\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Algebraic dependence in generating functions and expansion complexity

  • * Corresponding author: László Mérai

    * Corresponding author: László Mérai
Abstract Full Text(HTML) Related Papers Cited by
  • In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. Recently, a series of paper has been published for analysis of expansion complexity and for testing sequences in terms of this new measure of randomness. In this paper, we continue this analysis. First we study the expansion complexity in terms of the Gröbner basis of the underlying polynomial ideal. Next, we prove bounds on the expansion complexity for random sequences. Finally, we study the expansion complexity of sequences defined by differential equations, including the inversive generator.

    Mathematics Subject Classification: Primary: 11T71, 11Y16, 94A60, 94A55, 68Q25.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] S. BalasuriyaI. E. Shparlinski and A. Winterhof, An average bound for character sums with some counter-dependent recurrence sequences, Rocky Mountain J. Math., 39 (2009), 1403-1409.  doi: 10.1216/RMJ-2009-39-5-1403.
    [2] L. Carlitz, The distribution of irreducible polynomials in several indeterminates, Illinois Journal of Mathematics, 7 (1963), 371-375.  doi: 10.1215/ijm/1255644947.
    [3] C. Diem, On the use of expansion series for stream ciphers, LMS Journal of Computation and Mathematics, 15 (2012), 326-340.  doi: 10.1112/S146115701200109X.
    [4] D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-5350-1.
    [5] E. D. El-Mahassni and A. Winterhof, On the distribution and linear complexity of counter-dependent nonlinear congruential pseudorandom number generators, JP J. Algebra Number Theory Appl., 6 (2006), 411-423. 
    [6] J. C. FaugéreP. GianniD. Lazard and T. Mora, Efficient computation of zero-dimensional Gröbner bases by change of ordering, Journal of Symbolic Computation, 16 (1993), 329-344.  doi: 10.1006/jsco.1993.1051.
    [7] D. M. Goldschmidt, Algebraic Functions and Projective Curves, Graduate Texts in Mathematics, 215. Springer-Verlag, New York, 2003. doi: 10.1007/b97844.
    [8] D. Gomez, Multiplicative character sums with counter-dependent nonlinear congruential pseudorandom number generators, in Sequences and Their Applications—SETA 2010, Lecture Notes in Comput. Sci., Springer, Berlin, 6338 (2010), 188–195. doi: 10.1007/978-3-642-15874-2_15.
    [9] D. Gómez-PérezL. Mérai and H. Niederreiter, On the expansion complexity of sequences over finite fields, IEEE Trans. Inform. Theory, 64 (2018), 4228-4232.  doi: 10.1109/TIT.2018.2792490.
    [10] L. MéraiH. Niederreiter and A. Winterhof, Expansion complexity and linear complexity of sequences over finite fields, Cryptography and Communications, 9 (2017), 501-509.  doi: 10.1007/s12095-016-0189-2.
    [11] A. Shamir and B. Tsaban, Guaranteeing the diversity of number generators, Inform. and Comput., 171 (2001), 350-363.  doi: 10.1006/inco.2001.3045.
    [12] I. E. Shparlinski and A. Winterhof, On the discrepancy and linear complexity of some counter-dependent recurrence sequences, in Sequences and Their Applications—SETA 2006, Lecture Notes in Comput. Sci., Springer, Berlin, 4086 (2006), 295–303. doi: 10.1007/11863854_25.
    [13] J. von zur Gathen and  J. GerhardModern Computer Algebra, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139856065.
  • 加载中
SHARE

Article Metrics

HTML views(1216) PDF downloads(270) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return