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

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.

Folding-BSD Algorithm for Binary Sequence Decomposition

“…1. Given the binomial sequence ( n 2 r +i ) , with 0 ≤ i < 2 r and r being a non-negative integer, we have that [13]…”

“…are integer indexes, then its linear complexity is given by LC = i t + 1, see [13] (Corollary 14). 5.…”

“…In such a table, we point out the main contributions of our current work compared to past works: the introduction of the f-BSD algorithm, the complexity analysis of b-BSD (already introduced in [13]), backed up by our experiments on the number of sequences in the BSD, and the complexity analysis of f-BSD, which in the end shows how the f-BSD outperforms the previous algorithms. execution.…”

“…Applicable to To calculate sequences with the linear period a power of 2. Cardell et al [13] 2019 complexity Improvement in sequence Dependence on t Y (b-BSD algorithm) of binary length requirements. parameter.…”

“…This generator remains strong against different algorithms that analyze the cryptographic strength (e.g., linear complexity) of its sequences. Such algorithms can be enumerated as follows: (1) the well-known Berlekamp-Massey algorithm [12], (2) the sequence decomposition algorithm developed by Cardell et al in [13] and (3) the folding sequence decomposition algorithm introduced in this paper. To our knowledge, no other algorithms to calculate the parameter linear complexity of binary sequences are described in the literature.…”