Characterizations of generators for modified de Bruijn sequences

AbstractOrder n modified de Bruijn sequences are created by removing a single zero from the longest run of zeros in period 2n de Bruijn sequences. The M sequences are then the natural linear subset of modified de Bruijn sequences. Recursions which are the nonlinear duals to primitive polynomials over GF(2) are developed. Data is presented for 4 ≤ n ≤ 6

A clone-free, single molecule map of the domestic cow (Bos taurus) genome.

BackgroundThe cattle (Bos taurus) genome was originally selected for sequencing due to its economic importance and unique biology as a model organism for understanding other ruminants, or mammals. Currently, there are two cattle genome sequence assemblies (UMD3.1 and Btau4.6) from groups using dissimilar assembly algorithms, which were complemented by genetic and physical map resources. However, past comparisons between these assemblies revealed substantial differences. Consequently, such discordances have engendered ambiguities when using reference sequence data, impacting genomic studies in cattle and motivating construction of a new optical map resource--BtOM1.0--to guide comparisons and improvements to the current sequence builds. Accordingly, our comprehensive comparisons of BtOM1.0 against the UMD3.1 and Btau4.6 sequence builds tabulate large-to-immediate scale discordances requiring mediation.ResultsThe optical map, BtOM1.0, spanning the B. taurus genome (Hereford breed, L1 Dominette 01449) was assembled from an optical map dataset consisting of 2,973,315 (439 X; raw dataset size before assembly) single molecule optical maps (Rmaps; 1 Rmap = 1 restriction mapped DNA molecule) generated by the Optical Mapping System. The BamHI map spans 2,575.30 Mb and comprises 78 optical contigs assembled by a combination of iterative (using the reference sequence: UMD3.1) and de novo assembly techniques. BtOM1.0 is a high-resolution physical map featuring an average restriction fragment size of 8.91 Kb. Comparisons of BtOM1.0 vs. UMD3.1, or Btau4.6, revealed that Btau4.6 presented far more discordances (7,463) vs. UMD3.1 (4,754). Overall, we found that Btau4.6 presented almost double the number of discordances than UMD3.1 across most of the 6 categories of sequence vs. map discrepancies, which are: COMPLEX (misassembly), DELs (extraneous sequences), INSs (missing sequences), ITs (Inverted/Translocated sequences), ECs (extra restriction cuts) and MCs (missing restriction cuts).ConclusionAlignments of UMD3.1 and Btau4.6 to BtOM1.0 reveal discordances commensurate with previous reports, and affirm the NCBI's current designation of UMD3.1 sequence assembly as the "reference assembly" and the Btau4.6 as the "alternate assembly." The cattle genome optical map, BtOM1.0, when used as a comprehensive and largely independent guide, will greatly assist improvements to existing sequence builds, and later serve as an accurate physical scaffold for studies concerning the comparative genomics of cattle breeds

Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields

A maximal minor $M$ of the Laplacian of an $n$-vertex Eulerian digraph $\Gamma$ gives rise to a finite group $\mathbb{Z}^{n-1}/\mathbb{Z}^{n-1}M$ known as the sandpile (or critical) group $S(\Gamma)$ of $\Gamma$. We determine $S(\Gamma)$ of the generalized de Bruijn graphs $\Gamma=\mathrm{DB}(n,d)$ with vertices $0,\dots,n-1$ and arcs $(i,di+k)$ for $0\leq i\leq n-1$ and $0\leq k\leq d-1$, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs. Moreover, for a prime $p$ and an $n$-cycle permutation matrix $X\in\mathrm{GL}_n(p)$ we show that $S(\mathrm{DB}(n,p))$ is isomorphic to the quotient by $\langle X\rangle$ of the centralizer of $X$ in $\mathrm{PGL}_n(p)$. This offers an explanation for the coincidence of numerical data in sequences A027362 and A003473 of the OEIS, and allows one to speculate upon a possibility to construct normal bases in the finite field $\mathbb{F}_{p^n}$ from spanning trees in $\mathrm{DB}(n,p)$.Comment: I+24 page

On cross joining de Bruijn sequences

We explain the origins of Boolean feedback functions of nonlinear feedback shift registers (NLFSRs) of fixed order n generating de Bruijn binary sequences. They all come into existence by cross joining operations starting from one maximum period feedback shift register, e.g., a linear one which always exists for any order n. The result obtained yields some constructions of NLFSRs generating maximum period $2^n-1$ binary sequences

Design and Analysis of Cryptographic Pseudorandom Number/Sequence Generators with Applications in RFID

This thesis is concerned with the design and analysis of strong de Bruijn sequences and span n sequences, and nonlinear feedback shift register (NLFSR) based pseudorandom number generators for radio frequency identification (RFID) tags. We study the generation of span n sequences using structured searching in which an NLFSR with a class of feedback functions is employed to find span n sequences. Some properties of the recurrence relation for the structured search are discovered. We use five classes of functions in this structured search, and present the number of span n sequences for 6 <= n <= 20. The linear span of a new span n sequence lies between near-optimal and optimal. According to our empirical studies, a span n sequence can be found in the structured search with a better probability of success. Newly found span n sequences can be used in the composited construction and in designing lightweight pseudorandom number generators. We first refine the composited construction based on a span n sequence for generating long de Bruijn sequences. A de Bruijn sequence produced by the composited construction is referred to as a composited de Bruijn sequence. The linear complexity of a composited de Bruijn sequence is determined. We analyze the feedback function of the composited construction from an approximation point of view for producing strong de Bruijn sequences. The cycle structure of an approximated feedback function and the linear complexity of a sequence produced by an approximated feedback function are determined. A few examples of strong de Bruijn sequences with the implementation issues of the feedback functions of an (n+16)-stage NLFSR are presented. We propose a new lightweight pseudorandom number generator family, named Warbler family based on NLFSRs for smart devices. Warbler family is comprised of a combination of modified de Bruijn blocks (CMDB) and a nonlinear feedback Welch-Gong (WG) generator. We derive the randomness properties such as period and linear complexity of an output sequence produced by the Warbler family. Two instances, Warbler-I and Warbler-II, of the Warbler family are proposed for passive RFID tags. The CMDBs of both Warbler-I and Warbler-II contain span n sequences that are produced by the structured search. We analyze the security properties of Warbler-I and Warbler-II by considering the statistical tests and several cryptanalytic attacks. Hardware implementations of both instances in VHDL show that Warbler-I and Warbler-II require 46 slices and 58 slices, respectively. Warbler-I can be used to generate 16-bit random numbers in the tag identification protocol of the EPC Class 1 Generation 2 standard, and Warbler-II can be employed as a random number generator in the tag identification as well as an authentication protocol for RFID systems.1 yea

High Performance Computing for DNA Sequence Alignment and Assembly

Recent advances in DNA sequencing technology have dramatically increased the scale and scope of DNA sequencing. These data are used for a wide variety of important biological analyzes, including genome sequencing, comparative genomics, transcriptome analysis, and personalized medicine but are complicated by the volume and complexity of the data involved. Given the massive size of these datasets, computational biology must draw on the advances of high performance computing. Two fundamental computations in computational biology are read alignment and genome assembly. Read alignment maps short DNA sequences to a reference genome to discover conserved and polymorphic regions of the genome. Genome assembly computes the sequence of a genome from many short DNA sequences. Both computations benefit from recent advances in high performance computing to efficiently process the huge datasets involved, including using highly parallel graphics processing units (GPUs) as high performance desktop processors, and using the MapReduce framework coupled with cloud computing to parallelize computation to large compute grids. This dissertation demonstrates how these technologies can be used to accelerate these computations by orders of magnitude, and have the potential to make otherwise infeasible computations practical