Cryptography and Communications

, Volume 11, Issue 1, pp 21–39 | Cite as

On an algorithm generating 2-to-1 APN functions and its applications to “the big APN problem”

  • Valeriya Idrisova
Part of the following topical collections:
  1. Special Issue on Boolean Functions and Their Applications


Almost perfect nonlinear (APN) functions are of great interest to many researchers since they have the optimal resistance to the differential attack. The existence of bijective APN functions in even number of variables is an important open problem, and there is only one known example of such a function at present. In this paper we consider a special subclass of 2-to-1 vectorial Boolean functions that can allow us to search and construct APN permutations. We proved that each 2-to-1 function is potentially EA-equivalent to a permutation and proposed an algorithm that generates special symbol sequences for constructing 2-to-1 APN functions. Also, we described two methods for searching APN permutations, that are based on sequences generated by this algorithm.


Boolean function APN function 2-to-1 function APN permutation Differential uniformity S-box 

Mathematics Subject Classification (2010)

94A60 06E30 11T71 



We would like to cordially thank Anastasiya Gorodilova and Natalia Tokareva for useful observations and fruitful discussions all along this work. We sincerely thank Nikolay Kolomeec for his helpful comments and careful reading. We are much indebted to the reviewers for their valuable remarks and for providing the proof of Proposition 5.


  1. 1.
    Agievich, S., Gorodilova, A., Kolomeec, N., Nikova, S., Preneel, B., Rijmen, V., Shushuev, G., Tokareva, N., Vitkup, V.: Problems, solutions and experience of the first international student’s Olympiad in cryptography. Prikladnaya Diskretnaya Matematika 3(29), 5–28 (2015)Google Scholar
  2. 2.
    Berger, T., Canteaut, A., Charpin, P., Laigle-Chapuy, Y.: On almost perfect nonlinear mappings over \(\mathbb {F}_{2^{n}}\). IEEE Trans. Inform. Theory 52(9), 4160–4170 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beth, T., Ding, C.: On almost perfect nonlinear permutations. Advances in Cryptology, EUROCRYPT’93. Lect. Notes Comput. Sci 765, 65–76 (1993)CrossRefGoogle Scholar
  4. 4.
    Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Blondeau, C., Canteaut, A., Charpin, P.: Differential properties of \(x x^{2^{t}-1}\). IEEE Trans. Inf. Theory 57(12), 8127–8137 (2011)CrossRefGoogle Scholar
  6. 6.
    Blondeau, C., Nyberg, K.: Perfect nonlinear functions and cryptography. Finite Fields and Their Applications 32, 120–147 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brinkmann, M., Leander, G.: On the classification of APN functions up to dimension five. Des. Codes Cryptogr. 49(1–3), 273–288 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Browning, K.A., Dillon, J.F., McQuistan, M.T., Wolfe, A.J.: An APN permutation in dimension six. Post-proceedings of the 9-th International Conference on Finite Fields and Their Applications Fq’09. Contemporary Math. AMS 518, 33–42 (2010)CrossRefGoogle Scholar
  9. 9.
    Budaghyan, L.: Construction and Analysis of Cryptographic Functions, vol. VIII, p 168. Springer, Berlin (2014)zbMATHGoogle Scholar
  10. 10.
    Calderini, M., Sala, M., Villa, I.: A note on APN permutations in even dimension. Finite Fields Their Appl. 46, 1–16 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Canteaut, A., Charpin, P., Dobbertin, H.: Binary m-sequences with three-valued crosscorrelation: a proof of Welch conjecture. IEEE Trans. Inf. Theory. 46(1), 4–8 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Canteaut, A., Duval, S., Perrin, L.: A generalisation of Dillon’s APN permutation with the best known differential and linear properties for all fields of size \(2^{4k + 2}\). IACR Cryptology ePrint Archive 2016, 887 (2016)zbMATHGoogle Scholar
  13. 13.
    Carlet, C.: Open Questions on Nonlinearity and on APN Functions. In: Koç, Ç., Mesnager, S., Savaş, E. (eds.) Arithmetic of Finite Fields. WAIFI 2014. Lecture Notes in Computer Science, vol. 9061, pp. 83–107 (2015)Google Scholar
  14. 14.
    Carlet, C.: Vectorial Boolean Functions for Cryptography. Ch. 9 of the monograph Boolean Methods and Models in Mathematics, Computer Science, and Engineering, pp. 398–472. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  15. 15.
    Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15, 125–156 (1998)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Dobbertin, H.: One-to-one highly nonlinear power functions on \(GF(2^{n})\). Appl. Algebra Eng. Commun. Comput. 9(2), 139–152 (1998)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Dobbertin, H.: Almost perfect nonlinear power functions on \({{GF}}(2^{n})\): the Welch case. IEEE Trans. Inf. Theory. 45(4), 1271–1275 (1999)CrossRefGoogle Scholar
  18. 18.
    Dobbertin, H.: Almost perfect nonlinear functions over GFG F(2n): the Niho case. Inform. and Comput. 151, 57–72 (1999)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Dobbertin, H.: Almost perfect nonlinear power functions over \({{GF}}(2^{n})\): a new case for n divisible by 5. Proceedings of Finite Fields and Applications FQ5, 113–121 (2000)Google Scholar
  20. 20.
    Glukhov, M.M.: On the approximation of discrete functions by linear functions. Matematicheskie Voprosy Kriptografii 7(4), 29–50 (2016). (in Russian)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Glukhov, M.M.: On the matrices of transitions of differences for some modular groups. Matematicheskie Voprosy Kriptografii 4(4), 27–47 (2013). (in Russian)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gold, R.: Maximal recursive sequences with 3-valued recursive crosscorrelation functions. IEEE Trans. Inform. Theory 14, 154–156 (1968)CrossRefGoogle Scholar
  23. 23.
    Gorodilova, A.A.: Characterization of almost perfect nonlinear functions in terms of subfunctions. Diskr. Mat. 27(3), 3–16 (2016). Discrete Math. Appl. 26(4), 193–202MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hollmann, H., Xiang, Q.: A proof of the Welch and Niho conjectures on crosscorrelations of binary m-sequences. Finite Fields Their Appl. 7, 253–286 (2001)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hou, X.-D.: Affinity of permutations of \({F_{2}^{n}}\). Discrete Appl. Math. - Special issue: Coding and Cryptography Archive 154(2), 313–325 (2006)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Janwa, H., Wilson, R.: Hyperplane Sections of Fermat Varieties in \(p^{3}\) in char. 2 and some Applications to Cyclic Codes. Proceedings of AAECC-10, Lecture Notes in Computer Science, vol. 673, pp. 180–194. Springer, Berlin (1993)zbMATHGoogle Scholar
  27. 27.
    Kasami, T.: The weight enumerators for several classes of subcodes of the second order binary Reed-Muller codes. Inform. and Control. 18, 369–394 (1971)CrossRefGoogle Scholar
  28. 28.
    Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and its Applications, vol. 20, p 772. Addison-Wesley, Reading (1983)Google Scholar
  29. 29.
    Nyberg, K.: Differentially uniform mappings for cryptography. Advances in Cryptography, EUROCRYPT’93. Lect. Notes Comput. Sci 765, 55–64 (1994)CrossRefGoogle Scholar
  30. 30.
    Nyberg, K.: S-boxes and round functions with controllable linearity and differential uniformity, FSE’94. Lect. Notes Comput. Sci 1008, 111–130 (1994)CrossRefGoogle Scholar
  31. 31.
    Pasalic, E., Charpin, P.: Some results concerning cryptographically significant mappings over \({{GF}}(2^{n})\). Des. Codes Crypt. 57(3), 257–269 (2010)Google Scholar
  32. 32.
    Perrin, L., Udovenko, A., Biryukov, A.: Cryptanalysis of a theorem: decomposing the only known solution to the big APN problem. In: Robshaw, M., Katz, J. (eds.) Advances in Cryptology - CRYPTO 2016, Part II. Lect. Notes Comput. Sci, vol. 9815, pp. 93–122 (2016)Google Scholar
  33. 33.
    Pott, A.: Almost perfect and planar functions. Des. Codes Cryptography 78(1), 141–195 (2016)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Tuzhilin, M.E.: APN functions. Prikladnaya Diskretnaya Matematika 3, 14–20 (2009). (in Russian)Google Scholar
  35. 35.
    Vitkup, V.: On symmetric properties of APN functions. J. Appl. Ind. Math. 10(1), 126–135 (2016)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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