Abstract
Nonlinear n-stage feedback shift-register sequences over the finite field \(\mathbb{F}_q\) of period q n–1 are investigated under linear operations on sequences. We prove that all members of an easily described class of linear combinations of shifted versions of these sequences possess useful properties for cryptographic applications: large periods, large linear complexities and good distribution properties. They typically also have good maximum order complexity values as has been observed experimentally. A running key generator is introduced based on certain nonlinear feedback shift registers with modifiable linear feedforward output functions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
van Aardenne-Ehrenfest, T., de Bruijn, N.G.: Circuits and trees in oriented linear graphs. Simon Steven 28, 203–217 (1951)
Zong-Duo, D., Jun-Hui, Y.: Linear complexity of periodically repeated random sequences. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 168–175. Springer, Heidelberg (1991)
Fúster-Sabater, A., Caballero-Gil, P.: On the linear complexity on nonlinearly filtered PN-sequences. In: Safavi-Naini, R., Pieprzyk, J.P. (eds.) ASIACRYPT 1994. LNCS, vol. 917, pp. 80–90. Springer, Heidelberg (1995)
Gammel, B.M., Göttfert, R., Kniffler, O.: Status of Achterbahn and tweaks. In: SASC 2006—Stream Ciphers Revisited, Leuven, Belgium, February 2-3, 2006. Workshop Record, pp. 302–315 (2006)
Gammel, B.M., Göttfert, R., Kniffler, O.: An NLFSR-based stream cipher. In: IEEE International Symposium on Circuits and Systems — ISCAS 2006, Island of Kos, Greece, May 21-24 (2006)
Golomb, S.W.: Shift Register Sequences. Aegean Park Press, Laguna Hills (1982)
Groth, E.J.: Generation of binary sequences with controllable complexity. IEEE Trans. Inform. Theory IT-17, 288–296 (1971)
Jansen, C.J.A.: Investigations On Nonlinear Streamcipher Systems: Construction and Evaluation Methods, Ph.D. Thesis, Technical University of Delft, Delft (1989)
Key, E.: An analysis of the structure and complexity of nonlinear binary sequence generators. IEEE Trans. Inform. Theory IT-22, 732–736 (1976)
Laksov, D.: Linear recurring sequences over finite fields. Math. Scand. 16, 181–196 (1965)
Lam, C.C.Y., Gong, G.: A lower bound for the linear span of filtering sequences. In: A lower bound for the linear span of filtering sequences, Workshop Record of The State of the Art of Stream Ciphers, Brugge, October 2004, pp. 220–233 (2004)
Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and Its Applications, vol. 20. Addison-Wesley, Reading (1983) (Now Cambridge Univ. Press.)
Massey, J.L., Serconek, S.: A Fourier Transform Approach to the Linear Complexity of Nonlinearly Filtered Sequences. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 332–340. Springer, Heidelberg (1994)
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)
Mykkeltveit, J.: Nonlinear recurrences and arithmetic codes. Information and Control 33, 193–209 (1977)
Mykkeltveit, J., Siu, M.-K., Tong, P.: On the cycle structure of some nonlinear shift register sequences. Information and Control 43, 202–215 (1979)
Niederreiter, H.: Cryptology—The mathematical theory of data security. In: Mitsui, T., Nagasaka, K., Kano, T. (eds.) Prospects of Mathematical Science, pp. 189–209. World Sci. Pub., Singapore (1988)
Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NFS Regional Conference Series in Applied Mathematics, vol. 63. SIAM, Philadelphia (1992)
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)
Paterson, K.G.: Root counting, the DFT and the linear complexity of nonlinear filtering. Designs, Codes and Cryptography 14, 247–259 (1998)
Rueppel, R.A.: Analysis and Design of Stream Ciphers. Springer, Heidelberg (1986)
Selmer, E.S.: Linear Recurrence Relations over Finite Fields. Univ. of Bergen (1966)
Siegenthaler, T., Kleiner, A.W., Forré, R.: Generation of binary sequences with controllable complexity and ideal r-tuple distribution. In: Price, W.L., Chaum, D. (eds.) EUROCRYPT 1987. LNCS, vol. 304, pp. 15–23. Springer, Heidelberg (1988)
Willett, M.: The minimum polynomial for a given solution of a linear recursion. Duke Math. J. 39, 101–104 (1972)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gammel, B.M., Göttfert, R. (2006). Linear Filtering of Nonlinear Shift-Register Sequences. In: Ytrehus, Ø. (eds) Coding and Cryptography. WCC 2005. Lecture Notes in Computer Science, vol 3969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11779360_28
Download citation
DOI: https://doi.org/10.1007/11779360_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35481-9
Online ISBN: 978-3-540-35482-6
eBook Packages: Computer ScienceComputer Science (R0)