Advertisement

The Simplest Method for Constructing APN Polynomials EA-Inequivalent to Power Functions

  • Lilya Budaghyan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4547)

Abstract

In 2005 Budaghyan, Carlet and Pott constructed the first APN polynomials EA-inequivalent to power functions by applying CCZ-equivalence to the Gold APN functions. It is a natural question whether it is possible to construct APN polynomials EA-inequivalent to power functions by using only EA-equivalence and inverse transformation on a power APN mapping: this would be the simplest method to construct APN polynomials EA-inequivalent to power functions. In the present paper we prove that the answer to this question is positive. By this method we construct a class of APN polynomials EA-inequivalent to power functions. On the other hand it is shown that the APN polynomials constructed by Budaghyan, Carlet and Pott cannot be obtained by the introduced method.

Keywords

Affine equivalence Almost bent Almost perfect nonlinear CCZ-equivalence Differential uniformity Nonlinearity S-box Vectorial Boolean function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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 (1993)Google Scholar
  2. Biham, E., Shamir, A.: Differential Cryptanalysis of DES-like Cryptosystems. Journal of Cryptology 4(1), 3–72 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  3. Budaghyan, L., Carlet, C.: Classes of Quadratic APN Trinomials and Hexanomials and Related Structures. Preprint, available at http://eprint.iacr.org/2007/098
  4. Budaghyan, L., Carlet, C., Leander, G.: Constructing new APN functions from known ones. Preprint, available at http://eprint.iacr.org/2007/063
  5. Budaghyan, L., Carlet, C., Leander, G.: Another class of quadratic APN binomials over \(\mathbf{F}_{2^n}\): the case n divisible by 4. In: Proceedings of the Workshop on Coding and Cryptography (2007) (To appear) available at http://eprint.iacr.org/2006/428.pdf
  6. Budaghyan, L., Carlet, C., Leander, G.: A class of quadratic APN binomials inequivalent to power functions. Submitted to IEEE Trans. Inform. Theory, available at http://eprint.iacr.org/2006/445.pdf
  7. Budaghyan, L., Carlet, C., Felke, P., Leander, G.: An infinite class of quadratic APN functions which are not equivalent to power mappings. Proceedings of the IEEE International Symposium on Information Theory 2006, Seattle, USA (July 2006)Google Scholar
  8. 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)CrossRefMathSciNetGoogle Scholar
  9. Budaghyan, L., Carlet, C., Pott, A.: New Constructions of Almost Bent and Almost Perfect Nonlinear Functions. In: Charpin, P., Ytrehus, Ø., (eds.) Proceedings of the Workshop on Coding and Cryptography 2005, pp. 306–315 (2005)Google Scholar
  10. Canteaut, A., Charpin, P., Dobbertin, H.: A new characterization of almost bent functions. In: Knudsen, L.R. (ed.) FSE 1999. LNCS, vol. 1636, pp. 186–200. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. Canteaut, A., Charpin, P., Dobbertin, H.: Binary m-sequences with three-valued crosscorrelation: A proof of Welch’s conjecture. IEEE Trans. Inform. Theory 46(1), 4–8 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  12. Canteaut, A., Charpin, P., Dobbertin, H.: Weight divisibility of cyclic codes, highly nonlinear functions on \(\mathbf{F}_{2^m}\), and crosscorrelation of maximum-length sequences. SIAM Journal on Discrete Mathematics 13(1), 105–138 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  13. Carlet, C.: Vectorial (multi-output) Boolean Functions for Cryptography. In: Crama, Y., Hammer, P. (eds.) Chapter of the monography Boolean Methods and Models, Cambridge University Press, to appear soon. Preliminary version available at http://www-rocq.inria.fr/codes/Claude.Carlet/pubs.html
  14. Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Designs, Codes and Cryptography 15(2), 125–156 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  15. Chabaud, F., Vaudenay, S.: Links between differential and linear cryptanalysis. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 356–365. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  16. Daemen, J., Rijmen, V.: AES proposal: Rijndael (1999), http://csrc.nist.gov/encryption/aes/rijndael/Rijndael.pdf
  17. Dillon, J.F.: APN Polynomials and Related Codes. Polynomials over Finite Fields and Applications, Banff International Research Station (November 2006)Google Scholar
  18. Dobbertin, H.: One-to-One Highly Nonlinear Power Functions on GF(2n). Appl. Algebra Eng. Commun. Comput. 9(2), 139–152 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  19. Dobbertin, H.: Almost perfect nonlinear power functions over GF(2n): the Niho case. Inform. and Comput. 151, 57–72 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  20. Dobbertin, H.: Almost perfect nonlinear power functions over GF(2n): the Welch case. IEEE Trans. Inform. Theory 45, 1271–1275 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  21. Dobbertin, H.: Almost perfect nonlinear power functions over GF(2n): a new case for n divisible by 5. In: Jungnickel, D., Niederreiter, H. (eds.) Proceedings of Finite Fields and Applications FQ5, Augsburg, Germany, pp. 113–121. Springer, Heidelberg (2000)Google Scholar
  22. 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)CrossRefMathSciNetGoogle Scholar
  23. Gold, R.: Maximal recursive sequences with 3-valued recursive crosscorrelation functions. IEEE Trans. Inform. Theory 14, 154–156 (1968)zbMATHCrossRefGoogle Scholar
  24. Hollmann, H., Xiang, Q.: A proof of the Welch and Niho conjectures on crosscorrelations of binary m-sequences. Finite Fields and Their Applications 7, 253–286 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  25. Janwa, H., Wilson, R.: Hyperplane sections of Fermat varieties in P 3 in char. 2 and some applications to cyclic codes. In: Moreno, O., Cohen, G., Mora, T. (eds.) AAECC-10. LNCS, vol. 673, pp. 180–194. Springer, Heidelberg (1993)Google Scholar
  26. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  27. Lachaud, G., Wolfmann, J.: The Weights of the Orthogonals of the Extended Quadratic Binary Goppa Codes. IEEE Trans. Inform. Theory 36, 686–692 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  28. Matsui, M.: Linear cryptanalysis method for DES cipher. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 386–397. Springer, Heidelberg (1994)Google Scholar
  29. Nakagawa, N., Yoshiara, S.: A construction of differentially 4-uniform functions from commutative semifields of characteristic 2. In: Proceedings of WAIFI 2007, LNCS (2007)Google Scholar
  30. Nyberg, K.: Differentially uniform mappings for cryptography. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 55–64. Springer, Heidelberg (1994)Google Scholar
  31. Nyberg, K.: S-boxes and Round Functions with Controllable Linearity and Differential Uniformity. In: Preneel, B. (ed.) Fast Software Encryption. LNCS, vol. 1008, pp. 111–130. Springer, Heidelberg (1995)Google Scholar
  32. Sidelnikov, V.: On mutual correlation of sequences. Soviet Math. Dokl. 12, 197–201 (1971)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Lilya Budaghyan
    • 1
  1. 1.Department of Mathematics, University of Trento, I-38050 Povo (Trento)Italy

Personalised recommendations