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Attacking the Filter Generator over GF(2m)

  • Sondre Rønjom
  • Tor Helleseth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4547)

Abstract

We consider the filter generator over GF(2 m ) consisting of a linear feedback shift register of length k that generates a maximal length linear sequence of period 2 mk − 1 over GF(2 m ) and a Boolean function of degree d that combines bits from one element in the shift register (considered as an element in GF(2 m )) and creates a binary output bit z t at any time t. We show how to extend a recent attack by the authors on the binary filter generator to the filter generator over GF(2 m ). The attack recovers the initial state of the filter generator from L keystream bits with complexity O(L), after a pre-computation with complexity \(O(L (log_2 L)^3)\), where L is the linear complexity upper bounded by \(D=\sum_{i=1}^d {n \choose i}\) with n = mk, which is also the number of monomials of degree ≤ d over GF(2). In addition we explain why a function of only one element of the shift register reduces the linear complexity of the keystream significantly, compared to using the function freely on bits from several words in the initial state. We also discuss implications for the WG stream cipher [4].

Keywords

Filter generators m-sequences Boolean functions solving nonlinear equations 

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References

  1. Canteaut, A.: Open problems related to algebraic attacks on stream ciphers. In: Ytrehus, Ø. (ed.) WCC 2005. LNCS, vol. 3969, pp. 120–134. Springer, Heidelberg (2006)Google Scholar
  2. Golomb, S.W., Gong, G.: Gong Signal Design for Good Correlation: For Wireless Communication, Cryptography and Radar. Cambridge University Press, Cambridge (2005)Google Scholar
  3. Hawkes, P., Rose, G.: Rewriting variables: The complexity of fast algebraic attacks on stream ciphers. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 390–406. Springer, Heidelberg (2004)Google Scholar
  4. Nawaz, Y., Gong, G.: The WG stream cipher, eSTREAM project, http://www.cosic.esat.kuleuven.ac.be/ecrypt/stream/ciphers/wg/wg.pdf.
  5. Rønjom, S., Helleseth, T.: A New Attack on the Filter Generator, accepted by IEEE Transactions on Information TheoryGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Sondre Rønjom
    • 1
  • Tor Helleseth
    • 1
  1. 1.The Selmer Center, Department of Informatics, University of Bergen, PB 7800, N-5020 BergenNorway

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