Abstract
The joint linear complexity and the joint linear complexity profile are standard complexity measures for multisequences in the context of word-based stream ciphers. The last few years have seen major advances in the theory of these complexity measures, especially with regard to probabilistic results on the behavior of random (periodic and nonperiodic) multisequences. This paper presents a survey of these developments as well as the necessary background for the results.
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References
Armand, M.A.: Multisequence shift register synthesis over commutative rings with identity with applications to decoding cyclic codes over integer residue rings. IEEE Trans. Inform. Theory 50, 220–229 (2004)
Cusick, T.W., Ding, C., Renvall, A.: Stream Ciphers and Number Theory. Elsevier, Amsterdam (1998)
Daemen, J., Clapp, C.S.K.: Fast hashing and stream encryption with PANAMA. In: Vaudenay, S. (ed.) FSE 1998. LNCS, vol. 1372, pp. 60–74. Springer, Heidelberg (1998)
Dai, Z.-D., Feng, X., Yang, J.-H.: Multi-continued fraction algorithm and generalized B-M algorithm over F 2. In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 339–354. Springer, Heidelberg (2005)
Dai, Z.-D., Imamura, K., Yang, J.-H.: Asymptotic behavior of normalized linear complexity of multi-sequences. In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 129–142. Springer, Heidelberg (2005)
Dai, Z.-D., Yang, J.-H.: Linear complexity of periodically repeated random sequences. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 168–175. Springer, Heidelberg (1991)
Dawson, E., Simpson, L.: Analysis and design issues for synchronous stream ciphers. In: Niederreiter, H. (ed.) Coding Theory and Cryptology, pp. 49–90. World Scientific, Singapore (2002)
Ding, C.: Proof of massey’s conjectured algorithm. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 345–349. Springer, Heidelberg (1988)
Ding, C., Shan, W., Xiao, G.: The Stability Theory of Stream Ciphers. LNCS, vol. 561. Springer, Heidelberg (1991)
ECRYPT stream cipher project, available at: http://www.ecrypt.eu.org/stream
Feng, G.-L., Tzeng, K.K.: A generalized Euclidean algorithm for multisequence shift-register synthesis. IEEE Trans. Inform. Theory 35, 584–594 (1989)
Feng, G.-L., Tzeng, K.K.: A generalization of the Berlekamp-Massey algorithm for multisequence shift-register synthesis with applications to decoding cyclic codes. IEEE Trans. Inform. Theory 37, 1274–1287 (1991)
Feng, X., Dai, Z.-D.: Expected value of the linear complexity of two-dimensional binary sequences. In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 113–128. Springer, Heidelberg (2005)
Feng, X.T., Wang, Q.L., Dai, Z.D.: Multi-sequences with d-perfect property. J. Complexity 21, 230–242 (2005)
Fu, F.-W., Niederreiter, H., Su, M.: The expectation and variance of the joint linear complexity of random periodic multisequences. J. Complexity 21, 804–822 (2005)
Gustavson, F.G.: Analysis of the Berlekamp-Massey linear feedback shift-register synthesis algorithm. IBM J. Res. Develop. 20, 204–212 (1976)
Hawkes, P., Rose, G.G.: Exploiting multiples of the connection polynomial in word-oriented stream ciphers. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 303–316. Springer, Heidelberg (2000)
Kamiya, N.: On multisequence shift register synthesis and generalized-minimum-distance decoding of Reed-Solomon codes. Finite Fields Appl. 1, 440–457 (1995)
Massey, J.L., Serconek, S.: Linear complexity of periodic sequences: A general theory. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 358–371. Springer, Heidelberg (1996)
Meidl, W.: Discrete Fourier Transform, Joint Linear Complexity and Generalized Joint Linear Complexity of Multisequences. In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 101–112. Springer, Heidelberg (2005)
Meidl, W., Niederreiter, H.: On the expected value of the linear complexity and the k-error linear complexity of periodic sequences. IEEE Trans. Inform. Theory 48, 2817–2825 (2002)
Meidl, W., Niederreiter, H.: The expected value of the joint linear complexity of periodic multisequences. J. Complexity 19, 61–72 (2003)
Meidl, W., Winterhof, A.: On the joint linear complexity profile of explicit inversive multisequences. J. Complexity 21, 324–336 (2005)
Niederreiter, H.: Sequences with almost perfect linear complexity profile. In: Price, W.L., Chaum, D. (eds.) EUROCRYPT 1987. LNCS, vol. 304, pp. 37–51. Springer, Heidelberg (1988)
Niederreiter, H.: The probabilistic theory of linear complexity. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 191–209. Springer, Heidelberg (1988)
Niederreiter, H.: A combinatorial approach to probabilistic results on the linear-complexity profile of random sequences. J. Cryptology 2, 105–112 (1990)
Niederreiter, H.: Some computable complexity measures for binary sequences. In: Ding, C., Helleseth, T., Niederreiter, H. (eds.) Sequences and Their Applications, pp. 67–78. Springer, London (1999)
Niederreiter, H.: Linear complexity and related complexity measures for sequences. In: Johansson, T., Maitra, S. (eds.) INDOCRYPT 2003. LNCS, vol. 2904, pp. 1–17. Springer, Heidelberg (2003)
Niederreiter, H., Wang, L.-P.: Proof of a conjecture on the joint linear complexity profile of multisequences. In: Maitra, S., Veni Madhavan, C.E., Venkatesan, R. (eds.) INDOCRYPT 2005. LNCS, vol. 3797, pp. 13–22. Springer, Heidelberg (2005)
Niederreiter, H., Wang, L.-P.: The asymptotic behavior of the joint linear complexity profile of multisequences. Monatsh. Math. (to appear)
Niederreiter, H., Xing, C.P.: Rational Points on Curves over Finite Fields: Theory and Applications. London Math. Soc. Lecture Note Series, vol. 285. Cambridge University Press, Cambridge (2001)
Rueppel, R.A.: Analysis and Design of Stream Ciphers. Springer, Berlin (1986)
Rueppel, R.A., ciphers, S.: Contemporary Cryptology: The Science of Information Integrity. In: Simmons, G.J. (ed.), pp. 65–134. IEEE Press, New York (1992)
Sakata, S.: Extension of the Berlekamp-Massey algorithm to N dimensions. Inform. and Comput. 84, 207–239 (1990)
Schmidt, W.M.: Construction and estimation of bases in function fields. J. Number Theory 39, 181–224 (1991)
B. Smeets, The linear complexity profile and experimental results on a randomness test of sequences over the finite field F q , Technical Report, Department of Information Theory, University of Lund, 1988.
Wang, L.-P., Niederreiter, H.: Enumeration results on the joint linear complexity of multisequences. Finite Fields Appl. (to appear), available online as document: doi:10.1016/j.ffa.2005.03.005
Wang, L.-P., Zhu, Y.-F., Pei, D.-Y.: On the lattice basis reduction multisequence synthesis algorithm. IEEE Trans. Inform. Theory 50, 2905–2910 (2004)
Xing, C.P.: Multi-sequences with almost perfect linear complexity profile and function fields over finite fields. J. Complexity 16, 661–675 (2000)
Xing, C., Lam, K.-Y., Wei, Z.: A class of explicit perfect multi-sequences. In: Lam, K.-Y., Okamoto, E., Xing, C. (eds.) ASIACRYPT 1999. LNCS, vol. 1716, pp. 299–305. Springer, Heidelberg (1999)
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Niederreiter, H. (2006). The Probabilistic Theory of the Joint Linear Complexity of Multisequences. In: Gong, G., Helleseth, T., Song, HY., Yang, K. (eds) Sequences and Their Applications – SETA 2006. SETA 2006. Lecture Notes in Computer Science, vol 4086. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11863854_2
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DOI: https://doi.org/10.1007/11863854_2
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