On Verification of Restricted Extended Affine Equivalence of Vectorial Boolean Functions

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9061)

Abstract

Vectorial Boolean functions are used as substitution boxes in cryptosystems. Designing inequivalent functions resistant to known attacks is one of the challenges in cryptography. In doing this, finding a fast technique for determining whether two given functions are equivalent is a significant problem. A special class of the equivalence called restricted extended affine (REA) equivalence is studied in this paper. We update the verification procedures of the REA-equivalence types given in the recent work of Budaghyan and Kazymyrov (2012). In particular, we solve the system of linear equations simultaneously in the verification procedures to get better complexity. We also present the explicit number of operations of the verification procedures of these REA-equivalence types. Moreover, we construct two new REA-equivalence types and present the verification procedures of these types with their complexities.

Keywords

Vectorial Boolean functions EA-equivalence REA-equivalence 

References

  1. 1.
    Biryukov, A., De Canniere, C., Braeken, A., Preneel, B.: A tool-box for cryptanalysis: linear and affine equivalence algorithms. In: Biham, E. (ed.) Advances in Cryptology — EUROCRYPT 2003. LNCS, vol. 2656, pp. 33–50. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system, I. The user language. J. Symb. Comput. 24, 235–265 (1997)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Budaghyan, L., Kazymyrov, O.: Verification of restricted EA-equivalence for vectorial Boolean functions. In: Özbudak, F., Rodríguez-Henríquez, F. (eds.) WAIFI 2012. LNCS, vol. 7369, pp. 108–118. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  4. 4.
    Budaghyan, L., Carlet, C., Pott, A.: New classes of almost bent and almost perfect nonlinear polynomials. IEEE Trans. Inform. Theory 52, 1141–1152 (2006)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Crypt. 15(2), 125–156 (1998)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Carlet, C.: Vectorial Boolean functions for cryptography. Boolean Model. Methods Math. Comput. Sci. Eng. 134, 398–469 (2010)CrossRefGoogle Scholar
  7. 7.
    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) Google Scholar
  8. 8.
    Nyberg, K.: Differentially uniform mappings for cryptography. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 55–64. Springer, Heidelberg (1994) Google Scholar
  9. 9.
    Sınak, A.: On verification of restricted extended affine equivalence of vectorial Boolean functions. Master’s thesis, Middle East Technical University (2012)Google Scholar
  10. 10.
    Williams, V.V.: Breaking the Coppersmith-Winograd barrier, November 2011Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsMiddle East Technical UniversityAnkaraTurkey
  2. 2.Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  3. 3.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria
  4. 4.Department of Mathematics and Computer SciencesNecmettin Erbakan UniversityKonyaTurkey

Personalised recommendations