Advertisement

Cryptography and Communications

, Volume 11, Issue 6, pp 1165–1184 | Cite as

Changing APN functions at two points

  • Nikolay S. KaleyskiEmail author
Article
Part of the following topical collections:
  1. Special Issue on Boolean Functions and Their Applications

Abstract

We investigate a construction in which a vectorial Boolean function G is obtained from a given function F over \(\mathbb {F}_{2^{n}}\) by changing the values of F at two points of the underlying field. In particular, we examine the possibility of obtaining one APN function from another in this way. We characterize the APN-ness of G in terms of the derivatives and in terms of the Walsh coefficients of F. We establish that changing two points of a function F over \(\mathbb {F}_{2^{n}}\) which is plateaued (and, in particular, AB) or of algebraic degree deg(F) < n − 1 can never give a plateaued (and AB, in particular) function for any n ≥ 5. We also examine a particular case in which we swap the values of F at two points of \(\mathbb {F}_{2^{n}}\). This is motivated by the fact that such a construction allows us to obtain one permutation from another. We obtain a necessary and sufficient condition for the APN-ness of G which we then use to show that swapping two points of any power function over a field \(\mathbb {F}_{2^{n}}\) with n ≥ 5 can never produce an APN function. We also list some experimental results indicating that the same is true for the switching classes from Edel and Pott (Adv. Math. Commun. 3(1):59–81, 2009), and conjecture that the Hamming distance between two APN functions cannot be equal to two for n ≥ 5.

Keywords

Almost perfect nonlinear functions Differential uniformity Boolean functions 

Mathematics Subject Classification (2010)

94A60 06E30 

Notes

Acknowledgments

This research was co-funded by the Trond Mohn Foundation (formerly the Bergen Research Foundation). I would like to thank my supervisors Lilya Budaghyan and Claude Carlet, as well as Tor Helleseth and Nian Li for their ongoing interest and support.

References

  1. 1.
    Budaghyan, L.: The Equivalence of Almost Bent and Almost Perfect Nonlinear Functions and Their Generalizations. Ph.D. Thesis, Otto-von-Guericke-Universität Magdeburg, Universitätsbibliothek (2005)Google Scholar
  2. 2.
    Budaghyan, L., Carlet, C., Helleseth, T., Li, N., Sun, B.: On Upper Bounds for Algebraic Degrees of APN Functions. IEEE Trans. Inf. Theory 64(6), 4399–4411 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Budaghyan, L., Carlet, C., Pott, A.: New Classes of Almost Bent and Almost Perfect Nonlinear Polynomials. IEEE Trans. Inf. Theory 52(3), 1141–1152 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Carlet, C.: Boolean Models and Methods in Mathematics, Computer Science and Engineering: Vectorial Boolean Functions for Cryptography (2010)Google Scholar
  5. 5.
    Carlet, C.: Boolean and Vectorial Plateaued Functions and APN Functions. IEEE Trans. Inf. Theory 61(11), 6272–6289 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Carlet, C., Charpin, P., Zinoviev, V.A.: Codes, Bent Functions and Permutations Suitable for DES-like Cryptosystems. Des. Codes Crypt. 15(2), 125–156 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Carlitz, L.: Explicit Evaluation of Certain Exponential Sums. Math. Scand. 44, 5–16 (1979)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chabaud, F., Vaudenay, S.: Links between Differential and Linear Cryptanalysis. In: Workshop on the Theory and Application of Cryptographic Techniques, EUROCRYPT ’94, vol 950, pp. 356–365 (1994)CrossRefGoogle Scholar
  9. 9.
    Charpin, P., Mesnager, S., Sarkar, S.: Involutions over the Galois Field. IEEE Trans. Inf. Theory 62(4), 2266–2276 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Edel, Y., Pott, A.: A New Almost Perfect Nonlinear Function which is not Quadratic. Adv. Math. Commun. 3(1), 59–81 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Golomb, S.W., Gong, G.: Signal Design for Good Correlation: For Wireless Communication, Cryptography and Radar (2005)Google Scholar
  12. 12.
    Langevin, P.: Covering Radius of RM(1, 9) in RM (3, 9). In: EUROCODE’90, pp. 51–59. Springer (1991)Google Scholar
  13. 13.
    Li, Y., Wang, M., Yu, Y.: Constructing Differentially 4-uniform Permutations over gf(22k) from the Inverse Function Revisited. IACR Cryptol. ePrint Arch. 2013, 731 (2013)Google Scholar
  14. 14.
    Matsui, M.: Linear Cryptanalysis Method For DES Cipher. In: EUROCRYPT ’93 Workshop on the Theory and Application of Cryptographic Techniques on Advances in Cryptology, Pp. 386–397 (1994)CrossRefGoogle Scholar
  15. 15.
    Nyberg, K.: Perfect Nonlinear S-boxes. In: EUROCRYPT’91 Proceedings of the 10th annual international conference on Theory and application of cryptographic techniques, pp. 378–386 (1991)Google Scholar
  16. 16.
    Yu, Y., Wang, M., Li, Y.: Constructing Differentially 4 Uniform Permutations from Known Ones. Chin. J. Electron. 22(3), 495–499 (2013)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of InformaticsUniversity of BergenBergenNorway

Personalised recommendations