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Australasian Conference on Information Security and Privacy

ACISP 2018: Information Security and Privacy pp 207–224Cite as

Bounds on Differential and Linear Branch Number of Permutations

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10946))

Abstract

Nonlinear permutations (S-boxes) are key components in block ciphers. The differential branch number measures the diffusion power of a permutation, whereas the linear branch number measures resistance against linear cryptanalysis. There has not been much analysis done on the differential branch number of nonlinear permutations of \(\mathbb {F}_2^n\), although it has been well studied in case of linear permutations. Similarly upper bounds for the linear branch number have also not been studied in general. In this paper we obtain bounds for both the differential and the linear branch number of permutations (both linear and nonlinear) of \(\mathbb {F}_2^n\). We also prove that in the case of \(\mathbb {F}_2^4\), the maximum differential branch number can be achieved only by affine permutations.

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Notes

  1. 1.

    A matrix obtained by permuting rows (or columns) of an identity matrix.

References

  1. Bounds on the minimum distance of linear codes over GF(2). http://www.codetables.de/BKLC/Tables.php?q=2&n0=1&n1=256&k0=1&k1=256. Accessed 25 Aug 2017

  2. Banik, S., Pandey, S.K., Peyrin, T., Sasaki, Y., Sim, S.M., Todo, Y.: GIFT: a small present. In: Fischer, W., Homma, N. (eds.) CHES 2017. LNCS, vol. 10529, pp. 321–345. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66787-4_16

    Chapter  Google Scholar 

  3. Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. In: Menezes, A.J., Vanstone, S.A. (eds.) CRYPTO 1990. LNCS, vol. 537, pp. 2–21. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-38424-3_1

    Chapter  Google Scholar 

  4. Bogdanov, A., Knudsen, L.R., Leander, G., Paar, C., Poschmann, A., Robshaw, M.J.B., Seurin, Y., Vikkelsoe, C.: PRESENT: an ultra-lightweight block cipher. In: Paillier, P., Verbauwhede, I. (eds.) CHES 2007. LNCS, vol. 4727, pp. 450–466. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74735-2_31

    Chapter  Google Scholar 

  5. Carlet, C.: Vectorial Boolean functions for cryptography. In: Crama, P.H.Y. (ed.) Boolean Methods and Models. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  6. Daemen, J., Rijmen, V.: The wide trail design strategy. In: Honary, B. (ed.) Cryptography and Coding 2001. LNCS, vol. 2260, pp. 222–238. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45325-3_20

    Chapter  Google Scholar 

  7. Daemen, J., Rijmen, V.: The Design of Rijndael: AES - The Advanced Encryption Standard. Information Security and Cryptography. Springer, Heidelberg (2002). https://doi.org/10.1007/978-3-662-04722-4

    Book  MATH  Google Scholar 

  8. Griesmer, J.: A bound for error-correcting codes. IBM J. Res. Dev. 7, 532–542 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jean, J.: TikZ for Cryptographers (2016). https://www.iacr.org/authors/tikz/

  10. Leander, G., Poschmann, A.: On the classification of 4 bit S-boxes. In: Carlet, C., Sunar, B. (eds.) WAIFI 2007. LNCS, vol. 4547, pp. 159–176. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73074-3_13

    Chapter  MATH  Google Scholar 

  11. Macwilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes (North-Holland Mathematical Library). North Holland, January 1983

    Google Scholar 

  12. Matsui, M.: Linear Cryptanalysis method for DES cipher. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 386–397. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-48285-7_33

    Chapter  Google Scholar 

  13. Saarinen, M.-J.O.: Cryptographic analysis of all \(4 \times 4\)-bit s-boxes. In: Miri, A., Vaudenay, S. (eds.) SAC 2011. LNCS, vol. 7118, pp. 118–133. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-28496-0_7

    Chapter  Google Scholar 

  14. Shannon, C.E.: Communication theory of secrecy systems. Bell Syst. Tech. J. 28, 656–715 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  15. Shirai, T., Shibutani, K., Akishita, T., Moriai, S., Iwata, T.: The 128-bit blockcipher CLEFIA (extended abstract). In: Biryukov, A. (ed.) FSE 2007. LNCS, vol. 4593, pp. 181–195. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74619-5_12

    Chapter  Google Scholar 

  16. Siegenthaler, T.: Correlation-immunity of nonlinear combining functions for cryptographic applications (corresp.). IEEE Trans. Inf. Theory 30(5), 776–780 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, X., Zheng, Y., Imai, H.: Relating differential distribution tables to other properties of of substitution boxes. Des. Codes Crypt. 19(1), 45–63 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. TCS Innovation Labs, Hyderabad, India

    Sumanta Sarkar & Habeeb Syed

Authors
  1. Sumanta Sarkar
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  2. Habeeb Syed
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Correspondence to Sumanta Sarkar .

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Editors and Affiliations

  1. University of Wollongong, Wollongong, NSW, Australia

    Willy Susilo

  2. University of Wollongong, Wollongong, NSW, Australia

    Guomin Yang

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Sarkar, S., Syed, H. (2018). Bounds on Differential and Linear Branch Number of Permutations. In: Susilo, W., Yang, G. (eds) Information Security and Privacy. ACISP 2018. Lecture Notes in Computer Science(), vol 10946. Springer, Cham. https://doi.org/10.1007/978-3-319-93638-3_13

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  • DOI: https://doi.org/10.1007/978-3-319-93638-3_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-93637-6

  • Online ISBN: 978-3-319-93638-3

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