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doi: 10.3934/amc.2020022

## Algebraic dependence in generating functions and expansion complexity

 1 Department of Mathematics, University of Cantabria, Santander 39005, Spain 2 Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, A-4040 Linz, Austria

* Corresponding author: László Mérai

Received  July 2018 Revised  April 2019 Published  September 2019

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.

Citation: Domingo Gómez-Pérez, László Mérai. Algebraic dependence in generating functions and expansion complexity. Advances in Mathematics of Communications, doi: 10.3934/amc.2020022
##### References:
 [1] S. Balasuriya, I. 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.  Google Scholar [2] L. Carlitz, The distribution of irreducible polynomials in several indeterminates, Illinois Journal of Mathematics, 7 (1963), 371-375.  doi: 10.1215/ijm/1255644947.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [6] J. C. Faugére, P. Gianni, D. 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.  Google Scholar [7] D. M. Goldschmidt, Algebraic Functions and Projective Curves, Graduate Texts in Mathematics, 215. Springer-Verlag, New York, 2003. doi: 10.1007/b97844.  Google Scholar [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.  Google Scholar [9] D. Gómez-Pérez, L. 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.  Google Scholar [10] L. Mérai, H. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] J. von zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139856065.  Google Scholar

show all references

##### References:
 [1] S. Balasuriya, I. 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.  Google Scholar [2] L. Carlitz, The distribution of irreducible polynomials in several indeterminates, Illinois Journal of Mathematics, 7 (1963), 371-375.  doi: 10.1215/ijm/1255644947.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [6] J. C. Faugére, P. Gianni, D. 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.  Google Scholar [7] D. M. Goldschmidt, Algebraic Functions and Projective Curves, Graduate Texts in Mathematics, 215. Springer-Verlag, New York, 2003. doi: 10.1007/b97844.  Google Scholar [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.  Google Scholar [9] D. Gómez-Pérez, L. 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.  Google Scholar [10] L. Mérai, H. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] J. von zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139856065.  Google Scholar
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