On APN functions L1(x3) + L2(x9) with linear L1 and L2

Abstract

In a recent paper by L. Budaghyan, C. Carlet, and G. Leander (2009) it is shown that functions of the form L1(x3) + L2(x9), where L1 and L2 are linear, are a good source for construction of new infinite families of APN functions. In the present work we study necessary and sufficient conditions for such functions to be APN.

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References

  1. 1.

    Budaghyan, L., Carlet, C., Leander, G.: On a construction of quadratic APN functions. In: Proceedings of IEEE Information Theory workshop ITW’09, pp 374–378 (2009)

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 3.

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

    Article  Google Scholar 

  4. 4.

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

  5. 5.

    Janwa, H, Wilson, R.: Hyperplane sections of Fermat varieties in p 3 in char. 2 and some applications to cycle codes. In: Proceedings of AAECC-10, LNCS, vol. 673, pp 180–194. Springer, Berlin (1993)

    Google Scholar 

  6. 6.

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

    Article  Google Scholar 

  7. 7.

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

    MathSciNet  Article  Google Scholar 

  8. 8.

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

    MathSciNet  Article  Google Scholar 

  9. 9.

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

  10. 10.

    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)

    Google Scholar 

  11. 11.

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

    MathSciNet  Article  Google Scholar 

  12. 12.

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

  13. 13.

    Budaghyan, L., Carlet, C., Leander, G.: On Inequivalence between Known Power APN Functions. In: Proceedings of the International Workshop on Boolean Functions: Cryptography and Applications. BFCA 2008, Denmark (2008)

  14. 14.

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

    MathSciNet  Article  Google Scholar 

  15. 15.

    Edel, Y., Kyureghyan, G., Pott, A.: A new APN function which is not equivalent to a power mapping. IEEE Trans. Inform. Theory 52(2), 744–747 (2006)

    MathSciNet  Article  Google Scholar 

  16. 16.

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

    MathSciNet  Article  Google Scholar 

  17. 17.

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

    MathSciNet  Article  Google Scholar 

  18. 18.

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

    MathSciNet  Article  Google Scholar 

  19. 19.

    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)

    MathSciNet  Article  Google Scholar 

  20. 20.

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

    MathSciNet  Article  Google Scholar 

  21. 21.

    Edel, Y., Pott, A.: A new almost perfect nonlinear function which is not quadratic. In: IACR Cryptology ePrint Archive 2008, p 313 (2008)

  22. 22.

    Berger, T.P., Canteaut, A., Charpin, P., Lang-Chapuy, Y.: On Almost Perfect Nonlinear Functions Over F2n. IEEE Trans. Inform. Theory 52(9), 4160–4170 (2006)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Canteaut, A., Charpin, P., Kyureghyan, G.M.: A new class of monomial bent functions. Finite Fields Appl. 14(1), 221–241 (2008)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Browning, K.A., Dillon, J.F., Kibler, R.E., McQuistan, M.T.: APN polynomials and related codes. J. Comb. Inf. Syst. Sci. 34 (2009)

  25. 25.

    Yu, Y., Wang, M., Li, Y.: A matrix approach for constructing quadratic APN functions. Des. Codes Crypt. 73(27), 587–600 (2014)

    MathSciNet  Article  Google Scholar 

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Correspondence to Irene Villa.

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This article is part of the Topical Collection on Special Issue on Boolean Functions and Their Applications

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Villa, I. On APN functions L1(x3) + L2(x9) with linear L1 and L2. Cryptogr. Commun. 11, 3–20 (2019). https://doi.org/10.1007/s12095-018-0283-8

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Keywords

  • Boolean function
  • Almost perfect nonlinear
  • CCZ-equivalence
  • Nonlinearity

Mathematics Subject Classification (2010)

  • 94A60
  • 06E30
  • 20B40