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Binomial Representation of Cryptographic Binary Sequences and Its Relation to Cellular Automata

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  • Sara D. Cardell
  • Amparo Fúster-Sabater

Abstract

The binomial sequences are binary sequences that correspond to the diagonals of the binary Sierpinski’s triangle. They have fancy properties such that all the sequences with period equal to a power of 2 can be represented as the sum of a finite set of binomial sequences. Other structural properties of these sequences (period, linear complexity, construction rules, or relations among the different binomial sequences) have been analyzed in detail. Furthermore, this work enhances the close relation between the binomial sequences and a kind of Boolean networks, known as linear cellular automata. In this sense, the binomial sequences exhibit the same behavior as that of particular Boolean networks. Consequently, the binomial sequences can be considered as primary tools for generating other more complex Boolean networks with applications in communication systems and cryptography.

Suggested Citation

  • Sara D. Cardell & Amparo Fúster-Sabater, 2019. "Binomial Representation of Cryptographic Binary Sequences and Its Relation to Cellular Automata," Complexity, Hindawi, vol. 2019, pages 1-13, March.
    • Handle: RePEc:hin:complx:2108014
    DOI: 10.1155/2019/2108014
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    References listed on IDEAS

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    1. Juan A. Aledo & Silvia Martinez & Jose C. Valverde, 2015. "Parallel Dynamical Systems over Graphs and Related Topics: A Survey," Journal of Applied Mathematics, Hindawi, vol. 2015, pages 1-14, March.
    2. Desheng Zheng & Guowu Yang & Xiaoyu Li & Zhicai Wang & Feng Liu & Lei He, 2013. "An Efficient Algorithm for Computing Attractors of Synchronous And Asynchronous Boolean Networks," PLOS ONE, Public Library of Science, vol. 8(4), pages 1-7, April.
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    Cited by:

    1. Juan A. Aledo & Luis G. Diaz & Silvia Martinez & Jose C. Valverde, 2020. "Coexistence of Periods in Parallel and Sequential Boolean Graph Dynamical Systems over Directed Graphs," Mathematics, MDPI, vol. 8(10), pages 1-14, October.
    2. Jose Luis Martin-Navarro & Amparo Fúster-Sabater, 2021. "Review of the Lineal Complexity Calculation through Binomial Decomposition-Based Algorithms," Mathematics, MDPI, vol. 9(5), pages 1-22, February.
    3. Amparo Fúster-Sabater & Verónica Requena & Sara D. Cardell, 2022. "An Efficient Algorithm to Compute the Linear Complexity of Binary Sequences," Mathematics, MDPI, vol. 10(5), pages 1-23, March.

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