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Involutory differentially 4-uniform permutations from known constructions

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Abstract

Substitution boxes (S-boxes) are important components of block ciphers that can cause confusion in cryptosystems. The functions used as S-boxes should have low differential uniformity, high nonlinearity and high algebraic degree. When \(k>3\), due to the lack of knowledge about the existence of almost perfect nonlinear permutations over \(\mathbb {F}_{2^{2k}}\), which can offer optimal resistance to the differential cryptanalysis, S-boxes are often constructed from differentially 4-uniform permutations. To date, many infinite families of such functions have been constructed. In addition, the lower hardware implementation cost of S-boxes is an important criterion in the design of block ciphers. If the S-box is an involution, which means that the permutation is its own compositional inverse, then the implementation cost for its inverse can be saved. The same hardware circuit can thus be used for both encryption and decryption, which is an advantage in hardware implementation. In this paper, we investigate all of the differentially 4-uniform permutations that are known in the literature and determine whether they can be involutory. We find that some involutory differentially 4-uniform permutations with high nonlinearity and algebraic degree can be given from these known constructions. We also give some partial results and computer experiments to consider the problem of whether a permutation can be affine equivalent to an involution or it will become an involution upon adding an affine function. Some new families of differentially 4-uniform involutions constructed by composing the inverse function and cycles with length 3 are also given. This family of constructions has a high nonlinearity and a maximum algebraic degree.

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References

  1. Banik S., Bogdanov A., Isobe T., Shibutani K., Hiwatari H., Akishita T., Regazzoni F.: Midori: a block cipher for low energy. In: Advances in Cryptology—ASIACRYPT 2015—21st International Conference on the Theory and Application of Cryptology and Information Security, Auckland, New Zealand, November 29–December 3, 2015, Proceedings, Part II, pp. 411–436 (2015).

  2. Borghoff J., Canteaut A., Güneysu T., Kavun E.B., Knezevic M., Knudsen L.R., Leander G., Nikov V., Paar C., Rechberger C., Rombouts P., Thomsen S., Yalçin T.: PRINCE—a low-latency block cipher for pervasive computing applications—extended abstract. In: Advances in Cryptology—ASIACRYPT 2012—18th International Conference on the Theory and Application of Cryptology and Information Security, Beijing, China, December 2–6, 2012. Proceedings, pp. 208–225 (2012).

  3. Browning K.A., Dillon J.F., McQuistan M.T., Wolfe A.J.: An APN permutation in dimension six. In: Postproceedings of the 9th International Conference on Finite Fields and Their Applications Fq’9. Contemporary Mathematics, vol. 518, pp. 33–42. AMS (2010).

  4. Biryukov A.: Analysis of involutional ciphers: Khazad and Anubis. In: 10th International Workshop Fast Software Encryption, FSE 2003, Lund, Sweden, February 24–26, 2003, Revised Papers, pp. 45–53 (2003).

  5. Bracken C., Leander G.: A highly nonlinear differentially 4 uniform power mapping that permutes fields of even degree. Finite Fields Appl. 16(4), 231–242 (2010).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Bracken C., How Tan C., Tan Y.: Binomial differentially 4 uniform permutations with high nonlinearity. Finite Fields Appl. 18(3), 537–546 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  8. Carlet C.: Vectorial Boolean functions for cryptography. In: Crama Y., Hammer P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, vol. 134, pp. 398–471. Encyclopedia of Mathematics and its Applications, Chapter 9Cambridge University Press, New York (2010).

    Chapter  MATH  Google Scholar 

  9. Carlet C.: On known and new differentially uniform functions. In: Proceedings of the 16th Australasian Conference Information Security and Privacy, ACISP 2011, Melbourne, Australia, July 11–13, 2011, pp. 1–15 (2011).

  10. Carlet C., Charpin P., Zinoviev V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15(2), 125–156 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 \(2^{4k+2}\). IEEE Trans. Inf. Theory 63(11), 7575–7591 (2017).

    Article  MATH  MathSciNet  Google Scholar 

  12. Chen X., Deng Y., Zhu M., Qu L.: An equivalent condition on the switching construction of differentially 4-uniform permutations on from the inverse function. Int. J. Comput. Math. 94, 1–16 (2016).

    MathSciNet  Google Scholar 

  13. Canteaut A., Roué J.: On the behaviors of affine equivalent sboxes regarding differential and linear attacks. In: Proceedings of the 34th Annual International Conference on the Theory and Applications of Cryptographic Techniques Advances in Cryptology (EUROCRYPT 2015), Sofia, Bulgaria, April 26–30, 2015, Part I, pp. 45–74 (2015).

  14. Carlet C., Tang D., Tang X., Liao Q.: New construction of differentially 4-uniform bijections. In: Lin D. et al. (eds.) Proceedings of the 9th International Conference on Information Security and Cryptology (Inscrypt 2013), Guangzhou, China, November 27–30, 2013, pp. 22–38. Springer, New York (2014).

  15. Chabaud F., Vaudenay S.: Links between differential and linear cryptanalysis. In: Advances in Cryptology—EUROCRYPT’94, Workshop on the Theory and Application of Cryptographic Techniques, Perugia, Italy, May 9–12, 1994, Proceedings, pp. 356–365 (1994).

  16. Dobbertin H.: One-to-one highly nonlinear power functions on GF(\(2^n\)). Appl. Algebra Eng. Commun. Comput. 9(2), 139–152 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  17. Fu S., Feng X., Wu B.: Differentially 4-uniform permutations with the best known nonlinearity from butterflies. IACR Trans. Symmetric Cryptol. 2017(2), 228–249 (2017).

    Google Scholar 

  18. Grosso V., Leurent G., Standaert F.-X., Varici K., Durvaux F., Gaspar L., Kerckhof S.: SCREAM & iSCREAM side-channel resistant authenticated encryption with masking. Submission to CAESAR, 2014. https://competitions.cr.yp.to/round1/screamv1.pdf.

  19. Gold R.: Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory 14(1), 154–156 (1968).

    Article  MATH  Google Scholar 

  20. Hirschfeld J.W.P.: Projective Geometries Over Finite Fields, 2nd edn. Oxford Mathematical Monographs, Oxford University Press, Oxford (1998).

  21. Kasami T.: The weight enumerators for several classes of subcodes of the 2nd order binary reed-muller codes. Inf. Control 18(4), 369–394 (1971).

    Article  MATH  Google Scholar 

  22. Knudsen L.R.: Truncated and higher order differentials. In: Proceedings of the Second International Workshop on Fast Software Encryption, Leuven, Belgium, 14–16 December 1994, pp. 196–211 (1994).

  23. Kyureghyan G.M.M., Suder V.: On inverses of APN exponents. In: Proceedings of the 2012 IEEE International Symposium on Information Theory, ISIT 2012, Cambridge, MA, USA, July 1–6, 2012, pp. 1207–1211 (2012).

  24. Lai X.: Higher order derivatives and differential cryptanalysis. In: Blahut R.E., Costello D.J., Maurer U., Mittelholzer T. (eds.) Communications and Cryptography: Two Sides of One Tapestry, vol. 276, pp. 227–233. The Springer International Series in Engineering and Computer ScienceSpringer, Boston (1994).

    Chapter  Google Scholar 

  25. Li Y., Wang M.: On EA-equivalence of certain permutations to power mappings. Des. Codes Cryptogr. 58(3), 259–269 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  26. Li Y., Mingsheng W.: Permutation polynomials EA-equivalent to the inverse function over GF(\(2^n\)). Cryptogr. Commun. 3(3), 175–186 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  27. Li Y., Wang M.: Constructing differentially 4-uniform permutations over GF(\(2^{2m}\)) from quadratic APN permutations over GF(\(2^{2m+1})\). Des. Codes Cryptogr. 72(2), 249–264 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  28. Li Y., Wang M., Yu Y.: Constructing differentially 4-uniform permutations over GF(\(2^{2k}\)) from the inverse function revisited. IACR Cryptology ePrint Archive: Report 2013/731, 2013. https://eprint.iacr.org/2013/731.

  29. Matsui M.: Linear cryptanalysis method for DES cipher. In: Advances in Cryptology—EUROCRYPT’93, Workshop on the Theory and Application of of Cryptographic Techniques, Proceedings, Lofthus, Norway, May 23–27, 1993, pp. 386–397 (1993).

  30. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-correcting Codes. North-Holland Mathematical LibraryNorth-Holland Pub. Co., New York (1977).

    MATH  Google Scholar 

  31. Nyberg K.: Differentially uniform mappings for cryptography. InL Advances in Cryptology—EUROCRYPT’93, Workshop on the Theory and Application of of Cryptographic Techniques, Proceedings, Lofthus, Norway, May 23–27, 1993, pp. 55–64 (1993).

  32. Peng J., Tan C.H.: New explicit constructions of differentially 4-uniform permutations via special partitions of \(\mathbb{F}_{2^{2k}}\). Finite Fields Appl. 40, 73–89 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  33. Peng J., Tan C.H.: New differentially 4-uniform permutations by modifying the inverse function on subfields. Cryptogr. Commun. 9(3), 363–378 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  34. Peng J., Tan C.H., Wang Q.: A new family of differentially 4-uniform permutations over \(\mathbb{F}_{2^{2k}}\) for odd \(k\). Sci. China Math. 59(6), 1221–1234 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  35. Perrin L., Udovenko A., Biryukov A.: Cryptanalysis of a theorem: decomposing the only known solution to the big APN problem. In: Advances in Cryptology—CRYPTO 2016—36th Annual International Cryptology Conference, Proceedings, Santa Barbara, CA, USA, August 14–18, 2016, Part II, pp. 93–122 (2016).

  36. Qu L., Tan Y., Li C., Gong G.: More constructions of differentially 4-uniform permutations on \(\mathbb{F}_{2^{2k}}\). Des. Codes Cryptogr. 78(2), 391–408 (2016).

    MathSciNet  MATH  Google Scholar 

  37. Qu L., Tan Y., Tan C.H., Li C.: Constructing differentially 4-uniform permutations over \(\mathbb{F}_{2^{2k}}\) via the switching method. IEEE Trans. Inf. Theory 59(7), 4675–4686 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  38. Tang D., Carlet C., Tang X.: Differentially 4-uniform bijections by permuting the inverse function. Des. Codes Cryptogr. 77(1), 117–141 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  39. Yuyin Y., Wang M., Li Y.: Constructing differentially 4 uniform permutations from known ones. Chin. J. Electron. 22(3), 495–499 (2013).

    Google Scholar 

  40. Zha Z., Lei H., Sun S.: Constructing new differentially 4-uniform permutations from the inverse function. Finite Fields Appl. 25, 64–78 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  41. Zha Z., Lei H., Sun S., Shan J.: Further results on differentially 4-uniform permutations over \(\mathbb{F}_{2^{2m}}\). Sci. China Math. 58(7), 1577–1588 (2015).

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous reviewers and editors for their comments and suggestions, which significantly improved the quality and presentation of this paper. This work was supported by the National Natural Science Foundation of China (No. 61572491 and 11688101), the National Key Research and Development Program of China (No. 2016YFB0800401), and Science and Technology on Communication Security Laboratory (No. 6142103010701).

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Authors and Affiliations

  1. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Shihui Fu & Xiutao Feng

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Shihui Fu

  3. Science and Technology on Communication Security Laboratory, Chengdu, 610041, China

    Xiutao Feng

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  1. Shihui Fu
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  2. Xiutao Feng
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Correspondence to Xiutao Feng.

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Communicated by C. Carlet.

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Fu, S., Feng, X. Involutory differentially 4-uniform permutations from known constructions. Des. Codes Cryptogr. 87, 31–56 (2019). https://doi.org/10.1007/s10623-018-0482-5

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