, B. (2017). Some Results on the Known Classes of Quadratic APN Functions. In: El Hajji, S., Nitaj, A., Souidi, E. (eds) Codes, Cryptology and Information Security. C2SI 2017. Lecture Notes in Computer Science(), vol 10194. Springer, Cham. https://doi.org/10.1007/978-3-319-55589-8_1

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  • DOI: https://doi.org/10.1007/978-3-319-55589-8_1

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