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If a generalised butterfly is APN then it operates on 6 bits

  • Anne Canteaut
  • Léo PerrinEmail author
  • Shizhu Tian
Article
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Part of the following topical collections:
  1. Special Issue on Sequences and Their Applications

Abstract

Whether there exist Almost Perfect Non-linear permutations (APN) operating on an even number of bits is the so-called Big APN Problem. It has been solved in the 6-bit case by Dillon et al. in 2009 but, since then, the general case has remained an open problem. In 2016, Perrin et al. discovered the butterfly structure which contains Dillon et al.’s permutation over \(\mathbb {F}_{2^{6}}\). Later, Canteaut et al. generalised this structure and proved that no other butterflies with exponent 3 can be APN. Recently, Yongqiang et al. further generalized the structure with Gold exponent and obtained more differentially 4-uniform permutations with optimal nonlinearity. However, the existence of more APN permutations in their generalization was left as an open problem. In this paper, we adapt the proof technique of Canteaut et al. to handle all Gold exponents and prove that a generalised butterfly with Gold exponents over \(\mathbb {F}_{2^{n}}\) can never be APN when n > 3. More precisely, we prove that such a generalised butterfly being APN implies that the branch size is strictly smaller than 5. Hence, the only APN butterflies operate on 3-bit branches, i.e. on 6 bits in total.

Keywords

Boolean function Sbox APN Differential uniformity Butterflies 

Mathematics Subject Classification (2010)

94C10 11T71 94A60 

Notes

Acknowledgments

The work of Léo Perrin was supported by the Fondation Sciences Mathématiques de Paris. The work of Shizhu Tian was supported by the National Science Foundation of China (No. 61772517, 61772516). The authors thank the anonymous reviewers for their careful reading and for their valuable comments.

References

  1. 1.
    Berger, T.P., Canteaut, A., Charpin, P., Laigle-Chapuy, Y.: On almost perfect nonlinear functions over \(\mathbf {F}_{2}^{n}\). IEEE Trans. Inf. Theory 52(9), 4160–4170 (2006)CrossRefzbMATHGoogle Scholar
  2. 2.
    Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. In: Menezes, A.J., Vanstone, S.A. (eds.) CRYPTO’90, volume 537 of LNCS, pp 2–21. Springer, Heidelberg (1991)Google Scholar
  3. 3.
    Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Browning, K.A., Dillon, J.F., McQuistan, M.T., Wolfe, A.J.: An APN permutation in dimension six. In: Finite Fields: Theory and Applications - FQ9, volume 518 of Contemporary Mathematics, pp. 33–42. AMS (2010)Google Scholar
  5. 5.
    Canteaut, A., Duval, S., Perrin, L.: A generalisation of Dillon’s APN permutation with the best known differential and nonlinear properties for all fields of size 24k+ 2. IEEE Trans. Inf. Theory 63(11), 7575–7591 (2017)CrossRefzbMATHGoogle Scholar
  6. 6.
    Carlet, C., Charpin, P., Zinoviev, V.A.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15(2), 125–156 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chabaud, F., Vaudenay, S.: Links between differential and linear cryptanalysis. In: De Santis, A. (ed.) EUROCRYPT’94, volume 950 of LNCS, pp 356–365. Springer, Heidelberg (1995)Google Scholar
  8. 8.
    Fu, S., Feng, X., Wu, B.: Differentially 4-uniform permutations with the best known nonlinearity from butterflies. IACR Trans. Symm. Cryptol. 2017(2), 228–249 (2017)Google Scholar
  9. 9.
    Helleseth, T., Kholosha, A.: On the equation \(x^{2^{l}+ 1}+x+a = 0\) over G F(2k). Finite Fields Appl. 14(1), 159–176 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hou, X.d.: Affinity of permutations of \(\mathbb {F}_{2^{n}}\). Discret. Appl. Math. 154(2), 313–325 (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Li, Y., Tian, S., Yu, Y., Wang, M.: On the generalization of butterfly structure. IACR Trans. Symm. Cryptol. 2018(1), 160–179 (2018)Google Scholar
  12. 12.
    Li, Y., Wang, M.: Constructing S-boxes for lightweight cryptography with Feistel structure. In: Batina, L., Robshaw, M. (eds.) CHES 2014, volume 8731 of LNCS, pp 127–146. Springer, Heidelberg (2014)Google Scholar
  13. 13.
    Matsui, M.: Linear cryptanalysis method for DES cipher. In: Helleseth, T. (ed.) EUROCRYPT’93, volume 765 of LNCS, pp 386–397. Springer, Heidelberg (1994)Google Scholar
  14. 14.
    Nyberg, K.: Differentially uniform mappings for cryptography. In: Helleseth, T. (ed.) EUROCRYPT’93, volume 765 of LNCS, pp 55–64. Springer, Heidelberg (1994)Google Scholar
  15. 15.
    Nyberg, K., Knudsen, L.R.: Provable security against differential cryptanalysis (rump session). In: Brickell, E.F. (ed.) CRYPTO’92, volume 740 of LNCS, pp 566–574. Springer, Heidelberg (1993)Google Scholar
  16. 16.
    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.) CRYPTO 2016, Part II, volume 9815 of LNCS, pp 93–122. Springer, Heidelberg (2016)Google Scholar
  17. 17.
    Yu, Y., Wang, M., Li, Y.: A matrix approach for constructing quadratic APN, functions. Des Codes Cryptogr. 73(2), 587–600 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.InriaParisFrance
  2. 2.State Key Laboratory of Information SecurityInstitute of Information Engineering, Chinese Academy of SciencesBeijingChina
  3. 3.School of Cyber SecurityUniversity of Chinese Academy of SciencesBeijingChina

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