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Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 185–202 | Cite as

Two notions of differential equivalence on Sboxes

  • Christina BouraEmail author
  • Anne Canteaut
  • Jérémy Jean
  • Valentin Suder
Article
  • 184 Downloads
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography

Abstract

In this work, we discuss two notions of differential equivalence on Sboxes. First, we introduce the notion of DDT-equivalence which applies to vectorial Boolean functions that share the same difference distribution table (DDT). Next, we compare this notion to what we call the \(\gamma \)-equivalence, applying to vectorial Boolean functions whose DDTs have the same support. We discuss the relation between these two equivalence notions, demonstrate that the number of DDT- or \(\gamma \)-equivalent functions is invariant under EA- and CCZ-equivalence and provide an algorithm for computing the DDT-equivalence and the \(\gamma \)-equivalence classes of a given function. We study the sizes of these classes for some families of Sboxes. Finally, we prove a result that shows that the rows of the DDT of an APN permutation are pairwise distinct.

Keywords

Boolean function Sbox APN Difference distribution table Equivalence 

Mathematics Subject Classification

94A60 

Notes

Acknowledgements

The authors wish to thank Jean-Pierre Flori, Itai Dinur and Orr Dunkelman for helpful discussions.

References

  1. 1.
    Biham E., Shamir A.: Differential cryptanalysis of DES-like cryptosystems. In: Menezes A.J., Vanstone S.A. (eds.) CRYPTO’90, LNCS, vol. 537, pp. 2–21. Springer, Heidelberg (1991).Google Scholar
  2. 2.
    Blondeau C., Gérard B.: Multiple differential cryptanalysis: theory and practice. In: Joux A. (ed.) FSE 2011, LNCS, vol. 6733, pp. 35–54. Springer, Heidelberg (2011).Google Scholar
  3. 3.
    Blondeau C., Nyberg K.: New links between differential and linear cryptanalysis. In: Johansson T., Nguyen P.Q. (eds.) EUROCRYPT 2013, LNCS, vol. 7881, pp. 388–404. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38348-9_24.
  4. 4.
    Blondeau C., Canteaut A., Charpin P.: Differential properties of power functions. IJICoT 1(2), 149–170 (2010).  https://doi.org/10.1504/IJICOT.2010.032132.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brinkmann M., Leander G.: On the classification of APN functions up to dimension five. Des. Codes Cryptogr. 49(1–3), 273–288 (2008).  https://doi.org/10.1007/s10623-008-9194-6.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Browning K., Dillon J., Kibler R., McQuistan M.: APN polynomials and related codes. J. Comb. Inf. Syst. Sci. 34(1–4), 135–159 (2009).zbMATHGoogle Scholar
  7. 7.
    Browning K., Dillon J., McQuistan M., Wolfe A.: An APN permutation in dimension six. In: Finite Fields: Theory and Applications, Contemporary Mathematics, vol. 518, pp. 33–42. AMS (2010)Google Scholar
  8. 8.
    Canteaut A., Roué J.: On the behaviors of affine equivalent sboxes regarding differential and linear attacks. In: Oswald E., Fischlin M. (eds.) EUROCRYPT 2015, Part I, LNCS, vol. 9056, pp. 45–74. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46800-5_3.
  9. 9.
    Carlet C.: Partially-bent functions. Des. Codes Cryptogr. 3(2), 135–145 (1993).  https://doi.org/10.1007/BF01388412.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Carlet C.: Open questions on nonlinearity and on APN functions. In: Koç Ç.K., Mesnager S., Savaş E. (eds.) Arithmetic of Finite Fields - WAIFI 2014, pp. 83–107. Springer, New York (2015).  https://doi.org/10.1007/978-3-319-16277-5_5.Google Scholar
  11. 11.
    Carlet C., Charpin P., Zinoviev V.: Codes, bent functions and permutations suitable For DES-like cryptosystems. Des. Codes Cryptogr. 15(2), 125–156 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chabaud F., Vaudenay S.: Links between differential and linear cryptoanalysis. In: Santis A.D. (ed.) EUROCRYPT’94, LNCS, vol. 950, pp. 356–365. Springer, Heidelberg (1995).Google Scholar
  13. 13.
    Dobbertin H.: Almost perfect nonlinear power functions on GF(2\(^{\text{ n }}\)): the Welch case. IEEE Trans. Inf. Theory 45(4), 1271–1275 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Edel Y., Pott A.: A new almost perfect nonlinear function which is not quadratic. Adv. Math. Commun. 3(1), 59–81 (2009).  https://doi.org/10.3934/amc.2009.3.59.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Edel Y., Kyureghyan G.M., Pott A.: A new APN function which is not equivalent to a power mapping. IEEE Trans. Inf. Theory 52(2), 744–747 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Flori J.P., Jean J.: libapn C++ Library. https://github.com/ANSSI-FR/libapn (2018)
  17. 17.
    Gorodilova A.: On a remarkable property of APN Gold functions. Cryptology ePrint Archive, Report 2016/286 (2016)Google Scholar
  18. 18.
    Hernando F., McGuire G.: Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions. J. Algebr. 343(1), 78–92 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Knudsen L.R.: Truncated and higher order differentials. In: Preneel B. (ed.) FSE’94, LNCS, vol. 1008, pp. 196–211. Springer, Heidelberg (1995).Google Scholar
  20. 20.
    Knudsen L.R., Leander G., Poschmann A., Robshaw M.J.B.: PRINTcipher: a block cipher for IC-printing. In: Mangard S., Standaert F.X. (eds.) CHES 2010, LNCS, vol. 6225, pp. 16–32. Springer, Heidelberg (2010).Google Scholar
  21. 21.
    Langevin, P.: Classification of Boolean functions under the affine group. http://langevin.univ-tln.fr/project/agl/agl.html (2009)
  22. 22.
    Leander G., Poschmann A.: On the classification of 4 bit s-boxes. In: Carlet C., Sunar B. (eds.) Proceedings of Arithmetic of Finite Fields, First International Workshop, WAIFI 2007, Madrid, Spain, June 21–22, 2007, Lecture Notes in Computer Science, vol. 4547, pp. 159–176. Springer, New York (2007).  https://doi.org/10.1007/978-3-540-73074-3_13
  23. 23.
    Nyberg K.: Differentially uniform mappings for cryptography. In: Helleseth T. (ed.) EUROCRYPT’93, LNCS, vol. 765, pp. 55–64. Springer, Heidelberg (1994).Google Scholar
  24. 24.
    Nyberg K., Knudsen L.R.: Provable security against differential cryptanalysis (rump session). In: Brickell E.F. (ed.) CRYPTO’92, LNCS, vol. 740, pp. 566–574. Springer, Heidelberg (1993).Google Scholar
  25. 25.
    Park S., Sung S.H., Lee S., Lim J.: Improving the upper bound on the maximum differential and the maximum linear Hull probability for SPN structures and AES. In: Johansson T. (ed.) FSE 2003, LNCS, vol. 2887, pp. 247–260. Springer, Heidelberg (2003).Google Scholar
  26. 26.
    Rothaus O.S.: On “bent” functions. J. Comb. Theory Ser. A 20(3), 300–305 (1976).CrossRefzbMATHGoogle Scholar
  27. 27.
    Suder, V.: Antiderivative functions over \({\mathbb{F}}\_{2^n}\). In: Workshop on Coding and Cryptography—WCC 2015 (2015). https://hal.archives-ouvertes.fr/WCC2015/hal-01275708v1
  28. 28.
    Suder V.: Antiderivative functions over \({\mathbb{F}}\_{2^n}\). Des. Codes Cryptogr. 82(1–2), 435–447 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Yu Y., Wang M., Li Y.: A matrix approach for constructing quadratic APN functions. Des. Codes Cryptogr. 73(2), 587–600 (2014).  https://doi.org/10.1007/s10623-014-9955-3.MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of VersaillesVersaillesFrance
  2. 2.InriaParisFrance
  3. 3.ANSSIParisFrance

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