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Further Results on Multiples of Primitive Polynomials and Their Products over GF(2)

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Book cover Information and Communications Security (ICICS 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2513))

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Abstract

Recently the problem of analysing the multiples of primitive polynomials and their products has received a lot of attention. These primitive polynomials are basically the connection polynomials of the LFSRs (Linear Feedback Shift Registers) used in the stream cipher system. Analysis of sparse multiples of a primitive polynomial or product of primitive polynomials helps in identifying the robustness of the stream ciphers based on nonlinear combiner model. In this paper we first prove some important results related to the degree of the multiples. Earlier these results were only observed for small examples. Proving these results clearly identify the statistical behavior related to the degree of multiples of primitive polynomials or their products. Further we discuss a randomized algorithm for finding sparse multiples of primitive polynomials and their products. Our results clearly identify the time memory trade off for finding such multiples.

This work has been done as a part of M. Tech. (Computer Science) dissertation work at Indian Statistical Institute, Calcutta.

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References

  1. A. Canteaut and M. Trabbia. Improved fast correlation attacks using parity-check equations of weight 4 and 5. In Advances in Cryptology-EURO CRYPT 2000, number 1807 in LNCS, pages 573–588. Springer Verlag, 2000.

    Chapter  Google Scholar 

  2. C. Ding, G. Xiao, and W. Shan. The Stability Theory of Stream Ciphers. Number 561 in Lecture Notes in Computer Science. Springer-Verlag, 1991.

    MATH  Google Scholar 

  3. S. W. Golomb. Shift Register Sequences. Aegean Park Press, 1982.

    Google Scholar 

  4. K. C. Gupta and S. Maitra. Primitive polynomials over GF(2)-A cryptologic approach. In ICICS 2001, number 2229 in LNCS, Pages 23–34, November 2001.

    Google Scholar 

  5. K. C. Gupta and S. Maitra. Multiples of primitive polynomials over GF(2). INDOCRYPT 2001, number 2247 in LNCS, Pages 62–72, December 2001.

    Chapter  Google Scholar 

  6. K. Jambunathan. On choice of connection polynomials for LFSR based stream ciphers. INDOCRYPT 2000, number 1977 in LNCS, Pages 9–18, 2000.

    Google Scholar 

  7. G. A. Jones and J. M. Jones. Elementary Number Theory. Springer Verlag London Limited, 1998.

    Google Scholar 

  8. T. Johansson and F. Jonsson. Fast correlation attacks through reconstruction of linear polynomials. In Advances in Cryptology-CRYPTO 2000, number 1880 in Lecture Notes in Computer Science, pages 300–315. Springer Verlag, 2000.

    Google Scholar 

  9. R. Lidl and H. Niederreiter. Introduction to finite fields and their applications. Cambridge University Press, 1994.

    Google Scholar 

  10. F. J. MacWillams and N. J. A. Sloane. The Theory of Error Correcting Codes. North Holland, 1977.

    Google Scholar 

  11. A. J. Menezes, P. C. van Oorschot, and S. A. Vanstone. Handbook of Applied Cryptography. CRC Press, 1997.

    Google Scholar 

  12. S. Maitra, K. C. Gupta and A. Venkateswarlu. Multiples of Primitive Polynomials and Their Products over GF(2). In SAC 2002, August 2002, pages 218–234 in pre-proceedings, proceedings to be published in Lecture Notes in Computer Science.

    Google Scholar 

  13. W. Meier and O. Stafflebach. Fast correlation attacks on certain stream ciphers. Journal of Cryptology, 1:159–176, 1989.

    Article  MATH  Google Scholar 

  14. T. Siegenthaler. Correlation-immunity of nonlinear combining functions for cryptographic applications. IEEE Transactions on Information Theory, IT-30(5):776–780, September 1984.

    Google Scholar 

  15. T. Siegenthaler. Decrypting a class of stream ciphers using ciphertext only. IEEE Transactions on Computers, C-34(1):81–85, January 1985.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Dept. of Computer Sc. & Engineering, Indian Institute of Technology Madras, Chennai, 600 036, India

    Ayineedi Venkateswarlu

  2. Applied Statistics Unit, Indian Statistical Institute, 203, B T Road, Calcutta, 700 108, India

    Subhamoy Maitra

Authors
  1. Ayineedi Venkateswarlu
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  2. Subhamoy Maitra
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Editor information

Editors and Affiliations

  1. Labs for Information Technology, 21 Heng Mui Keng Terrace, Singapore, 119613

    Robert Deng , Feng Bao  & Jianying Zhou ,  & 

  2. Engineering Research Center for Information Security Technology, Chinese Academy of Sciences, P.O. Box 8718, Beijing, 100080, China

    Sihan Qing

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© 2002 Springer-Verlag Berlin Heidelberg

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Venkateswarlu, A., Maitra, S. (2002). Further Results on Multiples of Primitive Polynomials and Their Products over GF(2). In: Deng, R., Bao, F., Zhou, J., Qing, S. (eds) Information and Communications Security. ICICS 2002. Lecture Notes in Computer Science, vol 2513. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36159-6_20

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  • DOI: https://doi.org/10.1007/3-540-36159-6_20

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00164-5

  • Online ISBN: 978-3-540-36159-6

  • eBook Packages: Springer Book Archive

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