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Decoding a Perturbed Sequence Generated by an LFSR

  • Sara D. Cardell
  • Joan-Josep ClimentEmail author
  • Alicia Roca
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10495)

Abstract

Given a sequence of bits produced by a linear feedback shift register (LFSR), the Berlekamp-Massey algorithm finds a register of minimal length able to generate the sequence. The situation is different when the sequence is perturbed; for instance, when it is sent through a transmission channel. LFSRs can be described as autonomous systems. A perturbed sequence of bits generated by an LFSR can be interpreted as a codeword in the binary linear code generated by the corresponding observability matrix. The problem of finding the original sequence can then be stated as the decoding problem, “given the received codeword, find the information transmitted”. We propose two decoding algorithms, one based on a brute force attack and the other one based on the representation technique of the syndromes introduced by Becker, Joux, May, and Meurer (2012).

Keywords

LFSR Correlation attack Keystream sequence Companion matrix Autonomous system Syndrome decoding Decoding representation technique 

Notes

Acknowledgements

The first author was supported by FAPESP with number of process 2015/07246-0. The second author was partially supported by grants MIMECO MTM2015-68805-REDT and MTM2015-69138-REDT. The third author was partially supported by grants MINECO MTM2013-40960-P and MTM2015-68805-REDT.

References

  1. 1.
    Ågren, M., Löndahl, C., Hell, M., Johansson, T.: A survey on fast correlation attacks. Crypt. Commun. 4(3–4), 173–202 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Becker, A., Joux, A., May, A., Meurer, A.: Decoding random binary linear codes in 2n/20: how \(1+1=0\) improves information set decoding. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 520–536. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-29011-4_31 CrossRefGoogle Scholar
  3. 3.
    Canteaut, A., Naya-Plasencia, M.: Correlation attacks on combination generators. Crypt. Commun. 4(3–4), 147–171 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chepyzhov, V.V., Johansson, T., Smeets, B.: A simple algorithm for fast correlation attacks on stream ciphers. In: Goos, G., Hartmanis, J., Leeuwen, J., Schneier, B. (eds.) FSE 2000. LNCS, vol. 1978, pp. 181–195. Springer, Heidelberg (2001). doi: 10.1007/3-540-44706-7_13 CrossRefGoogle Scholar
  5. 5.
    Geffe, P.: How to protect data with ciphers that are really hard to break. Electronics 46(1), 99–101 (1973)Google Scholar
  6. 6.
    Golić, J.D.: Cryptanalysis of alleged A5 stream cipher. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 239–255. Springer, Heidelberg (1997). doi: 10.1007/3-540-69053-0_17 Google Scholar
  7. 7.
    Golomb, S.W.: Shift Register-Sequences. Aegean Park Press, Laguna Hill (1982)zbMATHGoogle Scholar
  8. 8.
    Johansson, T., Jönsson, F.: Theoretical analysis of a correlation attack based on convolutional codes. IEEE Trans. Inf. Theory 48(8), 2173–2181 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kailath, T.: Linear Systems. Prentice-Hall, Upper Saddle River (1980)zbMATHGoogle Scholar
  10. 10.
    Knuth, D.E.: The Art of Computer Programming. Sorting and Searching. Addison-Wesley, Boston (1998)zbMATHGoogle Scholar
  11. 11.
    Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, New York (1986)zbMATHGoogle Scholar
  12. 12.
    Lu, P., Huang, L.: A new correlation attack on LFSR sequences with high error tolerance. Prog. Comput. Sci. Appl. Logic 23, 67–83 (2004)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Massey, J.L.: Shift-register synthesis and BCH decoding. IEEE Trans. Inf. Theory 15(1), 122–127 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Meier, W.: Fast correlation attacks: methods and countermeasures. In: Joux, A. (ed.) FSE 2011. LNCS, vol. 6733, pp. 55–67. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-21702-9_4 CrossRefGoogle Scholar
  15. 15.
    Meier, W., Staffelbach, O.: Fast correlation attacks on stream ciphers. In: Barstow, D., Brauer, W., Brinch Hansen, P., Gries, D., Luckham, D., Moler, C., Pnueli, A., Seegmüller, G., Stoer, J., Wirth, N., Günther, C.G. (eds.) EUROCRYPT 1988. LNCS, vol. 330, pp. 301–314. Springer, Heidelberg (1988). doi: 10.1007/3-540-45961-8_28 Google Scholar
  16. 16.
    Meier, W., Staffelbach, O.: Fast correlation attacks on certain stream ciphers. J. Cryptology 1(3), 159–176 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Molland, H., Mathiassen, J.E., Helleseth, T.: Improved fast correlation attack using low rate codes. In: Paterson, K.G. (ed.) Cryptography and Coding 2003. LNCS, vol. 2898, pp. 67–81. Springer, Heidelberg (2003). doi: 10.1007/978-3-540-40974-8_7 CrossRefGoogle Scholar
  18. 18.
    Siegenthaler, T.: Decrypting a class of stream ciphers using ciphertext only. IEEE Trans. Comput. 34(1), 81–85 (1985)CrossRefGoogle Scholar
  19. 19.
    Zhang, B., Wu, H., Feng, D., Bao, F.: A fast correlation attack on the shrinking generator. In: Menezes, A. (ed.) CT-RSA 2005. LNCS, vol. 3376, pp. 72–86. Springer, Heidelberg (2005). doi: 10.1007/978-3-540-30574-3_7 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Sara D. Cardell
    • 1
  • Joan-Josep Climent
    • 2
    Email author
  • Alicia Roca
    • 3
  1. 1.Instituto de Matemática, Estatística e Computação CientíficaUniversidade Estadual de Campinas (UNICAMP)CampinasBrazil
  2. 2.Departament de MatemàtiquesUniversitat d’AlacantAlacantSpain
  3. 3.Departamento de Matemática Aplicada, IMMUniversitat Politècnica de ValènciaValènciaSpain

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