Verification of Restricted EA-Equivalence for Vectorial Boolean Functions

  • Lilya Budaghyan
  • Oleksandr Kazymyrov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7369)


We present algorithms for solving the restricted extended affine equivalence (REA-equivalence) problem for any m-dimensional vectorial Boolean functions in n variables. The best of them has complexity O(22n + 1) for REA-equivalence F(x) = M 1 ·G(x ⊕ V 2) ⊕ M 3 ·x ⊕ V 1. The algorithms are compared with previous effective algorithms for solving the linear and the affine equivalence problem for permutations by Biryukov et. al [1].


EA-equivalence Matrix Representation S-box Vectorial Boolean Function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Biryukov, A., De Canniere, C., Braeken, A., Preneel, B.: A Toolbox for Cryptanalysis: Linear and Affine Equivalence Algorithms. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 33–50. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Daemen, J., Rijmen, V.: The Design of Rijndael. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  3. 3.
    Kwon, D.: New Block Cipher: ARIA. In: Lim, J.-I., Lee, D.-H. (eds.) ICISC 2003. LNCS, vol. 2971, pp. 432–445. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Oliynykov, R., Gorbenko, I., Dolgov, V., Ruzhentsev, V.: Symmetric block cipher ”Kalyna”. Applied Radio Electronics 6, 46–63 (2007) (in Ukrainian)Google Scholar
  5. 5.
    Oliynykov, R., Gorbenko, I., Dolgov, V., Ruzhentsev, V.: Results of Ukrainian National Public Cryptographic Competition. Tatra Mt. Math. Publ. 47, 99–113 (2010), MathSciNetzbMATHGoogle Scholar
  6. 6.
    Nyberg, K.: Differentially Uniform Mappings for Cryptography. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 55–64. Springer, Heidelberg (1994)Google Scholar
  7. 7.
    Biham, E., Shamir, A.: Differential Cryptanalysis of DES-like Cryptosystems. Journal of Cryptology 4(1), 3–72 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    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
  10. 10.
    Carlet, C.: Vectorial Boolean Functions for Cryptography. In: Crama, Y., Hammer, P. (eds.) Chapter of the Monography Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 398–469. Cambridge University Press (2010)Google Scholar
  11. 11.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Williams, V.V.: Breaking the Coppersmith-Winograd barrier (November 2011),
  13. 13.
    Stein, W.A., et al.: Sage Mathematics Software (Version 4.8.2), The Sage Development Team (2012),

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Lilya Budaghyan
    • 1
  • Oleksandr Kazymyrov
    • 1
  1. 1.Department of InformaticsUniversity of BergenBergenNorway

Personalised recommendations