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Algebraic cryptanalysis of stream ciphers using decomposition of Boolean function

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Abstract

Algebraic attack is an important attack strategy against symmetric ciphers, particularly stream ciphers. The most vital issue of this attack strategy is to reduce the degree of the algebraic equations as much as possible in order to obtain a lower time complexity. This paper first presents one such means of obtaining low degree equations using the decomposition of Boolean functions. This method overcomes the three major drawbacks of fast algebraic attack. We discuss the general attack strategy using decomposable Boolean function. We also demonstrate the decomposition of some Boolean function used in practical stream ciphers. Then we find a bound on the degree of a function to be multiplied with a given function so that the product has low degree decomposition. The second major contribution of this paper is a new probabilistic algebraic attack for LFSR based stream cipher by using decomposition of Boolean function. Finally we apply our method to the stream cipher Grain-v1, which is one of the finalist of estream call for stream cipher proposals, by injecting fault in one bit of NFSR.

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References

  1. Armknecht, F.: Improving fast algebraic attacks. In: Fast Software Encryption, pp. 65–82. Springer, Berlin (2004)

  2. Braeken, A., Preneel, B.: Probabilistic algebraic attacks. In: Cryptography and Coding, pp. 290–303. Springer, Berlin (2005)

  3. Cid, C., Kiyomoto, S., Kurihara, J.: The RAKAPOSHI stream cipher. In: Information and Communications Security, pp. 32–46 (2009)

  4. Courtois, N.: Fast algebraic attacks on stream ciphers with linear feedback. In: Advances in Cryptology—CRYPTO 2003, pp. 176–194 (2003)

  5. Courtois, N., Meier, W.: Algebraic attacks on stream ciphers with linear feedback. In: Advances in Cryptology—EUROCRYPT 2003, pp. 345–359 (2003)

  6. Courtois, N., Klimov, A., Patarin, J., Shamir, A.: Efficient algorithms for solving overdefined systems of multivariate polynomial equations. In: Advances in Cryptology—EUROCRYPT 2000, pp. 392–407. Springer, Heidelberg (2000)

  7. Courtois, N., O’Neil, S., Quisquater, J.J.: Practical algebraic attacks on the Hitag2 stream cipher. In: Information Security, pp. 167–176 (2009)

  8. Crama, Y., Hammer, P.L.: Boolean Models and Methods in Mathematics, Computer Science, and Engineering. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  9. Cusick, T.W., Stănică, P.: Cryptographic Boolean Functions and Applications. Academic Press, Amsterdam (2009)

    Google Scholar 

  10. Dawson, E., Clark, A., Golic, J., Millan, W., Penna, L., Simpson, L.: The LILI-128 keystream generator. In: Proceedings of First NESSIE Workshop. Citeseer, Leuven (2000)

  11. Faugre, J.C.: A new efficient algorithm for computing Gröbner bases (F4). J. Pure Appl. Algebra 139(1), 61–88. http://www-salsa.lip6.fr/~jcf/Papers/F99a (1999)

  12. Faugre, J.C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F5). In: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation. ISSAC ’02, pp. 75–83. ACM, New York. http://www-salsa.lip6.fr/~jcf/Papers/F02a (2002)

  13. Hell, M., Johansson, T., Maximov, A., Meier, W.: A stream cipher proposal: Grain-128. In: 2006 IEEE International Symposium on Information Theory, pp. 1614–1618. IEEE, New York (2006)

  14. Hell, M., Johansson, T., Meier, W.: Grain: a stream cipher for constrained environments. Int. J. Wirel. Mob. Comput. 2(1), 86–93 (2007)

    Article  Google Scholar 

  15. Karmakar, S., Chowdhury, D.R.: Fault analysis of Grain-128 by targeting NFSR. In: Progress in Cryptology—AFRICACRYPT 2011, pp. 298–315. Springer, Berlin (2011)

  16. Meier, W., Pasalic, E., Carlet, C.: Algebraic attacks and decomposition of Boolean functions. In: Advances in Cryptology—EUROCRYPT 2004, pp. 474–491. Springer, Heidelberg (2004)

  17. Mihaljević, M.J., Gangopadhyay, S., Paul, G., Imai, H.: Internal state recovery of keystream generator LILI-128 based on a novel weakness of the employed Boolean function. Inf. Process. Lett. 112(21), 805–810 (2012a)

    Article  MATH  Google Scholar 

  18. Mihaljević, M.J., Gangopadhyay, S., Paul, G., Imai, H.: Internal state recovery of grain-v1 employing normality order of the filter function. Inf. Secur. IET 6(2), 55–64 (2012b)

    Article  Google Scholar 

  19. Segers, A.: Algebraic attacks from a Gröbner basis perspective. Master’s Thesis (2004)

Download references

Acknowledgments

The authors would like to thank the anonymous reviewers of this paper for their valuable comments and suggestions to improve the paper.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    Dibyendu Roy, Pratish Datta & Sourav Mukhopadhyay

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  1. Dibyendu Roy
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  2. Pratish Datta
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  3. Sourav Mukhopadhyay
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Correspondence to Dibyendu Roy.

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Roy, D., Datta, P. & Mukhopadhyay, S. Algebraic cryptanalysis of stream ciphers using decomposition of Boolean function. J. Appl. Math. Comput. 49, 397–417 (2015). https://doi.org/10.1007/s12190-014-0845-7

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  • DOI: https://doi.org/10.1007/s12190-014-0845-7

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