Abstract
Grain is one of eSTREAM hardware-oriented finalists. It uses a 160-bit NFSR as a primitive which is a cascade connection of an 80-bit LFSR into an 80-bit NFSR. In this paper, we study affine sub-families of the family of output sequences of this 160-bit NFSR and prove that it includes no affine sub-families of order greater than 31 or less than 28 except for an affine sub-family of order 2.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Please note that [10] made a clerical mistake in the definition of the notation \(\mu _{k}(t)\) in (5) by writing \(\mathrm T (x_{0},x_{1},\ldots ,x_{m}) \) for \(\mathrm T (x_{0},x_{1},\ldots ,x_{k})\).
References
Bard G., Courtois N., Jefferson C.: Efficient methods for conversion and solution of sparse systems of low-degree multivariate polynomials over GF(2) via SAT solvers. Cryptology ePrint Archiv, Report 2007/024 (2007).
De Cannière C., Preneel B.: Trivium. In: New Stream Cipher Designs: The eSTREAM Finalists. Lecture Notes in Computer Science, vol. 4986, pp. 244–266. Springer-Verlag, New York (2008).
Een N., Sörensson N.: MiniSat-A SAT solver with conflict-clause minimization. In: Proceedings of the Theory and Applications of Satisfiability Testing (SAT’05). http://www.cs.chalmers.se/Cs/Research/FormalMethods/MiniSat/MiniSat.html (2005).
Golomb S.W.: Shift Register Sequences. Holden-Dan Inc, San Francisco (1967).
Green D.H., Dimond K.R.: Nonlinear product-feedback shift registers. Proc. IEEE 117, 681–686 (1970).
Hu H.G., Gong G.: Periods on two kinds of nonlinear Feedback shift registers with time varying feedback functions. Int. J. Found. Comput. Sci. 22(6), 1317–1329 (2011).
Hell M., Johansson T., Meier W.: The grain family of stream ciphers. New Stream Cipher Designs: The eSTREAM Finalists. Lecture Notes in Computer Science, vol. 4986, pp. 179–190. Springer-Verlag, New York (2008).
Lidl R., Niederreiter H.: Finite Fields. Addison-Wesley, Reading (1983).
Mykkeltveit J., Siu M.K., Tong P.: On the cycle structure of some nonlinear shift register sequences. Inf. Control 43, 202–215 (1979).
Tian T., Qi W.F.: On the largest affine sub-families of a family of NFSR sequences. Des. Codes Cryptogr. (2012). doi:10.1007/s10623-012-9723-1.
Acknowledgments
This research is supported by NSF of China under Grant No. (61272042, 61100202).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V. Rijmen.
Rights and permissions
About this article
Cite this article
Ma, Z., Qi, WF. & Tian, T. On affine sub-families of the NFSR in Grain. Des. Codes Cryptogr. 75, 199–212 (2015). https://doi.org/10.1007/s10623-013-9901-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-013-9901-9