New Construction of Differentially 4-Uniform Bijections

  • Claude Carlet
  • Deng Tang
  • Xiaohu Tang
  • Qunying Liao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8567)

Abstract

Block ciphers use Substitution boxes (S-boxes) to create confusion into the cryptosystems. For resisting the known attacks on these cryptosystems, the following criteria for functions are mandatory: low differential uniformity, high nonlinearity and not low algebraic degree. Bijectivity is also necessary if the cipher is a Substitution-Permutation Network, and balancedness makes a Feistel cipher lighter. It is well-known that almost perfect nonlinear (APN) functions have the lowest differential uniformity 2 (the values of differential uniformity being always even) and the existence of APN bijections over \(\mathbb {F}_{2^n}\) for even \(n\ge 8\) is a big open problem. In real practical applications, differentially 4-uniform bijections can be used as S-boxes when the dimension is even. For example, the AES uses a differentially 4-uniform bijection over \(\mathbb {F}_{2^8}\). In this paper, we first propose a method for constructing a large family of differentially 4-uniform bijections in even dimensions. This method can generate at least \(\big (2^{n-3}-\lfloor 2^{(n-1)/2-1}\rfloor -1\big )\cdot 2^{2^{n-1}}\) such bijections having maximum algebraic degree \(n-1\). Furthermore, we exhibit a subclass of functions having high nonlinearity and being CCZ-inequivalent to all known differentially 4-uniform power bijections and to quadratic functions.

Keywords

Block cipher Substitution box Differential uniformity CCZ-equivalence Nonlinearity 

References

  1. 1.
    Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bracken, C., Leander, G.: A highly nonlinear differentially 4 uniform power mapping that permutes fields of even degree. Finite Fields Appl. 16(4), 231–242 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Browning, K.A., Dillon, J.F., McQuistan, M.T., Wolfe, A.J.: An APN permutation in dimension six. In: Postproceedings of the 9th International Conference on Finite Fields and their Applications Fq’9. Contemporary Mathematics Journal of American Mathematical Society, vol. 518, pp. 33–42 (2010)Google Scholar
  4. 4.
    Carlet, C.: On known and new differentially uniform functions. In: Parampalli, U., Hawkes, P. (eds.) ACISP 2011. LNCS, vol. 6812, pp. 1–15. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. 5.
    Carlet, C.: Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions. Des. Codes Cryptogr. 59(1–3), 89–109 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Carlet, C.: More constructions of APN and differentially 4-uniform functions by concatenation. Sci. China Math. 56(7), 1373–1384 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Knudsen, L.: Truncated and higher order differentials. In: Preneel, B. (ed.) FSE 1994. LNCS, vol. 1008, pp. 196–211. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  8. 8.
    Li, Y., Wang, M.: Constructing differentially 4-uniform permutations over \(GF(2^{2m+1})\) from quadratic APN permutations over \(GF(2^{2m})\). To appear in Des. Codes Cryptogr. (2012). doi:10.1007/s10623-012-9760-9
  9. 9.
    MacWilliams, F.J., Sloane, N.J.: The Theory of Error-Correcting Codes. North Holland, Amsterdam (1977)MATHGoogle Scholar
  10. 10.
    Matsui, M.: Linear cryptanalysis method for DES cipher. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 386–397. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  11. 11.
    Nyberg, K.: Differentially uniform mappings for cryptography. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 55–64. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  12. 12.
    Shannon, C.E.: Communication theory of secrecy systems. Bell Syst. Tech. J. 28, 656–715 (1949)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lachaud, G., Wolfmann, J.: The weights of the orthogonals of the extended quadratic binary Goppa codes. IEEE Trans. Inf. Theory 36(3), 686–692 (1990)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Qu, L., Tan, Y., Tan, C., Li, C.: Constructing Differentially 4-Uniform Permutations over \({\mathbb{F}}_{2}^{2k}\) via the Switching Method. IEEE Trans. Inf. Theory 59(7), 4675–4686 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Claude Carlet
    • 1
  • Deng Tang
    • 1
    • 2
  • Xiaohu Tang
    • 2
  • Qunying Liao
    • 3
  1. 1.LAGA, Department of MathematicsUniversity of Paris 8, CNRS, UMR 7539Saint-Denis Cedex 02France
  2. 2.Provincial Key Lab of Information Coding and TransmissionInstitute of Mobile Communications, Southwest Jiaotong UniversityChengduChina
  3. 3.Institute of Mathematics and Software ScienceSichuan Normal UniversityChengduChina

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