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.
Highlights
Pseudorandom binary sequences are simple successions of bits with applications in fields so different as spreadspectrum communications, circuit testing, error-correcting codes, numerical simulations, or cryptography
The family of binary sequences considered in this work, sequences whose period is a power of 2, has good cryptographic properties such as long period and large linear complexity
E.g., irregular decimation, are introduced to break the linearity of the Linear Feedback Shift Registers (LFSRs)-based sequence generators, this linearity is still visible in their output sequences
Summary
Pseudorandom binary sequences are simple successions of bits with applications in fields so different as spreadspectrum communications, circuit testing, error-correcting codes, numerical simulations, or cryptography (stream cipher). Most generators producing such sequences are based on Boolean functions and Linear Feedback Shift Registers (LFSRs) [1]. The binomial sequences correspond to the diagonals of the Sierpinski’s triangle modulo 2 In this way, the binomial sequences exhibit many attractive properties that can be very useful in the analysis and generation of cryptographic sequences. Cellular automata with two-state cells is a special kind of Boolean network where all the nodes use the same function and the links are all arranged in a regular bounded integer lattice structure.
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