De Bruijn Sequences Research Articles

СПОСОБИ СИНТЕЗУ ТОПОЛОГІЧНИХ ОРГАНІЗАЦІЙ НА ОСНОВІ КОДОВИХ ПЕРЕТВОРЕНЬ ДЕ БРУЙНА

Urgency of the research. Today, topological organizations such as trees, fat trees, 3D-mesh, 6D-tor, DragonFly are used as a way of network organization in data centers. All these topologies have fault-tolerant properties due to the presence of several alternative routing paths. Therefore, the task of developing new topologies that have many balanced routes (one cost path) is relevant. One of the best options for research on tree topology improvement is the combination with de Bruijn graphs in different number systems, which allows varying the degree of vertices, diameter and number of alternative paths for routing. This paper proposes a new way to obtain specified characteristics when combining tree-based topologies with a de Bruijn graph. Target setting. When creating high-performance computing systems, it is important to develop their topological organizations in advance, which allows in time to anticipate the characteristics of the system. Actual scientific researches and issues analysis. At present there are a number of works devoted to topologies based on de Bruijn sequences. There is a detailed review of the synthesis of hybrid topologies based on de Bruijn graphs and hypercube. Uninvestigated parts of general matters defining. The possibility of using hybrid topologies based on de Bruijn sequences and different types of trees remains unexamined at present. The research objective. The tasks are to develop a way to synthesize new topologies based on trees and de Bruijn sequences and to analyze the characteristics of the synthesized topological organizations for different levels of fault tolerance. The statement of basic materials. The synthesis of five topologies based on trees and de Bruijn sequences is described. A comparative analysis of all synthesized topologies is performed. Coclusions. An analysis of the characteristics is performed, the main advantages and disadvantages of the proposed topological structures are highlighted, and suggestions for their improvement are made.

OPPORTUNITIES TO MINIMIZE HARDWARE AND SOFTWARE COSTS FOR IMPLEMENTING BOOLEAN FUNCTIONS IN STREAM CIPHERS

Currently, nonlinear Boolean functions are actively investigated worldwide. However, many questions remain unanswered. The theory of nonlinear Boolean functions that are suitable for use in cryptographically strong algorithms is significantly incomplete. Despite the existence of numerous publications on these themes, many issues related to the interconnection of design characteristics affecting the generator’s performance and its cryptographic characteristics still remain unsolved. The possibility of generating a special type of sequence, called de Bruijn sequence, at minimal hardware and software costs to implement nonlinear Boolean functions in stream encryption systems, is the main subject of this work. The paper presents the possible structure boundaries (algebraic degree of a Boolean function, the number of monomials in a function) of iterative de Bruijn sequence bitrate generators for various generated sequence characteristics, such as linear complexity and autocorrelation function. The profile of the linear complexity of the studied sequences is close to the expected value of the linear complexity, as well as for a truly random sequence.

Absolute time encoding for temporal super-resolution using de Bruijn coded exposures

Many target tracking tasks require high spatial and temporal precision. High frame rate imaging at high spatial resolution is commonly used in these applications, but this approach is expensive and generates large amounts of data which can complicate implementation. When tracking a single object in motion, almost all of this information is unused. A technique has been developed to exploit this sparsity and track motion with a long exposure where absolute timing is encoded by modulating the exposure over time according to a de Bruijn sequence. This technique has been implemented in the Desert Fireball Network to track bright meteors entering the Earth’s atmosphere for orbit determination and successful meteorite recovery. An alternate proof of concept implementation was also developed demonstrating tracking at 36 megapixels and 1000 Hz using a consumer camera with an inexpensive modulated light source and retroreflective target. The technique could be applied to other tracking problems requiring high temporal and spatial precision such as particle image velocimetry and space surveillance and tracking.

An efficient generalized shift-rule for the prefer-max De Bruijn sequence

One of the fundamental ways to construct De Bruijn sequences is by using a shift-rule. A shift-rule receives a word as an argument and computes the symbol that appears after it in the sequence. An optimal shift-rule for an (n,k)-De Bruijn sequence runs in time O(n). We propose an extended notion we name a generalized-shift-rule, which receives a word, w, and an integer, c, and outputs the c symbols that comes after w. An optimal generalized-shift-rule for an (n,k)-De Bruijn sequence runs in time O(n+c). We show that, unlike in the case of a shift-rule, a time optimal generalized-shift-rule allows to construct the entire sequence efficiently. We provide a time optimal generalized-shift-rule for the well-known prefer-max and prefer-min De Bruijn sequences.

The Adjacency Graphs of a Class of LFSRs and Their Applications*

Nonlinear feedback shift registers (NFSRs) are widely used in communication and cryptography. How to construct more NFSRs with maximal periods, which can generate sequences with maximal periods, i.e., de Brujin sequences, is an attractive problem. Recently many results on constructing de Bruijn sequences from adjacency graphs of Linear feedback shift registers (LFSRs) by means of the cycle joining method have been obtained. In this paper we discuss a class of LFSRs with characteristic polynomial p2(x), where p(x) is a primitive polynomial of degree n ≥ 2 over the finite field F2. As results, we determine their cycle structures and adjacency graphs, and further construct a class of new de Bruijn sequences from these LFSRs.

Permutation on binary de bruijn sequence

Binary de bruijn sequences are studied by many researchers since it has many applications such as in DNA sequencing, robotics, and radar. In this paper we study the properties of binary de bruijn sequences in term of permutation. We showed that it has a nice property that we hope it will be useful in further research.

New results on the minimal polynomials of modified de Bruijn sequences

Modified de Bruijn sequences are generated by removing a single zero from the longest run of zeros of de Bruijn sequences. It is known that the minimal polynomial of a modified de Bruijn sequence of order n at least has an irreducible factor of degree n. Based on this observation, we give some new results on the minimal polynomial of a modified de Bruijn sequence in this paper. First, it is shown that the minimal polynomial of a modified de Bruijn sequence of order n cannot be the product of an irreducible polynomial of degree n and the irreducible polynomial of degree 2. Second, it is proved that the minimal polynomial of a modified de Bruijn sequence of order n cannot be the product of an irreducible polynomial of degree n and a primitive polynomial of degree k with n≥8k. This is a generalization of the main result in the paper Kyureghyan (2008) [3] which only considered products of two primitive polynomials. Third, it is proved that the minimal polynomial of a modified de Bruijn sequence of order n cannot be a product of an irreducible polynomial f(x) of degree n and a polynomial of order t dividing 2k−1 with gcd⁡(ord(f(x)),t)=1 and n≥4k. As an application, for the cases n=2pe and n=p1⋅p2 where p,p1,p2 are prime numbers and 2pi−1 is also a prime number for i=1,2, a non-trivial lower bound is given for the linear complexity of a modified de Bruijn sequence of order n distinct from m-sequences, which could not be proved by the previous techniques.

Single-Shot Colored Speckle Pattern for High Accuracy Depth Sensing

Structured light techniques are widely used in the depth measurement. In this paper, we propose a colored speckle method for high-accuracy depth sensing. The projected pattern is a composite of the discrete phase information and the colored De Bruijn sequence information, which can prove the feasibility, in practice, and the robustness to the environment. A Gabor filter is used to estimate the wrapped phase. To alleviate the phase ambiguity of conventional measurement methods, the period of the phase is distributed in two channels of the colored pattern by the De Bruijn sequence decoding. A morphological closure is adopted to determine the De Bruijn code by the phase neighborhoods back and forth. The proposed method is suitable for a dynamic scene. The quantitative and qualitative experiments are conducted, in practice, to evaluate the performance of the proposed method. The results of the experiments prove that not only can our proposed method achieve high accuracy and dense depth map, but it is also suitable for color object and dynamic scenes indoors.

The cycle structure of NFSR(fd) and its applications

Let NFSR(f ) denote the nonlinear feedback shift register (NFSR) with characteristic function f = x0 ⊕ g(x1,x2,…,xn− 1) ⊕ xn. In this paper, the cycle structure of NFSR(fd) is discussed, where $f^{d}=x_{0}\oplus g(x_{d},x_{2d},{\ldots } ,x_{(n-1)d})\oplus x_{nd}$ is also a characteristic function determined by f and a given integer d. If the cycle structure of NFSR(f ) is known, then it is shown that the cycle structure of NFSR(fd) can be completely determined. Moreover, with these results, three applications of the cycle structure of NFSR(fd) are presented: Firstly, the cycle structure of NFSR(fd) is discussed when f belongs to a class of symmetric characteristic functions. Compared with the previous work, our result can cover more cases while the proof is more straightforward. Secondly, we show the cycle structure of NFSR(fd) when f is a characteristic function of de Bruijn sequences and d = 2k. At last, a new necessary condition for f to be a characteristic function of de Bruijn sequences is presented, which can partially support the observation proposed in Calik et al. (IEICE Trans. Fund. Electron. Commun. Comput. Sci. E93-A,(6), 1226–1231 2012) and Chan et al. (Lect. Notes Comput. Sci. 809, 166–173 1993).

Constructions of de Bruijn sequences from a full-length shift register and an irreducible LFSR

In this paper, we study the construction of de Bruijn sequences. A method is given to stretch a de Bruijn sequence of order n to a de Bruijn sequence of order n+k with the help of a k-stage irreducible LFSR. It is shown that there is a correspondence between the cycle structure of an LFSR(l) and that of an NFSR(f⁎l), where f is the characteristic function of a de Bruijn sequence of order n and the LFSR(l) is a k-stage irreducible LFSR. Besides, an efficient algorithm is given to construct a class of de Bruijn sequences whose time complexity is 2n+1O(k) and memory requirement is O(2n+1).

A Successor Rule Framework for Constructing $k$ -Ary de Bruijn Sequences and Universal Cycles

We present a simple framework for constructing $k$ -ary de Bruijn sequences, and more generally, universal cycles, via successor rules. The framework is based on the often used method of joining disjoint cycles. It generalizes several previously known de Bruijn sequence constructions based on the pure cycling register and is applied to derive a new construction that is perhaps the simplest of all successors. Furthermore, it generalizes an algorithm to construct binary de Bruijn sequences based on any arbitrary nonsingular feedback function. The framework is applied to derive and prove the correctness of successors to efficiently construct 1) universal cycles for $k$ -ary strings of length $n$ whose weight is bounded by some $w$ and 2) universal cycles for permutations. It has also been subsequently applied to find the first universal cycle constructions for weak orders.

Code scale based on cyclically rearranged de Bruijn sequence

Code scale based on cyclically rearranged de Bruijn sequence

On a Construction of Easily Decodable Sub-de Bruijn Arrays

We consider the two-dimensional generalizations of de Bruijn sequences; i.e., the integer-valued arrays whose all fragments of a fixed size (windows) are different. For these arrays, dubbed sub-de Bruijn, we consider the complexity of decoding; i.e., the determination of a position of a window with given content in an array. We propose a construction of arrays of arbitrary size with arbitrary windows where the number of different elements in the array is of an optimal order and the complexity of decoding a window is linear.

Normal numbers and nested perfect necklaces

M. B. Levin used Sobol–Faure low discrepancy sequences with Pascal triangle matrices modulo 2 to construct, a real number x such that the first N terms of the sequence (2nxmod1)n≥1 have discrepancy O((logN)2∕N). This is the lowest discrepancy known for this kind of sequences. In this note we characterize Levin’s construction in terms of nested perfect necklaces, which are a variant of the classical de Bruijn sequences. Moreover, we show that every real number x whose binary expansion is the concatenation of nested perfect necklaces of exponentially increasing order satisfies that the first N terms of (2nxmod1)n≥1 have discrepancy O((logN)2∕N). For the order being a power of 2, we give the exact number of nested perfect necklaces and an explicit method based on matrices to construct each of them. The computation of the nth digit of the binary expansion of a real number built from nested perfect necklaces requires O(logn) elementary mathematical operations.

On shortening u-cycles and u-words for permutations

This paper initiates the study of shortening universal cycles (u-cycles) and universal words (u-words) for permutations either by using incomparable elements, or by using non-deterministic symbols. The latter approach is similar in nature to the recent relevant studies for the de Bruijn sequences. A particular result we obtain in this paper is that u-words for n-permutations exist of lengths n!+(1−k)(n−1) for k=0,1,…,(n−2)!.

A method for constructing artificial DNA libraries based on generalized de Bruijn sequences

This work presents an algorithm for constructing artificial DNA libraries that consist of chains having minimal tendency to hybridize with each other. Such libraries can be applied in DNA computing (Adleman, 1994) for encoding problem instances. The algorithm is based on an extension of de Bruijn graphs and de Bruijn sequences. In general, the algorithm can be used to build a chain over any alphabet of even size with defined binary symmetric irreflexive relation of complementarity — the obtained chain is a modified de Bruijn sequence with shifted windows (there are no repetitive k-long subchains that start on positions divisible by a given parameter) where half of these windows are removed to eliminate relation of complementarity among them.

On distinct consecutive r-differences

Suppose A⊂R of size k has distinct consecutive r-differences, that is for 1≤i≤k−r, the r-tuples(ai+1−ai,…,ai+r−ai+r−1) are distinct. Then for any finite B⊂R, one has|A+B|≫r|A||B|1/(r+1). Utilizing de Bruijn sequences, we show this inequality is sharp up to the constant.Moreover, for the sequence {nα}, a sharp upper bound for the size of the distinct consecutive r-differences is obtained, which generalizes Steinhaus' three gap theorem. A dual problem on the consecutive r-differences of the returning times for some ϕ∈R defined by {T:{Tθ}<ϕ} is also considered, which generalizes a result of Slater.

Joker de Bruijn: Covering k-Mers Using Joker Characters.

Sequence libraries that cover all k-mers enable universal and unbiased measurements of nucleotide and peptide binding. The shortest sequence to cover all k-mers is a de Bruijn sequence of length nvert Sigma { vert ^k} + k - 1n. Researchers would like to increase k to measure interactions at greater detail, but face a challenging problem: the number of k-mers grows exponentially in k, while the space on the experimental device is limited. In this study, we introduce a novel advance to shrink k-mer library sizes by using joker characters, which represent all characters in the alphabet. Theoretically, the use of joker characters can reduce the library size tremendously, but it should be limited as the introduced degeneracy lowers the statistical robustness of measurements. In this work, we consider the problem of generating a minimum-length sequence that covers a given set of k-mers using joker characters. The number and positions of the joker characters are provided as input. We first prove that the problem is NP-hard. We then present the first solution to the problem, which is based on two algorithmic innovations: (1) a greedy heuristic and (2) an integer linear programming (ILP) formulation. We first run the heuristic to find a good feasible solution, and then run an ILP solver to improve it. We ran our algorithm on DNA and amino acid alphabets to cover all k-mers for different values of k and k-mer multiplicity. Results demonstrate that it produces sequences that are very close to the theoretical lower bound.

An efficient shift rule for the prefer-max De Bruijn sequence

A shift rule for the prefer-max De Bruijn sequence is formulated, for all sequence orders, and over any finite alphabet. An efficient algorithm for this shift rule is presented, which has linear (in the sequence order) time and memory complexity.

An interleaved method for constructing de Bruijn sequences

This paper presents an efficient method for constructing 22n−2 cyclic different de Bruijn sequences of order 2n from the feedback function of a de Bruijn sequence of order n. This is done by extending an n-stage nonlinear feedback shift register (NFSR) to a 2n-stage NFSR, and then recursively joining all cycles of the 2n-stage NFSR to form a full cycle.