Abstract
Different binary sequence generators produce sequences whose period is a power of 2. Although these sequences exhibit good cryptographic properties, in this work it is proved that such sequences can be obtained as output sequences from simple linear structures. More precisely, every one of these sequences is a particular solution of a linear difference equation with binary coefficients. This fact allows one to analyze the structural properties of the sequences with such a period from the point of view of the linear difference equations. In addition, a new application of the Pascal’s triangle to the cryptographic sequences has been introduced. In fact, it is shown that all these binary sequences can be obtained by XORing a finite number of binomial sequences that correspond to the diagonals of the Pascal’s triangle reduced modulo 2.
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Acknowledgements
The first author was supported by FAPESP with number of process 2015/07246-0. The second author was supported by both Ministerio de Economía y Competitividad, Spain under grant TIN2014-55325-C2-1-R (ProCriCiS), and Comunidad de Madrid, Spain, under grant S2013/ICE-3095-CM (CIBERDINE). Both authors thank European FEDER funds.
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Cardell, S.D., Fúster-Sabater, A. (2017). Linear Models for High-Complexity Sequences. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2017. ICCSA 2017. Lecture Notes in Computer Science(), vol 10404. Springer, Cham. https://doi.org/10.1007/978-3-319-62392-4_23
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DOI: https://doi.org/10.1007/978-3-319-62392-4_23
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