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.
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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)
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)
Beierle, C., Carlet, C., Leander, G., Perrin, L.: A further study of quadratic APN permutations in dimension nine, arXiv:2104.08008 (2021)
Beierle, C., Leander, G.: New instances of quadratic APN functions, arXiv:2009.07204 (2020)
Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991)
Budaghyan, L., Calderini, M., Villa, I.: On equivalence between known families of quadratic APN functions. Finite Fields Appl. 101704, 66 (2020)
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)
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)
Canteaut, A., Perrin, L., Tian, S.: If a generalised butterfly is APN then it operates on 6 bits. Cryptogr. Commun. 11, 1147—1164 (2019)
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)
Carlet, C.: Boolean Functions for Cryptography and Coding Theory. Cambridge University Press, Cambridge (2021)
Carlet, C., Kim, K.H., Mesnager1, S.: A direct proof of APN-ness of the Kasami function. Des. Codes Cryptogr. 89, 441–446 (2021)
Delgado, M.: The state of the art on the conjecture of exceptional APN functions. Note Mat. 37, 41–51 (2017)
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)
Hartshorne, R., Geometry, Algebraic: Graduate Texts in Mathematics. Springer-Verlag, New York (1977)
Hirschfeld, J.W.P., Korchmáros, G., Torres, F.: Algebraic curves over a finite field, Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2008)
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)
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)
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.
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Bartoli, D., Timpanella, M. On a conjecture on APN permutations. Cryptogr. Commun. 14, 925–931 (2022). https://doi.org/10.1007/s12095-022-00558-7