Skip to main content

On a conjecture on APN permutations


The single trivariate representation proposed in [C. Beierle, C. Carlet, G. Leander, L. Perrin, A Further Study of Quadratic APN Permutations in Dimension Nine, arXiv:2104.08008] of the two sporadic quadratic APN permutations in dimension 9 found by Beierle and Leander (2020) is further investigated. In particular, using tools from algebraic geometry over finite fields, we prove that such a family does not contain any other APN permutation for larger dimensions.

This is a preview of subscription content, access via your institution.


  1. Aubry, Y., McGuire, G., Rodier, F.: A few more functions that are not APN infinitely often. Finite Fields: Theory and Applications 518, 23–31 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Bartoli, D.: Hasse-Weil type theorems and relevant classes of polynomial functions. In: London Mathematical Society Lecture Note Series, Proceedings of 28th British Combinatorial Conference. Cambridge University Press, to appear (2020)

  3. Beierle, C., Carlet, C., Leander, G., Perrin, L.: A further study of quadratic APN permutations in dimension nine, arXiv:2104.08008 (2021)

  4. Beierle, C., Leander, G.: New instances of quadratic APN functions, arXiv:2009.07204 (2020)

  5. Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991)

    Article  MathSciNet  Google Scholar 

  6. Budaghyan, L., Calderini, M., Villa, I.: On equivalence between known families of quadratic APN functions. Finite Fields Appl. 101704, 66 (2020)

    MathSciNet  MATH  Google Scholar 

  7. Budaghyan, L., Carlet, C., Leander, G.: Two classes of quadratic APN binomials inequivalent to power functions. IEEE Trans. Inf. Theory 54(9), 4218–4229 (2008)

    Article  MathSciNet  Google Scholar 

  8. Budaghyan, L., Calderini, M., Carlet, C., Coulter, R.S., Villa, I.: Constructing APN functions through isotopic shifts. IEEE Trans. Inf. Theory 66, 5299–5309 (2020)

    Article  MathSciNet  Google Scholar 

  9. Canteaut, A., Perrin, L., Tian, S.: If a generalised butterfly is APN then it operates on 6 bits. Cryptogr. Commun. 11, 1147—1164 (2019)

    Article  MathSciNet  Google Scholar 

  10. Cafure, A., Matera, G.: Improved explicit estimates on the number of solutions of equations over a finite field. Finite Fields Appl. 12, 155–185 (2006)

    Article  MathSciNet  Google Scholar 

  11. Carlet, C.: Boolean Functions for Cryptography and Coding Theory. Cambridge University Press, Cambridge (2021)

    MATH  Google Scholar 

  12. Carlet, C., Kim, K.H., Mesnager1, S.: A direct proof of APN-ness of the Kasami function. Des. Codes Cryptogr. 89, 441–446 (2021)

  13. Delgado, M.: The state of the art on the conjecture of exceptional APN functions. Note Mat. 37, 41–51 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Browning, K., Dillon, J.F., McQuistan, M., Wolfe, A.J.: An APN permutation in dimension six. In: Post-proceedings of the 9-th International conference on finite fields and their applications, american mathematical society, vol. 518, pp 33–42 (2010)

  15. Hartshorne, R., Geometry, Algebraic: Graduate Texts in Mathematics. Springer-Verlag, New York (1977)

    Google Scholar 

  16. Hirschfeld, J.W.P., Korchmáros, G., Torres, F.: Algebraic curves over a finite field, Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2008)

    Book  Google Scholar 

  17. Homma, M., Kim, S.J.: Sziklai’s conjecture on the number of points of a plane curve over a finite field III. Finite Fields Appl. 16, 315–319 (2010)

    Article  MathSciNet  Google Scholar 

  18. Perrin, L., Udovenko, A, Biryukov, A.: Cryptanalysis of a theorem: Decomposing the only known solution to the big APN problem. In: M. Robshaw, J. Katz (eds.) Advances in Cryptology - CRYPTO 2016, Proceedings, Part II, volume 9815 of LNCS, pp 93–122. Springer (2016)

Download references


The research of D. Bartoli and M. Timpanella was partially supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM). The second author was supported by the project ”VALERE: VAnviteLli pEr la RicErca” of the University of Campania ”Luigi Vanvitelli”. The authors are grateful to C. Beierle, C. Carlet, G. Leander, and L. Perrin for a number of valuable comments on an earlier draft.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Daniele Bartoli.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bartoli, D., Timpanella, M. On a conjecture on APN permutations. Cryptogr. Commun. 14, 925–931 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: