\r\n\tIn this book, the authors will present the highlights of basic research of biomechanical and biochemical pathways of bone homeostasis and the developing clinical methods for treatment of bone loss, either following trauma or systemic disease.
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1. Introduction
The discovery of genome-wide genetic variation was central to the field of genomics [1,2]. Now, recent advances in second-generation sequencing technologies and better methods of targeted enrichment mean the detection of genome-wide patterns of genetic variation will soon be a routine operation [3,4]. Yet these advances in DNA sequencing have revealed a new bottleneck: the functional classification and interpretation of newly discovered genetic variation.
The scale of this problem is enormous. The high throughput and low cost of second-generation sequencing platforms now allow geneticists to routinely perform single experiments that identify tens of thousands to millions of variant sites in a single individual, but the methods that exist to annotate these variant sites using information from publicly available databases are too slow to be useful for the large sequencing datasets being generated. Because sequence annotation of variant sites is required before functional characterization can proceed, the lack of a high-throughput pipeline to annotate variant sites efficiently can be a major bottleneck in genetics research and clinical applications of genomics technologies.
To address this problem, we developed the Sequence Annotator (SeqAnt, http://seqant.genetics.emory.edu/), an open source web service and software package that rapidly annotates DNA sequence variants and identifies recessive or compound heterozygous loci in human, mouse, fly, and worm genome sequencing experiments [5]. Variants are characterized with respect to their functional type, frequency, and evolutionary conservation. Annotated variants can be viewed on a web browser, downloaded in a tab-delimited text file, or directly uploaded in a Browser Extensible Document (BED) format to the UCSC Genome Browser. To demonstrate the speed of SeqAnt, we annotated a series of publicly available datasets that ranged in size from 37 to 3,439,107 variant sites; the total time to annotate these data completely ranged from 0.17 seconds to 28 minutes 49.8 seconds.
1.1. Sequence annotation tools
Genome databases accessible via web browsers are very useful in the search for annotation information for DNA sequences. The UCSCGenome Browser web application has been a huge development of great value in analyzing and characterizing sequence information [6]. The application includes a variety of genomic tracks, assemblies, and browsers with genetic information from a host of species. The UCSC Genome Browser, with its various functionalities and annotation options, offers a one-stop shop for researchers, who can work directly on the web application by uploading their data, or they can download source codes of interest from the UCSC Genome Browser and run those locally. Despite its power, however, the main limitation we see in using the UCSC browser for sequence annotation lies in the limited amount of data that can be accessed at a given time, along with the need for human intervention. For example, it is time-consuming for geneticists who want annotation across multiple variant sites at once over different functional classes to use the browser comfortably. Ensembl is yet another superb broad-based web application with an expansive database, offering researchers choices on extracting specific regions of interest and annotating particular regions in the genome [7]. This application has various functionalities and tools that can accept uploaded data, convert formats of documents, and search for sequences of interest; still, like the UCSC browser, it is not the best choice for performing high-throughput sequencing annotation.
SNPnexus is a genetic variation tool developed to help determine functionally relevant SNPs for a given genomic region [8]. It has a user-friendly web interface that accepts inputs in the form of genomic positions, dbSNP id, or chromosomal region. The application database includes two different human genome assemblies: the hg19 and hg18 builds. SNPnexus generates calls on genomic mapping of variant sites, protein function consequences of such variants in the genome, the regulatory elements conserved within the region, and the conservation score of the variant site. The application also provides the genotype and allele frequencies estimation for known SNPs using data from the HapMap Project. This annotation tool, like so many others, is very useful for human variant annotation; however, it does not characterize variants in other species.
Since the development of SeqAnt in 2010, other software tools have come along to perform sequence annotation. Segtor is a tool designed to annotate large sets of genomic coordinates, intervals, single nucleotide variants (SNVs), indels, and translocations [9]. A more recent and very closely related annotation tool is AnnTools [10]. This is an open source web application that accepts user Inputs and queries their database for a full spectrum of variant site annotation, including single nucleotide variants, insertions and deletions, structural variants, and copy number variants. The application has a minimal memory footprint and likewise annotates variants quite rapidly. Nevertheless, AnnTools is restricted to human genome variant annotations and in this sense differs from SeqAnt, which annotates other species besides humans. There are also a number of other variant site annotation tools available either as downloadable command line applications or user interface web applications; these include snpEff (http://snpeff.sourceforge.net), MU2A[11], and Snat [12].
1.2. The distinction of SeqAnt
The uniqueness of SeqAnt versus all the other annotation tools we mentioned lies in three factors, which had been the key considerations for developing this technology to begin with. First, SeqAnt delivers annotations for multiple different species, ranging from primates to mammals, and now zebrafish and nematodes. Second, the web application has its own database updated from the UCSC website, which is a collection of binary files that drive the record speed with which large genomic data are annotated. Third, in addition to speed, the memory footprint is quite minimal, as data stored in binary files enable individuals from the public to download both the source file and database and locally run the application without elaborate computing apparatus. Some of the other tools mentioned have one or two of these unique features, but none have the robustness that comes from combining all three approaches to efficiently annotate variants and make meaningful functional calls across species, like SeqAnt does. Overall, we believe these represent important changes to SeqAnt that will be of broad utility to researchers using next-generation sequencing platforms in a wide variety of systems. SeqAnt will continue to be a fully open source web service and software package, and we believe it will prove especially useful for those investigators who lack dedicated bioinformatics personnel or infrastructure in their laboratories.
2. Upgraded features of SeqAnt 2.0
Since the initial publication of SeqAnt, we made a number of improvements that have been incorporated into SeqAnt 2.0 [5]. These modifications fall into four main categories. The first focused on updating the SeqAnt website (http://seqant.genetics.emory.edu). The second includes major changes made to the content and structure of the underlying binary databases that hold the annotation information. The third involves a significant redesign of the directory structure holding the output files. Finally, the last modification included substantial revisions to the number and content of output files themselves. Each of these updates will be described in greater detail in the sections that follow.
2.1. SeqAnt 2.0 - website updates
We undertook a major redesign of the SeqAnt web interface to make it more user-friendly. On the home page, we eliminated redundant tabs and buttons, simplified the overall design, and upgraded the graphic interface’s color scheme (Figure 1). This page includes basic information about the original publication of SeqAnt [5], a link to contact the Zwick laboratory, and the web URL for the theSourceForge website (http://seqant.sourceforge.net), where the source code and associated binary libraries can be freely downloaded. From this page, the user is able to quickly access the three main types of input data accepted by SeqAnt. These include SEQUENCE FILE, LIST OF VARIANTS, and SINGLE VARIANT. In addition, the user can choose to view a TUTORIAL or select a set of SAMPLE FILES to gain experience performing analyses with the SeqAnt.
Figure 1.
Screenshot of new SeqAnt 2.0 home page
Selecting the SEQUENCE FILE option returns the web interface shown in Figure 2. A typical use of this feature is when the user wants variation annotation information in a genomic region from a particular chromosome. Three different input files are accepted. The first is a reference sequence file in FASTA format of the entire genomic region being annotated. The second is a sequence file containing multiple FASTA sequences from a sequencing experiment, with each FASTA sequence representing a chromosomal region. The third is a genomic position file in the BED format which represents the coordinates for each of the chromosomal regions in the sequence file. The sequences in both the reference file and the sequence file should be in the positive orientation to ensure accurate annotation. The user is provided the option to choose a reference genome and assembly that will be used for annotating variant sites.
Selecting the LIST OF VARIANTS option returns the web interface shown in Figure 3. Only one input file is required to use this feature, the variations list file, which contains a listing of variant sites and the chromosomal regions of these sites, the minor allele and the reference allele. The variant list file is basically a pileup file, with a \'.snp\' or a \'.txt\' extension. If the PEMapper option were selected in this interface, the variation list file would be modified to include the sample ID for each individual within the experimental study where the sequence data was generated, if multiple individual samples were being analyzed. This particular (List of Variants) feature is very useful for researchers who want to perform genetic variation analysis (such as whole exome annotation) over a wide expanse of the genome.
Figure 2.
Screenshot of the SEQUENCE FILE page
Selecting the SINGLE VARIANT option returns the web interface shown in Figure 4. The user is provided the option to choose a reference genome and assembly that will be used for annotating a single variant site.The user then only needs to provide a chromosome and base position to obtain the annotation information.
Figure 3.
Screenshot of the LIST OF VARIANTS page
Figure 4.
Screenshot of the SINGLE VARIANT page
2.2. SeqAnt 2.0 - Binary database upgrades
One of the unique features of SeqAnt is the ease and speed with which variant information is accessed from a set of customized binary databases. The SeqAnt binary databases are created from flat text table files obtained from the UCSC Genome Browser website [6]. Five main types of data constitute the SeqAnt binary databases. These include:
Reference Genome Sequence
RefGene Annotation
dbSNP Variation Data
PhastCons Evolutionary Conservation Scores
PhyloP Evolutionary Conservation Score
Standard queries, implemented through the web interfaces described above, are able to extract the annotation information from the binary databases. The actual structure of the binary databases is not directly visible to a SeqAnt user, but is worth examining in greater detail. The Reference Genome Sequence provides the basic backbone for other annotation information. Reference sequences for a given species are organized by different builds (i.e. human genome 18, human genome 19). Within each build, data are organized by chromosome, which reflects the structure of the flat files obtained from UCSC. The RefGene Annotation is the collection of information pertaining to known genes for a given species and build. This information is also organized by chromosome. The collection of variant sites in a given species is contained within the dbSNP Variation Data that is also organized by chromosome. Finally, the SeqAnt 2.0 binary databases include two different measures of evolutionary conservation for all sites in a given reference genome sequence. The PhastCons score is best used to detect functional elements in noncoding sequences, whereas the phyloP score provides a measure of the evolutionary conservation of single sites and is most useful for evaluating sites located in coding regions of genes.
Binary files are significantly smaller than their corresponding flat files, so querying binary files uses less memory than the same analysis performed with a flat file. Considering the vast amount of data that has to be accessed during sequence annotation of large genomic regions, the significant difference in the size of the binary files versus flat files helps to account for the speed with which information is processed using binary files. SeqAnt 2.0 updated a number of these specific binary files; a detailed description of the changes follows in the next sections.
2.2.1. Upgrade of dbSNP to SNP132 Track for hg19 Assembly (Homo sapiens)
The original goal of the dbSNP database (http://www.ncbi.nlm.nih.gov/projects/SNP/) was to develop a comprehensive catalog of common (>5% frequency) human genetic variation [13,14]. These variants were subsequently validated by genotyping in multiple human populations, and their patterns of statistical correlation among variants, known as linkage disequilibrium, were revealed in the HapMap project [15,16]. SeqAnt 1.0 included data from the SNP131 track from the dbSNP [17]. SeqAnt 2.0 was updated to the SNP132 build, which was characterized and uploaded to the UCSC Genome Browser in the summer of 2011. SNP132 has an expanded collection of variant sites that can help researchers determine whether an identical variant has been seen before in a different individual.
Figure 5.
Contents of SeqAnt Output Directory. Directories are in bold; individual files shown in a standard font face.
2.2.2. Addition of PhyloP46way Conservation Score Database for hg19 Assembly (Homo sapiens)
The phyloP Evolutionary Conservation Score data type is a new addition to SeqAnt 2.0. Binary databases, including phylopP scores from a 46-way alignment of vertebrate species to the human genome, were included to complement the PhastCons Evolutionary Conservation Scores previously included in the application. The phyloP scores predict the probability of a given variant site having undergone evolution over time. The absolute phyloP values represent negative log p-values for the null hypothesis that there was no evolution across the regions annotated [18]. Regions that are more conserved tend to have more positive values, whereas sites believed to be fast evolving have negative values. The medium range of these scores for the 46-way alignment from the UCSC Genome Browser is between approximately -3 and +3. It should be noted that, unlike PhastCons, which takes into account flanking bases on a sequence in arriving at its final score for a given variant site, phyloP scores are computed by basically comparing the particular base in the sequence with aligned bases from other species [18]. Variations in highly conserved regions often suggest a significant change that could have functional implications. The PhyloP46way dataset we have on the upgraded SeqAnt web application is the most recent phyloP track in the UCSC, released in December 2009.
2.2.3. Addition of Full Genome Data Set by Chromosome of Zebrafish (danRer6 Assembly)
We selected zebrafish (Daniorerio) as the next species to be incorporated into the SeqAnt database because of its emergence as a model organism for a wide range of scientific studies, from behavioral genetics to drug modeling studies and integrative physiology [19,20]. SeqAnt 2.0 has now been updated to include binary files for the genome sequence of zebrafish. We derived binary databases for the first four data types from flat table files on the UCSC Genome Browser website. Flat table files for the phyloP evolutionary conservation score were not available and were therefore not included. The reference genome binaries use the danRer6 assembly, which annotated the datasets by chromosome and was released in December of 2008. The RefGene annotation and dbSNP variation data are relative to the danRer6 assembly. PhastCons evolutionary conservation scores were derived from multiple alignment between seven species and zebrafish. Including the zebrafish in SeqAnt 2.0 should prove valuable for researchers who work with this species.
2.3. SeqAnt 2.0: output directory structure and files
Significant changes to the number and types of output files are reflected in a new output directory structure in SeqAnt 2.0. The output from SeqAnt is contained within a Results directory that includes three subdirectories (Figure 5). This Results directory has the name of the original SeqAnt input file and a subscript \'_Annotation_Files\'. Within this directory, there are three distinct directories (All_Variations, BED_Annotation, Unique_Variations) holding the output of SeqAnt, which will be described in detail below. This directory also contains three other files of interest to a user. The first is a *.summary.txt file that provides a summary of all the variants annotated by SeqAnt. The second is a Compound Replacement file that identifies variants, genes, and sample identifiers for those loci with two or more replacement variants. The collected list of variants includes those that could be compound recessive in a given individual, although since the phase of the variants is not determined, this would have to be validated by other means. This file may be useful when looking for genes that harbor variants that may fit a recessive loss-of-function model. The last is a *.log file generated by SeqAnt that records the major events that occur when SeqAnt processes a dataset.
2.3.1. All_variations directory
This directory contains the complete variant annotation files obtained from annotating input files with SeqAnt 2.0 (Figure 5). Two main types of genetic variation are annotated by SeqAnt: single nucleotide variants (SNPs) and insertions/deletions (INDELs). For SNPs, a given variant site when annotated belongs in one of five functional classifications. These include exonic.replacement, exonic.silent, untranslated region (UTR), intronic, or intergenic. For INDELs, a given variant when annotated belongs in one of four functional classifications. These include exonic, UTR, intronic, or intergenic. Overall, there are a total of nine files that contain the variants and their associated annotation information. These annotation files include all possible splice variants impacted by a given variant site. Thus, a given variant site may be listed multiple times in one of the nine output files.
2.3.2. BED_annotationdirectory
This directory contains files in BED format (http://genome.ucsc.edu/FAQ/FAQformat) that can be visualized on the UCSC Genome Browser or other viewer able to process files in this format. There are ten files total in this directory. Nine of the files include the variants and annotation information as described above; the tenth file (*.ucsc.bed) contains all the annotation information from each of the nine files in a single BED file for the entire genomic region to be visualized. These files can be uploaded to the UCSC browser as custom tracks to be visualized. They can also be visualized in other software packages that process BED files, such as the Integrative Genomics Viewer (Version 2.1) [21].
2.3.3. Unique_variations directory
In contrast to the annotation in the All_Variations directory, the Unique_Variations directory contains nine files that contain a single variant annotation for each SNP or INDEL. Thus, each variant is listed just once, regardless of the number of different splice variants it is predicted to impact. These files allow the user to quickly determine the total number of variants for any specific functional class.
2.4. SeqAnt 2.0 - Output files
2.4.1. Redesign of Result Columns for Annotation Files
We introduced a number of changes to the annotation fields contained within the SeqAnt output files. First, we rearranged the order of columns in the output files to aid users in evaluating their results. Second, we introduced additional feature columns to the output files. These included row 10, which depicts the transcript change that occurs for a coding sequence variant, row 14, which shows the concomitant amino-acid change for a coding sequence variant, and rows 21 and 22, which report the phyloP conservation score values for each variant position annotated. A summary of the annotation information provided by SeqAnt 2.0 is shown below in Table 1. A representation of an example output file is shown in Figure 6 below.
Field ID
Annotation Field
Description
1
Variation_Type
Type of variant
2
Functional Class
Annotated functional category for variant site
3
Chromosome
Chromosome containing variant site
4
Position
Absolute position of variant site on a chromosome
5
Gene_Name
Name of locus containing variant site
6
RefSeq_ID
Ref_Seq ID from UCSC track
7
Gene_Strand
Orientation of locus
8
Reference_Base
Reference allele at variant site
9
Input_Base
Minor allele at variant site
10
Transcript Change
Nucleotide base change on transcript
11
Original_Amino_Acid
Reference amino acid at variant site
12
Amino_Acid_Number
Position of amino acid on peptide chain
13
Modified_Amino_Acid
Modified amino acid due to variant site
14
Amino_Acid_Change
Amino acid change on peptide chain
15
dbSNP_IDs
dbSNP ID If variant site has been reported
16
Het_Rates
dbSNPheterozygosity of reported variant site
17
Orientation
dbSNP orientation of reported variant site
18
PhastCons_placentals
Placental PhastCons score for variant site (46way)
19
PhastCons_primates
Primate PhastCons score for variant site (46way)
20
PhastCons_vertebrate
Vertebrate PhastCons score for variant site (46way)
21
PhyloP_placental
Placental phyloP score for variant site (46way)
22
PhyloP_primates
Primate phyloP score for variant site (46way)
23
PhyloP_vertebrate
Vertebrate phyloP score for variant site (46way)
Table 1.
Annotation information output by SeqAnt 2.0
Figure 6.
Snapshot of Exonic Replacement Annotation Output File. The top half shows the data for fields 1 - 13. The bottom half of the figure shows the data from fields 14 - 23. The last four columns report the number of homozygous and heterozygous SNPs and associated sample IDs.
3. An application of SeqAnt 2.0: Targeted next-generation sequencing of NLGN3 and NLGN4X in humans
The targeted sequencing of specific genes or genomic regions is a common experimental design that can benefit from the use of SeqAnt. Here we describe such a study. We sequenced the NLGN3 and NLGN4X loci in a sample of 144 males with a diagnosis of autism. All the patient samples were obtained from the multiplex Autism Genetic Resource Exchange (AGRE) [22]. Raw base-calling data generated with an Illumina Genome Analyzer (IGA) were used as input for mapping and alignment. The total amount of sequence generated was 7.04 GB. Paired-end reads were mapped and variants were called using PEMapper (Cutler DJ et al, personal communication). In total, 99.7% of target bases had at least 8X coverage, with a median depth of coverage of 452. We identified a total of 208 sites of variation, with 176 single nucleotide polymorphisms and 32 insertions or deletions. Overall levels of variation were estimated at 5.8 x 10-4 (Θwper site [23]), which matched our expectation for loci from the human X chromosome. We also observed an excess of rare variants, as evidenced by a negative value for the Tajima’s D test statistic (-0.27,[24]).
Single nucleotide variants (SNVs) and small insertions and deletions (INDELs) were annotated using SeqAnt [5]. For the SNPs, a total of 68, or 39%, had not been reported before (31 in NLGN3 and 37 in NLGN4X,\n\t\t\t\tTable 2). For the INDELs, a total of 24, or 75%, had not been reported before (5 in NLGN3 and 19 in NLGN4X, Table 3). As summarized in Figure 7, almost all common variation (>5% frequency in our sample) is contained in dbSNP, whereas most rare variants (<5%) have not been cataloged there.
Functional class
Total SNPs
SNPs in dbSNP
Novel SNPs
Novel SNPs at Evolutionary Conserved Sites
Replacement
1
1
0
0
Silent
3
3
0
0
UTR
18
10
8
2
Intron
134
78
56
9
Intergenic
20
16
4
0
Total
176
108
68
11
Table 2.
Functional annotation of SNPs at the NLGN3 and NLGN4X loci identified by next-generation sequencing of 144 males with autism.
Using SeqAnt to rapidly annotate our sequence data allows us to quickly draw four main conclusions. First, most common variation is already contained in dbSNP, while much of the rare variation remains undiscovered. Second, we did not see any novel replacement variants at either NLGN3 or NLGN4X, suggesting that mutations at these loci are rare causes of autism. Third, we identified novel UTR variants at highly evolutionarily conserved sites, which could contribute to autism susceptibility. We focused on this set of variants for direct functional testing. Finally, we identified novel intronic variants at evolutionarily conserved sites that appear to be located in transcription factor binding sites. These variants are being followed up to determine whether they have a regulatory role that impacts the expression of NLGN3 or NLGN4X. In summary, SeqAnt 2.0 allowed us to rapidly annotate all the sites of variation in our sample and rapidly focus attention on those variants most likely to be autism susceptibility alleles.
Functional Class
Total Indels
Indels in dbSNP
Novel Indels
Novel Indels at Evolutionary Conserved Sites
Coding
0
0
0
0
UTR
1
0
1
1
Intron
25
7
18
0
Intergenic
6
1
5
0
Total
32
8
24
1
Table 3.
Functional annotation of INDELs at the NLGN3 and NLGN4X loci identified by next-generation sequencing of 144 males with autism.
Figure 7.
Summary of SNV and indel variation discovered at the NLGN3 and NLGN4X loci in males with ASD. The frequency of SNVs and INDELs (minor alleles) in cases is plotted against their level of evolutionary conservation. Most common variation has already been discovered and exists in public databases (blue; circles and diamonds); most of the rare variation at both loci was discovered in our study and not contained in public databases (red; circles and diamonds).
4. An application of SeqAnt 2.0: Sequencing the AFF2locus and X chromosome exome in patients with autism
With improvements in methods of targeted enrichment and next-generation sequencing, the targeted sequencing of all genes on a specific chromosome has become feasible. Specific genes/genomic regions is a common experimental design that benefits from the use of SeqAnt [25]. Here we performed an experiment that combined targeted sequencing with chromosomal exome sequencing. We selected 127 males from the Autism Genetic Resource Exchange (AGRE) multiplex collection and 75 males from the Simons Foundation Autism Research Initiative (SFARI) Simplex Collection, New York, NY, USA (SSC) for target DNA amplification and DNA sequencing. From the AGRE collection, we chose multiplex families with two or more male affected sib-pairs who shared >99% of 76 genotyped SNPs in the AFF2 genomic region [22]. One male was randomly chosen if both affected siblings were equally affected; otherwise, the male with autism was chosen over those boys with a diagnosis of not quite autism (NQA) or broad spectrum. From the SSC collection, we chose only those boys who were described as autistic and not reported to have any other syndromes. From the SSC collection, we chose 75 male children from different families with a diagnosis of ASD [26].
For the AGRE samples, we prepared target DNA for sequencing the AGRE samples by performing long PCR (LPCR) amplification of the AFF2 genomic region, followed by sequencing on an Illumina Genome Analyzer. For the SSC samples, we prepared target DNA for Illumina sequencing by using RainDance Technology’s (RDT) microdroplet-based technology to enrich for the human X chromosome exome, as described previously [25]. Following enrichment we performed 70-bp single-end multiplex sequencing on an Illumina Genome Analyzer (IGA). Nearly 20 GB of sequence was generated for AGRE samples, while ~55 GB of sequence was generated for the SSC samples. The AFF2 reference sequence used for the AGRE samples consists of 10 discontiguous fragments covering 84.8 kb, and the SSC reference sequence consisted of the entire human X chromosome, which spanned 5748 discontiguous fragments covering 4.7 Mb. Raw base-calling data generated with the IGA were mapped and variants called using PEMapper (Cutler DJ et al, personal communication). For AGRE samples, 99% of the bases had more than 8X coverage. Median depth of coverage was in the range of 388-1548. For the SSC samples, between 83% and 97% of the targeted reference bases had more than 8X coverage. Median depth of coverage was in the range of 20-607. We identified a total of 286 sites of variation, with 269 single nucleotide polymorphisms (SNPs) and 17 insertions or deletions (INDELs). Overall levels of variation were similar between the two datasets (Θwper site [23]; AGRE - 6.0 x 10-4, SSC - 6.7 x 10-4), with an excess of rare variants as evidenced by a negative value for the Tajima’s D test statistics for both sets of samples ([24]; AGRE: -1.46, SSC: -1.41).
We used SeqAnt to annotate the variants found at the AFF2 locus in the total sample of 202 males with a diagnosis of autism (Mondal et al, in revision). We sought to test the hypothesis that rare variants at the AFF2 locus can act as autism susceptibility alleles. Annotating our variants using the other web-based tools, like the UCSC Genome Browser or the Ensembl Genome Browser, would have been time-consuming and laborious. SeqAnt helped us rapidly annotate these SNPs and INDELs into different functional classes, as well as reported whether a variant had already been cataloged in the dbSNP database (Tables 4, 5). SeqAnt also reported the PhastCons and phyloP conservation scores, which are important in helping to determine whether a variant might cause a deleterious change in the protein structure/function, since variants in the well-conserved sites are likely to cause such changes. By using this feature of SeqAnt, we could easily identify our list of candidate variants that were rare, as well as likely to cause a damaging change.
Functional Class
Total SNPs
SNPs in dbSNP
Novel SNPs
Novel SNPs atConserved Sites
Replacement
5
0
5
5
Silent
8
4
4
4
UTR
33
20
13
1
Intron
223
129
94
6
Total
269
153
116
16
Table 4.
Functional annotation of single nucleotide polymorphisms at the AFF2 locus identified by next-generation sequencing of 202 males with autism
Functional Class
Total Indels
Indels in dbSNP
Novel Indels
Novel Indels atConserved Sites
Exonic
0
0
0
0
UTR
2
0
2
1
Intron
15
7
8
1
Total
17
7
10
2
Table 5.
Functional annotation of indels at the AFF2 locus identified by next-generation sequencing of 202 males with autism
As expected, almost all common variation (>5% frequency in our population) is contained in dbSNP, whereas most rare variants (<5%) are not cataloged in dbSNP (Figure 8). We found that, in our cases, there were five (2.5% of total cases sequenced) singleton nonsynonymous variants. This level of variation in our cases was significantly higher than that seen in a set of 5400 controls. Furthermore, we used SeqAnt to rapidly annotate 1006 X chromosome genes that had been sequenced in the 75 SSC samples, and ultimately showed that the excess mutations at AFF2 were unusual compared to other X chromosome loci. Thus, the ability to rapidly annotate our sequence variants discovered from sequencing the entire X chromosome exome had a major impact on our ability to assess the role of AFF2 as an autism susceptibility locus. Finally, SeqAnt helped us identify three rare noncoding UTR sequence variants, one of which was at an evolutionarily conserved site. Subsequent functional testing suggested that the variant at the conserved site acts to influence the level of AFF2 expression. Thus, for this experiment, SeqAnt allowed us to rapidly focus on those sites of greatest interest for both statistical analyses and direct functional testing.
Figure 8.
Summary of SNV and indel variation discovered at the AFF2 locus in males with ASD. The frequency of SNVs and indels (minor alleles) in cases is plotted against their level of evolutionary conservation. Most common variation has already been discovered and exists in public databases (blue; circles and diamonds). Most of the rare variation at AFF2 was discovered in our study and not contained in public databases (red; circles and diamonds).
5. An application of SeqAnt 2.0: Discovering new mutations from forward genetic screens in the mouse
Forward genetic screens in Musmusculus have been very informative, revealing unsuspected mechanisms governing basic biological processes [27-32]. In this approach, a potent chemical mutagens, such as N-ethyl-N-nitrosourea (ENU), is used to randomly induce mutations in mice. The mice are then bred and phenotypically screened to identify lines that disrupt a specific biological process of interest. Although identifying a mutation using the rich resources of mouse genetics is straightforward, it is unfortunately neither fast nor cheap.
To solve this problem, we developed a methodology that combines multiplex chromosome-specific exome capture, next-generation sequencing, rapid mapping, sequence annotation, and variation filtering to detect newly induced causal variants in a dramatically accelerated way [33]. Rapid sequence annotation and variation filtering are critical to this approach. We used SeqAnt as a part of this methodology for rapid annotation of variations obtained from mutant, parental, and background strains in a single experiment. By using SeqAnt, we first annotated all the variants into different functional classes. Next, by comparing variants identified in mutant offspring to those found in dbSNP, the unmutagenized background strains, and parental lines, we could immediately distinguish the induced putative causative mutations from preexisting variations or experimental artifacts (Table 6).
Mutant Line
Functional Classes
Total Homozygous Variants
In dbSNP
In Background Strains, Not in dbSNP
Remaining Variants
Replacement Variants Within Mapped Region
AB5
Replacement
96
80
13
3
1
AB5
Silent
157
143
12
2
-
AB5
UTR
331
191
135
5
-
AB5
Intronic
106
87
17
2
-
AB5
Intergenic
54
50
4
0
-
M2
Replacement
43
8
31
4
2
M2
Silent
19
11
7
1
-
M2
UTR
73
16
55
2
-
M2
Intronic
46
18
20
8
-
M2
Intergenic
40
4
36
0
-
X5
Replacement
128
59
63
6
2
X5
Silent
192
128
63
1
-
X5
UTR
387
231
152
4
-
X5
Intronic
205
116
86
3
-
X5
Intergenic
89
34
55
0
-
Y1
Replacement
17
1
14
2
1
Y1
Silent
5
0
4
1
-
Y1
UTR
14
2
11
1
-
Y1
Intronic
34
0
31
3
-
Y1
Intergenic
7
0
7
0
-
Table 6.
Results of filtering homozygous variants sites for each mouse mutant line sequenced.
We demonstrated the use of this approach to find the causative mutations induced in four novel ENU lines identified from a recent ENU screen. In all four cases, after applying our method and combining with standard mapping data used to initially localize the variant to a chromosome, we found two or fewer putative mutations (and sometimes only a single one). Confirming that the variant was in fact causative was then easily achieved via standard segregation approaches. SeqAnt gave us the ability to rapidly annotate and screen variants of lesser interest (silent, UTR, intronic, intergenic), so we could instead focus our attention on those variants (replacement) that were most likely to account for the mutant phenotype.
6. An application of SeqAnt 2.0: Exomesequencing to discover mutations affecting neutrophil function in very-early-onset pediatricCrohn’sdisease
Children with very-early-onset (VEO) pediatricCrohn’s disease (CD) are found to have high levels of neutrophil dysfunction. Neutrophils are an abundant type of white blood cell that play an essential role in innate immunity. We therefore hypothesized that children with very-early-onset Crohn’s disease would exhibit an increased frequency of genetic mutations affecting neutrophil function. For an initial study we selected 45 VEO CD patients (median (range) age: 8.5 (5-10) years) with CBir1 sero-reactivity and moderate-to-severe clinical disease activity at diagnosis. We used the Roche NimbleGenSeqCap EZ Human Exome Library v2.0 on genomic DNA extracted from whole blood to capture the whole exome for each patient. Barcodes were used to prepare the libraries for whole-exome capture, which allowed us to sequence two whole exomes per lane of next-generation sequencing. We performed multiplexed 100 base-pair paired-end sequencing on an IlluminaHiSeq 2000 instrument. We used PEMapper (Cutler and Zwick, in revision) to map raw sequence reads and identify variants sites relative to the ~30.8 Mb human exome reference sequence (NCBI37/hg19).
We then used SeqAnt to annotate all variant sites for functional significance, frequency, presence in databases like dbSNP, and measures of evolutionary conservation. Our central hypothesis was that early-onset (pediatric) forms of IBD would be substantially influenced by deleterious mutations found in the neutrophil pathway. If true, a straightforward evolutionary model of mutation-selection balance predicts that these variants ought to be rare in the general population, found at highly evolutionarily conserved sites, and have large effects on gene function. Thus, variants found in coding regions (replacement, nonsense, exonic insertions/deletions) that putatively alter protein structure and function will be the strongest candidates as contributors to IBD in pediatric patients. A number of lines of evidence specifically implicate loci involved in neutrophil functional pathways. We therefore proposed a strategy of first discovering variation in genes known to function in the neutrophil pathway, followed by direct functional testing of alleles from specific patients.
Gene
Location
Variants
Type
Position
Function
Frequency in VEO CD Patients
Frequency in Control Population
CSF2RA
chrX (p22.33)
0
-
-
GM-CSF signaling
-
-
CSF2RB
chr22 (q12.3)
1
SNP
37331455
GM-CSF signaling
0.02
0.0024
CYBB
chrX (p11.4)
1
SNP
37663322
oxidative burst
0.02
0.0032
DUOX1
chr15 (q21.1)
2
SNP SNP
45448069 45431655
enterocyte, H202
0.02 0.02
0.0003 0.0002
DUOX2
chr15 (q21.1)
1
Indel
45393428-30
enterocyte, H2O2
0.02
-
FCGR1A
chr1 (q21.2)
0
-
-
phagocytosis
-
-
FCGR2A
chr1(q23.3)
0
-
-
phagocytosis
-
-
FCGR2B
chr1 (q23.3)
0
-
-
phagocytosis
-
-
FCGR3A
chr1 (q23.3)
0
-
-
phagocytosis
-
-
FCGR3B
chr1 (q23.3)
0
-
-
phagocytosis
-
-
IL27RA
chr19 (p13.12)
1
Indel
14159807
IL-27 signaling
0.02
-
JAK2
chr9 (p24.1)
0
-
-
GM-CSF signaling
-
-
MPO
chr17 (q22)
0
-
-
bacterial killing
-
-
NCF1
chr7 (q11.23)
0
-
-
oxidative burst
-
-
NCF2
chr1 (q25.3)
0
-
-
oxidative burst
-
-
NCF4
chr22 (q12.3)
1
SNP
37273825
oxidative burst
0.02
0.0001
NLRP12
chr19 (q13.42)
0
-
-
chemotaxis
-
-
NOS2
chr17 (q11.2-q12)
3
Indel Indel Indel
26087106 26096042 26085975-76
reactive nitrogen intermediates
0.2 0.2 0.57
- - -
NOX1
chrX (q21.1)
0
-
-
oxidative burst
-
-
NOX3
chr6 (q25.3)
0
-
-
oxidative burst
-
-
NOX4
chr11 (q14.3)
2
SNP SNP
89088208 89182666
oxidative burst
0.02 0.02
- 0.0022
NOX5
chr15 (q23)
0
-
-
oxidative burst
-
-
PRAM1
chr19 (p13.2)
1
Indel
8564497-500
adhesion
0.02
-
RAC1
chr7 (p22.1)
0
-
-
oxidative burst
-
-
RAC2
chr22 (q12.3)
0
-
-
oxidative burst
-
-
SELPLG
chr12 (q24.11)
1
SNP
109017468
adhesion
0.11
-
SLC11A1
chr2 (q35)
2
Indel SNP
219247739 219254723
bacterial killing
0.02 0.02
- -
STAT3
chr17 (q21.2)
2
SNP SNP
40481429 40477064
IL-27 signaling
0.02 0.02
- 0.0002
STAT5A
chr17 (q21.2)
1
SNP
40461109
GM-CSF signaling
0.02
-
STAT5B
chr17 (q21.2)
0
-
-
GM-CSF signaling
-
-
VAV1
chr19 (p13.3)
0
-
-
oxidative burst
-
-
VAV2
chr9 (q34.2)
0
-
-
oxidative burst
-
-
VAV3
chr1 (p13.3)
0
-
-
oxidative burst
-
-
Table 7.
Genetic variants found in genes that regulate neutrophil function.
We used SeqAnt to annotate all the sequence variations from the 45 exomes and identified a total of 60,682 variant sites of interest in coding regions (54,313 replacement SNPs, 2953 indels covering 6369 bases). For our exploratory genome-wide analysis of SNPs, we restricted our analysis to those variants with phyloP scores greater than 2.0, which corresponds to the top 1% of conserved sites in the human genome. Remaining were 12,575, of which 51% (6490) were not cataloged in dbSNP 132 and might constitute novel mutations contributing to early-onset IBD. We then restricted our analysis to 33 neutrophil genes. Table 6contains a list of these 33 neutrophil genes with the number of rare putative functional variants (replacement SNPs or exonicindels). These variants are to be followed up using direct functional assays to assess function. Again, SeqAnt enabled us to rapidly annotate all variants, ignore those variants of lesser interest, and focus our attention on those most likely to contribute to the VEO CD in our sequenced patients.
7. Future directions
We have shown many useful features of SeqAnt and how it can be applied in a variety of experiments, yet we continue to develop SeqANt and plan to expand its functionalities going forward. Our goal is to create a one-stop online tool that readily accepts raw sequencing data and generates output through the annotation and functional characterization stages. Moreover, because our software and libraries are open source, they can be downloaded and optimized locally as part of a next-generation sequencing pipeline. SeqAnt is a truly dynamic application that is updated regularly to keep up with the constant flow of new sequencing data, genome assemblies, and improved annotation information available from public databases like those found at the UCSC Genome Browser.
Genomic sequence annotation requires an up-to-date and comprehensive database of DNA sequence information for a given organism. Our first aim is to continue adding to our database organisms whose genomic information could be annotated. We plan on including several other mammals, vertebrates, invertebrates, and ultimately bacteria strains in the near future. This will give researchers a web application they can use to speed their genetic studies of such organisms. We are also in the process of updating the dbSNP information contained in the SeqAnt database.
Another area of future focus is to broaden the types of input and output files that SeqAnt could work with, while embracing standards in broad use in the bioinformatics community. We intend to include the capability to directly annotate.vcf files as a standard input file format. Presently, all our output files are either text files or BED files. We also plan to provide the option of having the annotation output in.vcf format. Furthermore, we intend to modify SeqAnt to make the.map and.ped files (PLINK formats) from the snp variant file, which will be beneficial for substructure analysis and several other analyses that can be done using PLINK.
The inclusion of additional custom tracks from the UCSC browser to annotate for conserved and putatively functional sites will also be a future area of SeqAnt development. Our hope is that this will improve the effectiveness of downstream functional analysis. We also plan to have the application hosted in a cloud computing environment, side by side with other bioinformatics tools. This is relevant not only because of the wider accessibility it guarantees, but there is often the added ease of using other tools in the same environment to generate and modify input and output files from SeqAnt for further analysis.
SeqAnt was set up to be a dynamic application, and our improvements to this software make it possible to apply SeqAnt to different genomic variant analysis situations. Inevitable advances in sequencing technologies will spur continued demand for tools that can make sense out of the enormous raw sequence data generated, and we will work continually to make SeqAnt adaptable to these improvements and even more accessible to the wider public.
8. Conclusion
Great advances in targeted enrichment methods and DNA sequencing are beginning to allow individual investigators to sequence significant portions of many genomes; the bottleneck this has revealed lies with the annotation and interpretation of the resulting genomic variation data. SeqAnt is a software tool that directly addresses this bottleneck in a wide variety of potential applications. SeqAnt is an open source application that contains a number of unique features. The first is its ability to annotate data from many organisms, not just humans. Second, it is able to perform this analysis with a minimal memory footprint. Third, it completes this analysis in record time, thereby removing a significant bottleneck facing a researcher using the latest next-generation sequencing platforms.
The modifications we made to the application ensure we have the latest data tracks for the species we currently have in the SeqAnt binary databases. Furthermore, we have expanded the number of species that can now be annotated. Finally, with the addition of the PhyloP46Way conservation track, researchers can more confidently assess the evolution and significance of a particular variant site when the phyloP scores are viewed side by side with the PhastCons score values.
We have applied SeqAnt to various studies in our lab, from the work analysis of data on targeted sequencing of particular genes to the analysis of whole-exome data. We also used SeqAnt in the variant annotation of mouse genome and the adaptation of HapMap data for analyzing human exomes. The results from these various applications establish SeqAnt as a user-friendly tool that could help researchers in their work over a wide range of endeavors.
SeqAnt will continue to be an open source web application, which we will constantly update to meet the demands of changing and improving genomic and sequencing technologies. The future of genomics and variation studies lies in our ability to properly use the massive amounts of information we have obtained from DNA sequencing. Sequence annotation tools like SeqAnt that can efficiently turn such data into useable information will play a key role in this future.
Acknowledgement
This work was supported by the National Institutes of Health/National Institutes of Mental Health (NIH/NIMH) and Gift Fund (grant number: MH076439, MEZ); the Simons Foundation Autism Research Initiative (MEZ); and the Training Program in Human Disease Genetics (grant number: 1T32MH087977, DR). We thank members of the Cutler and Zwick labs and Jennifer G. Mulle for discussion, Cheryl T. Strauss for editing, and the Emory-Georgia Research Alliance Genome Center (EGC), supported in part by PHS Grant UL1 RR025008 from the Clinical and Translational Science Award program, National Institutes of Health, National Center for Research Resources, for performing the Illumina sequencing discussed in this chapter. The ELLIPSE Emory High Performance Computing Cluster was used for the development of SeqAnt.
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Sequence annotation tools",level:"2"},{id:"sec_2_2",title:"1.2. The distinction of SeqAnt",level:"2"},{id:"sec_4",title:"2. Upgraded features of SeqAnt 2.0",level:"1"},{id:"sec_4_2",title:"2.1. SeqAnt 2.0 - website updates",level:"2"},{id:"sec_5_2",title:"2.2. SeqAnt 2.0 - Binary database upgrades",level:"2"},{id:"sec_5_3",title:"2.2.1. Upgrade of dbSNP to SNP132 Track for hg19 Assembly (Homo sapiens)",level:"3"},{id:"sec_6_3",title:"2.2.2. Addition of PhyloP46way Conservation Score Database for hg19 Assembly (Homo sapiens)",level:"3"},{id:"sec_7_3",title:"2.2.3. Addition of Full Genome Data Set by Chromosome of Zebrafish (danRer6 Assembly)",level:"3"},{id:"sec_9_2",title:"2.3. SeqAnt 2.0: output directory structure and files",level:"2"},{id:"sec_9_3",title:"2.3.1. All_variations directory",level:"3"},{id:"sec_10_3",title:"2.3.2. BED_annotationdirectory",level:"3"},{id:"sec_10_4",title:"2.3.3. Unique_variations directory",level:"4"},{id:"sec_13_2",title:"2.4. SeqAnt 2.0 - Output files",level:"2"},{id:"sec_13_3",title:"Table 1.",level:"3"},{id:"sec_16",title:"3. An application of SeqAnt 2.0: Targeted next-generation sequencing of NLGN3 and NLGN4X in humans",level:"1"},{id:"sec_17",title:"4. An application of SeqAnt 2.0: Sequencing the AFF2locus and X chromosome exome in patients with autism",level:"1"},{id:"sec_18",title:"5. An application of SeqAnt 2.0: Discovering new mutations from forward genetic screens in the mouse",level:"1"},{id:"sec_19",title:"6. An application of SeqAnt 2.0: Exomesequencing to discover mutations affecting neutrophil function in very-early-onset pediatricCrohn’sdisease",level:"1"},{id:"sec_20",title:"7. Future directions",level:"1"},{id:"sec_21",title:"8. Conclusion",level:"1"},{id:"sec_22",title:"Acknowledgement",level:"1"}],chapterReferences:[{id:"B1",body:'LanderE. 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Rivest et al. patented (US) RSA, which forms the basis for most public encryption systems. RSA describes a public key encryption algorithm and certification process, which protects user data over networks. The patent expired in September 2000 and now is available for general use. According to Marketsandmarkets.com [1], the global network encryption market size is expected to grow from USD 2.9 billion in 2018 to USD 4.6 billion by 2023, at a compound annual growth rate (CAGR) of 9.8%. Major growth drivers for the market include increasing adoption of optical transmission, an increasing demand to meet various regulatory compliances and a growing focus on shielding organizations from network security breaches. In short, RSA forms the basis of almost all public encryption systems. This, however, is not without risk. This chapter explores some of these vulnerabilities in a mathematical context and provides the reader with an appreciation of the strength of RSA.
RSA is secure and difficult to factorize in polynomial time. Conventional sequential computing machines, running in polynomial time, take an unfeasible amount of CPU cycles to find factorization solutions to RSA keys. Quantum computing holds great promise; this, however, is realistically still some way off. Opportunities exist using conventional computing (sequential and parallel) using better mathematical techniques. A discussion on exploiting implementation flaws is also considered.
Of keen interest is our lack of understanding of prime numbers and their structure. The current perception is that there appears to be some underlying structure, but essentially, primes are randomly distributed. This is explored in Sections 8 and 12. Vulnerabilities in the selection of primes are exploited in Section 5 using Euler’s factorization.
Poor RSA key design and their exploits are considered in Section 6 using Wiener’s method and in Sections 15–17 using a combination of LLL, Coppersmith and Pohlig-Hellman. All of these attacks can be mitigated by designing the RSA keys with these exploits in mind. RSA key design (Section 2) consists of two parts, a private key Nd and a public key Ne. A composite number N, is derived from two prime numbers. The de numbers are selected in an ad hoc manner using Euler’s totient.
Development of quantum computing is continuing at breakneck speed; however useful machines are yet to appear. Parallel computing however is here and now, and whilst factorizing RSA keys is not achievable on conventional computers in polynomial time, parallel computing has allowed for multiple solutions to be tested simultaneously. This is an area where research continues and new algorithms as shown in Sections 20 and 14 lend themselves well to GPU parallel processing systems.
RSA100 is a 100 binary bit number made up of two 50 binary bit prime numbers. The motivation in breaking this composite number allows us to find the Euler’s totient number φn. Once this is known, using the public key PU=Ne, it is possible to derive the private key PR=Nd, and hence all cypher-text encrypted (e) messages can thus be decrypted back to plain text, using (d).
3. A simple RSA encryption/decryption example
Using two primes P1 and P2 to generate a composite number N,
Encrypt a message m, into cipher text C, with public key PU. Let the message m = 1461989. C=memodN=14619891313mod2137458620009=1912018123454. To recover the original message, decrypt using Private Key, PR= (N, d) = (1912018123454, 1973036027077)m=CdmodN=19120181234541973036027077mod2137458620009=1461989.
From this simple example, consider the following: How can we use a known public key PU = (N,e) to decrypt the original message? To decrypt the message, the private key is used: PR=Nd. How can d, be discovered? d is derived using Euler’s totient function [φn = (P1 – 1) (P2 – 1)], and the extended Euclidean algorithm edmodφn=1. However when a public key is transmitted, the totient φn and the two primes P1 and P2 remain secret. If φn, P1 or P2 can be determined, the private key will be compromised and the cypher-text will no longer be secure.
When the totient φn is known, d can be determined through the normal key generation processes, so the determination of the two primes (P1, P2) is not required to recover the message from the cypher-text. The following proof is provided for completeness and shows how the two primes P1, P2 can be recovered if the composite N and the totient φn are known.
4. If the composite N and the totient φn are known, the original primes can be recovered
The quadratic formula can be used to find P1 and P2φn=P1−1P2−1,N=P1,P2. General quadratic form: ax2+bx+c=0=>x=−b±b2−4ac2a
Maurice Kraitchik, a Belgian mathematician, considered only values of a and b:a2≡b2modN.
a2≡b2modN⟹Δ14620052mod2137458620009≡16
6. Euler’s factorization method
Gaussian primes are of the form 4x−1, and primes of the form 4x+1 are Pythagorean. Fermat’s Christmas theorem on sum of two squares states that an odd prime can be expressed as P=x2+y2 iff P≡1mod4.
Gaussian primes are of the form P≡3mod4 and are not representable as the sum of two squares.
Consider a composite number N:N = P1P2 and P1: P1=a2+b2, P2:P2=c2+d2.
Wiener’s theorem. Let N=P1P2 and P1<P2<2P1 and a private key PR=Nd and a public key PU=Ne. Let d<13N14, given a public key PU=Ne, with ed≡1modφn. The attacker can efficiently recover d [2]. The attack uses the continued fraction method to expose the private key d, when d is small. It assumes eN≈kd⇒φn=ed−1k. Consider a public key PU=Ne:PU=21374586200091973036027077
Continued fraction 19730360270772137458620009=0111146841125110121111237117=
As per Section 2, if the composite N and the totient φn are known, the original primes P1 and P2 can be recovered.
8. Sum of squares
Overmars [3] showed that all Pythagorean triples could be represented as N=n2+n+2m−12. If the composite number N, is constructed using two Pythagorean primes (4x + 1) then two representations as the sum of two squares can be found. Euler’s Factorization Method (Section 4) can be applied. Finding these two representations is non-trivial and CPU-intensive. The equation Nmn=n2+n+2m−12 provides a course search using increments of n and fine convergence using m. In this way n is incremented and m is decremented about N to find the two solutions along the diagonal of a field of Nmn≈N.
Once the two sum of two squares has been found, Euler’s factorization method (Section 4), can be used to find the prime constructions of N:N=P1P2.
If the composite number (N) is constructed using Pythagorean primes (4x+1), then a solution exists as two sums of two squares and Euler’s factorization method can be applied.
9. Gaussian and Pythagorean primes
As shown in Section 4, if Pythagorean primes (4x+1≡4x−3) are used to construct the composite number (N), a solution exists as two sums of two squares. However, if N is constructed using Gaussian primes (4x−1≡4x+3), then Euler’s sum of two squares method cannot be used. Is there a test that we can use to see if the composite has been constructed using Pythagorean primes? (Table 1)
4x − 1
4x + 1
x,y=3,15
11
13
4y – 1
16xy−4x+y+1
16xy−4x−y−1
59
649
767
4y + 1
16xy−4y−x−1
16xy+4x+y+1
61
671
793
Table 1.
Possible composite constructs using Pythagorean and Gaussian primes.
Consider the following composite constructions:
N=4x+14y+1 using Pythagorean primes
N=4x−14y−1 using Gaussian primes
N=4x+14y−1 using a mix of Pythagorean and Gaussian primes
i. Pythagorean prime construction N=4x+14y+1=16xy+4x+y+1 Two sum of two squares representations exist and Euler’s factorization can be used. 1≡Pmod4. 9≡Pmod16. See Section 4. 793=13∗61=32+282=82+272
ii. Gaussian prime construction N=4x−14y−1=16xy−4x+y+1≡4m−3≡4n+1 Sums of three squares exist. 1≡Pmod4. 9≡Pmod16.
649=11∗59=12+182+182=32+82+242=62+172+182=82+122+212=102+152+182=122+122+192 Legendre’s three-square theorem can test the composite: N=x2+y2+z2true ifN≠4a8b+7a,b∈Z,
iii. Mixed Pythagorean-Gaussian prime construction N=4x+14y−1=16xy−4x−y−1,N=4x−14y+1=16xy+4x−y−1. Sums of four squares exist. 3≡Pmod4. 13∗59=767
In summary, a composite whose construction is based upon both Pythagorean and Gaussian primes can easily be identified when Pmod4≡3 is true. However, sums of four squares exist and Euler’s factorization cannot be used. When Pmod4≡1 is true, the composite could be constructed using Pythagorean primes or Gaussian primes. Use the Legendre test to further discriminate. When the Pythagorean construct is confirmed, the two sums of two squares can be found, and Euler’s factorization can be used. If the composite construction is both Pythagorean and Gaussian, sums of three squares exist and Euler’s factorization cannot be used.
10. Overmars factorization method
Another classification of the composite number uses a different construct for primes and seeks to define the composite number as follows: Let N=P1P2 and test N:N±1mod4=0. Two cases are considered in the classification, and this determines the constructs of the primes used. Note the sign of ±1 determines the case used, and the test is both simple and concise [4].
N + 1 is the better candidate, as it has more factors to try. So cases (1,2) are considered.
Case (2) N=am−n−1am+n+1=a2m2−n2−2an−1N+1a=am2−n2−2nTrya:a=25:N+1a=am2−n2−2n,N+110=10m2−n2−2n=N+120=5m2−n2−nm≥Na=3902057185540126551228957333948437101890500690019
When a is small, this method becomes computationally unfeasible.
12. Overmars factorization using smooth factors
Consider the construction of primes (Sections 8 and 9), P=am±n±1. More generally, P:P=am±n±x Consider N=P1P2⇒8079781=1249×6469 (Table 2).
Case (1) ⊕⊝N+x2moda2=0,P1=am−n+x,P2=am+n−x
x
N − x2
±x
a
m
n
gcd(m,n)
Smoothness
1
2235311433
⊝⊝
10
386
261
1
5-smooth
3
224794217
⊝⊝
2
1931
1305
1
5
223673313
⊝⊝
6
644
435
1
7
22321032179
⊕⊕
18
214
145
1
3-smooth
11
2232544887
⊝⊝
90
43
29
1
5-smooth
13
2232113191
⊕⊕
6
641
435
1
17
223673291
⊝⊝
6
646
435
1
19
223517892
⊕⊕
30
128
87
1
5-smooth
23
223673271
⊝⊝
6
647
435
1
29
223454987
⊝⊝
18
216
145
1
5-smooth
Table 2.
N − x2.
N=am−n+xam+n−x=a2m2−n2+2anx−x2
N=am2−an2−2anx+1=am2−an−x2
Case (2) ⊝⊕N+x2moda2=0,P1=am−n−x,P2=am+n+x
N=am−n−1am+n+1=a2m2−n2−2anx−x2
N=am2−an2+2anx+1=am2−an+x2
Case (1,2) N+x2a=am2−n2±2nxa:aisafactor ofN+x2
n=am2−N±xa,m≥N+a∓x2a,m=N+an±x2a2
Case (3) ⊝⊝N−x2moda2=0,P1=am−n−x,P2=am+n−x
N=am−n−xam+n−x=a2m2−n2−2amx+x2
N=am2−2amx+x2−an2=am−x2−an2
Case (4) ⊕⊕N−x2moda2=0,P1=am−n+x,P2=am+n+x
N=am−n+xam+n+x=a2m2−n2+2amx+x2
N=am2+2amx+x2−an2=am+x2−an2
Case (3,4) N−x2a=am2−n2±2mxa:aisafactor ofN−x2
n=am±x2−Na2,m≥N∓x2a,m=N+an2±x2a
N=9043−29−119043+29−11=1249×6469
When a smooth x can be found, larger a values allow for faster convergence to a solution. The selection of x and a is somewhat arbitrary and prime constructs are a modification of Fermat’s a2 − b2. Smooth factors of N±x2 produce larger a values and convergence faster to a solution.
13. Primes
The current state of the art in prime number generation is Atkin’s sieve [5, 6].
The algorithm completely ignores any numbers with remainder mod 60 that is divisible by 2, 3 or 5, since numbers with a mod 60 remainder divisible by one of these three primes are themselves divisible by that prime. Atkin stated three theorems given below:
All numbers n with mod 60 remainder 1, 13, 17, 29, 37, 41, 49 or 53 are mod 4 ≡ 1. These numbers are prime if the number of solutions to 4x2 + y2 = n is odd and the number is squarefree.
All numbers n with mod 60 remainder 7, 19, 31 or 43 have a mod 6 ≡ 1. These numbers are prime if and only if the number of solutions to 3x2 + y2 = n is odd and the number is squarefree.
All numbers n with mod 60 remainder 11, 23, 47 or 59 have a mod 12 ≡ 11. These numbers are prime if and only if the number of solutions to 3x2 − y2 = n is odd and the number is squarefree.
None of the primes are divisible by 2, 3 or 5 and are not divisible by their squares (22, 32, and 52). For a thorough analysis of “primes of the Form x2 + ny2” the reader is referred to a text by Cox [7].
The often overlooked works of Dubner, who is credited with the term “primorial” [8] are now considered [9, 10]. The primorial is a factorial of primes: 1#=2,2#=2x3=6,3#=2x3x5=30,4#=#3x7=210 and so on. 0#=1. The primorial is by definition squarefree.
The nth primorial is the product of n primes, where πn is the prime counting function.
n#=∏i=1πnpi=pπn#
Using this structure, Dubner was able to create series of primes in a particular primorial.
It can be shown that the structure of primes is palindromic in the primorials [11].
For example, in Figure 1, take the discrete derivative of the numbers in the third primorial, 3#. The following palindromic sequence can be added to #3=30 and subtracted from #4=210 to determine all of the primes in that primorial:
This describes the second table in Figure 1. All of the primes in the third primorial can be found using 24 small numbers. Mod 7 is used to sieve and eliminate composite multiples of 7. Mod 11 and 13 are used to highlight further composites, but these are kept and used to generate primes in the next primorial.
Modulo testing: Pmodm=0,Pk<m<k+1#
For k=3,Pk:P3=5,Pk+1:P4=7,#3=30,#4=210,210≈14,m=7,11,13, eliminate Pk+1=7
As shown in Figure 2, 24 small numbers are used to derive 482 new values. This uses 10 modulo tests to identify composites and 1 modulo test to eliminate factors of 11 (Figure 3).
Figure 2.
Primes in the 4th primorial.
Figure 3.
Gaps between primes of each successive primorial.
Pn#, ΔPn−1# Current primorial and the difference between primes from the previous. Simple array descriptor provides rich prime fields of higher densities. Small numbers describe primes of higher magnitude. Large arrays of primes can be stored in much less memory.
14. Number systems
Conventional numbering systems consist of a base (or radix).
The primorial number system is said to be ‘primoradic’; having a primorial base. The primorial number system is a mixed radix numeral system adapted to the numbering of the primorials (Table 3).
n
…
7
6
5
4
3
2
1
pn
n…
17
13
11
7
5
3
2
n#
…
510510
30030
2310
210
30
6
2
highest
Pn+1−1
18
16
12
10
6
4
1
Table 3.
Primorial radix number system.
General properties of mixed radix number systems apply to the base primorial system. The primorial number system OEIS A000040 is denoted by a subscript “∏”.
The (LLL) forms the basis of the Coppersmith attack (Section 15), and a brief explanation is given here with further reading and references for the reader. The Lenstra-Lenstra-Lavász (LLL) lattice basis reduction algorithm [13] calculates an LLL-reduced, short, nearly orthogonal lattice basis, in time Od5nlog3B, where B is the largest length of bi under the Euclidean norm, given a basis B=b1b2…bd with n-dimensional integer coordinates, for a lattice L (a discrete subgroup of Rn) with d≤n and giving polynomial-time factorization of polynomials with rational coefficients.
A thorough explanation is given by Bosma [14], and a summary of the example contained in the reference is given below.
INPUT: Let lattice basis b1,b2,b3∈Z3 be given by the columns of 1−13105126
OUTPUT: LLL-reduced basis 01−1100012
Using the Lenstra-Lenstra-Lavász lattice reduction (LLL), the short vectors in a lattice can be found. This is used by the Coppersmith attack. Coppersmith\'s algorithm uses the LLL to construct polynomials with small coefficients that all have the same root modulo. When a linear combination is found to meet inequality conditions, standard factorization methods can find the solutions over integers.
17. Coppersmith attack
When d is small andeis largeviathe Euler totient rule, the Wiener attack (Section 5) can be used. Conversely, when d is large, e is small. Particular applications of the Coppersmith method for attacking RSA include cases when the public exponent e is small or when partial knowledge of the secret key is available (Section 13) [15].
A small public exponent e, reduces the encryption time. Common choices for e are 3, 17 and 65537216+1 [16]. These are Fermat primes Fx:Fx=22x+1 and are chosen because the modular exponent derivation is faster. The Coppersmith method reduces the solving of modular polynomial equations to solving polynomial equations over integers.
Let Fx=xn+an−1xn−1+…+a1x+a0 and Fx0≡0modM for an integer x0<M1n. Coppersmith can find the integer solution for x0 by finding a different polynomial f related to F that has the root x0modM but only has small coefficients. The small coefficients are constructed using the LLL (Section 14). Given F, the LLL constructs polynomials p1x,p2x,…pnx that all have same root x0modMa,a∈Z.a depends on the degree of F and the size of x0. Any linear combination has the same root x0modMa.
The next step is to use LLL to construct a linear combination fx=∑cipix of the pix so that the inequality fx0<Ma holds. Then standard factorization provides the zeroes of fx over Z.
Let N be an integer and f∈Zx be a monic polynomial of degree d, over integers such that xd+cn−1xd−1+…+c2x2+c1x+c0. Set X=N1d−∈ for 1d>∈>0. Given Nf then all integers x0<X:fx0≡0modN can now be found. All roots of fmodN, smaller than X=N1d can be found.
18. Pohlig-Hellman
The Pohlig-Hellman [17] algorithm is a method to compute a discrete logarithm (which is a difficult problem) on a multiplicative group. The order of which is a smooth number (also called friable), meaning its order can be factorized into small primes. A positive integer is called B-smooth if none of its prime factors is greater than B. For example, 1620 has prime factorization 22 × 34 × 5; therefore 1620 is 5-smooth because none of its prime factors are greater than 5. This is similar to that of the Overmars factorization method (Section 10). The Pohlig-Hellman [17] algorithm applies to groups whose order is a prime power. The basic idea is to iteratively compute the p-adic digits of the logarithm by repeatedly “shifting out” all but one unknown digit in the exponent and computing that digit by elementary methods. This is a similar idea to Section 13.
INPUT: A cyclic group G of order n with a generator g, an element h∈G, and a prime factorization n=∏i=1rpiei OUTPUT: The unique integer x∈0…n−1:gx=h
Example: Let p=41,α=7,β=12 solve 12=7xmod41
Find the prime factors of p−1⇒41−1=40=235⇒gs=2,5. Find one x for each g.
For g=2,x=20x0+21x1+22x223⇒cubic→three terms
i. x0:βp−1g0=αp−1gX0⇒12402=7402x0−1mod41=−1x0mod41test forx0:x0=0,1,2,…
By the Chinese remainder theorem, x=13mod40since the exponentsarep−1=41−1=40hence12≡713mod41. So the solution to 12=7xmod41⇒x=13.
19. Shor’s algorithm
Shor’s algorithm [18], factors composite numbers, N=P1P2, consisting of two primes in polynomial time using quantum computing techniques. The algorithm evaluates the period of axmodn where gcdan=1. This is inefficient using sequential computing on a conventional computer. When run on a quantum computer, a congruence of squares with probability 0.5 occurs in polynomial time. For two co-prime sinusoids of period P1 and P2, at what point do they zero-cross each other? The phase of each sinusoid at any given point is observed, and if they are factors of N then the phase of P1 and P2 is zero. Shor’s algorithm tests the phase of P1=P2=N=0 (Figure 4).
Figure 4.
N as a composite of two Sinusoids P1 and P2 [19].
Phase estimation is well suited to quantum computers and hence this factorization technique produces solutions in polynomial time. For further information on quantum phase estimation, the reader is directed to WIKI [20]. The impact of this type of attack is discussed in detail by Mosca [21].
Choose a<N
Find the period r of anmodN (using Quantum computing)
Public infrastructure cryptographic hardware uses a library RSALib. This is found in both NIST FIPS 140-2 and CC EAL 5+. These are certified devices for use in identity cards, passports, Trusted Platform Modules, PGP and tokens for authentication and software signing. This is in use in tens of millions of devices worldwide. Nemec et al. [22] have identified a vulnerability that allows for the factorization of 1024 and 2048 bit keys in less than 3 CPU months.
RSALib primes are of the form p=k∗M+65537amodM.
These can be fingerprinted using the discrete logarithm log65537NmodM.
The public modulus N is generated by 65537 in the multiplicative group ZM∗. The public modulus of RSALib can thus be fingerprinted with the discrete logarithm c=log65537NmodM. This can be factorized using Pohlig-Hellman (Section 16). The group G=65537 is smooth G=24∗34∗52∗7∗11∗13∗17∗23∗29∗37∗41∗53∗83 for RSA512 keys. The smoothness of G is due to the smoothness of M being Primorial.
Factorization is achieved using the Coppersmith algorithm with a known pmodM:65537amodM. Nemec et al used the Howgrave-Graham[23] implementation of the Coppersmith’s algorithm to find a small solution x0 of: fx=x+Mp−1modN∗65537a′modM′modN
A summary of RSALib vulnerability and its impact is now given and the reader is directed to Memec et al. [22] for further detail. eIDs used in passports for citizens are affected. Code signing is vulnerable. Twenty-four percent of TPMs used in laptops are affected (sample size 41). A third of PGP, used in email systems could be factorizable. There was no observable impact on TLS/HTTPS. One hundred percent of SCADA systems sampled were affected (sample 15). E-health and EMV payment cards were also likely to be susceptible.
Mitigating the impact of the RSALib vulnerability requires changing the algorithm. This requires a firmware replacement which is not possible in already deployed devices such as smartcards and TPMs whose code is stored in read-only memory. Key lengths not of 512, 1024, 2048 and 4096, such as RSA3936 appear to be resilient. The use of key pairs outside of vulnerable devices could be deployed using another library. Changes to RSALib are required so that proveable safe primes are constructed not using the vulnerability.
21. Overmars factorization, bringing it together
Section 11 considered the following cases. The following discussion generalizes these cases and provides the structure for algorthmic solutions to be found. The palindromic nature of primes (Section 12) can be exploited further to explore solutions in a particular Primorial range. Recall;
Now we need to develop the methodology for finding (selecting) a and x. This brings together the concepts of primorials [9], Smooth [24], small factors [17], factorization (Fermat), modulo testing as per Atkin’s Sieve [5] and the structure of primes (Sections 12 and 18), to find as large an a as possible so that Overmars Factorization [4] converges more rapidly to a solution.
Recall the following (Section 12). Primes are of the form P=4x±1andP=6x±1. Composite numbers, constructed from these primes: N=P1P2, are a combination of Pythagorean and Gaussian primes. The following test N±1mod4≡0 can be used to determine which combination of primes was used to construct the composite. If N+1mod4≡0 is true a mix of Pythagorean and Gaussian primes was used. If N−1mod4≡0 is true then the composite consists of only Gaussian or only Pythagorean primes. The Sieve of Atkin [5] uses mod12≡0 and mod60≡0. This is now applied as per Overmars [4] in the following manner, if mod12≡0 is true then a=6, if mod60≡0 is true let a=30. The ideas of Atkin are further extended in both directions: mod4≡0⇒a=2,mod420≡0⇒a=210,mod4620≡0⇒a=2310,mod60060≡0⇒a=30030…
This is Primorial, Pk#:Pk#,kthPrimorial is”Smooth”. The general form (Section 19) is now given: Case (1 ⊕⊝, 2 ⊝⊕) N+x2a=am2−n2±2nx, N+x2moda≡0,a:a=2Pk#,x:1≤x≤Na Case (3 ⊝⊝, 4 ⊕⊕) N−x2a=am2−n2∓2mx,N−x2moda≡0,a:a=2Pk#,x:1≤x≤Na
If a:a=2Pk# can be choosen, then we search x in the primes to find solutions to N±x2mod2Pk#≡0 A solution is found for P1m, when P1∈Z. Case (1 ⊕⊝, 2 ⊝⊕) NmodP1≡0,P1:P1=am−am2−N Case (3 ⊝⊝,4⊕⊕) NmodP1≡0, P1:P1=am∓x−am∓x2−N
Consider Section 11 example, N=P1P2⇒8079781=1249∗6469
Integer solutions x=N−2bPk#. From Table 5, determining which x value should be used is not clear. Whilst x=1 should work, no solutions will be found if a:a=30. From Table 5 only when x=11or19 do we find solutions. Ranking the possible solutions in terms of factors 29 (8) would be first, 19 (7) second and 11 (6) third.
x
mod60
mod180
mod1620
N − x2
±x
b
a
m
n
gcd(m,n)
Smoothness
1
0
2235311433
⊝⊝
10
386
261
1
5-smooth
1
0
2235311433
⊕⊕
6
643
435
1
5-smooth
11
0
0
2232544887
⊝⊝
3
90
43
29
1
5-smooth
19
0
223517892
⊕⊕
1
30
128
87
1
5-smooth
29
0
0
0
223454987
⊝⊝
18
216
145
1
5-smooth
Table 5.
Smooth candidates of the factors of N − x2.
Based upon low order factors the rankings would be 29 2234 first and 11 2232 second. Setting a=30,x=29 will not find solutions for m,n. Setting a=30,x=11⇒m=129,n=57,gcd12957=3, so the optimal value for a=90.P1=30m−11−30m−112−N. Look for solutions to 30m−112−N which are a perfect square. In this case, m=129⇒30∗129−112−8079781=6812100=26102.
Recall that the starting value for m:N+a2±x2a≤m<N−12a⇒99≤m<134663, 30 iterations.
Whilst this is quite a good result the first failure needs also to be taken into account. This would be bound by the Primorial and P1:1<P1<N:am−am2−N=1⇒m<N+12a
Here m:N+a2±x2a≤m<N+12a≤123≤m<134663⇒134540iterations.
This can be further bound by the Primorial. In the case of RSA numbers, the binary bits available to represent a particular prime number range can also be used to bound the range (Table 6).
x
Modulo testing
N − x2
a
m
n
gcd(m,n)
Smoothness
60
420
4620
60060
1
0
233251014611423
30
5-smooth
11
0
25351397157251
30
5524
2002
2
5-smooth
19
0
0
2433571577531
210
789
286
1
7-smooth
61
0
0
0
2435711310667
2310
11-smooth
401
0
0
0
0
2335721113191493
30030
13-smooth
1601
0
0
0
0
233357111321697
30030
13-smooth
45281
0
0
0
0
233571113181501
30030
13-smooth
45589
0
0
0
0
25357111345317
30030
4
2
2
13-smooth
Table 6.
Smooth candidates of the factors of N − x2.
Consider N=23852269081.
In this case, solutions using modulo testing generate good candidates to solve for (m, n), however for a=30030, three of the candidates have no solution. Using sequential programing, each possible candidate is considered one after another, until the maximum m value. However, using parallel programing techniques on GPUs (such as nVIDIA P100s), all of the candidates can be tested simultaneously and the processes are all terminated when one of the processes finds a solution. This is very efficient and effective in finding P1, P2. Once these are known, along with the public key Pu=Ne, using Euler’s totient, the private key PR=Nd can be determined. Once the private key is known the cypher-text is no longer secure.
22. Conclusion
In short RSA is secure and difficult to factorise. Conventional sequential computing machines, running in polynomial time, take an infeasible amount of CPU cycles to find factorization solutions to RSA keys. Quantum computing holds great promise and Shor’s algorithm [18] demonstrates how this can be achieved. However, quantum computing is realistically still some way off. Opportunities exist using conventional computing (sequential and parallel) with better mathematical techniques. Section 18 showed how implementation vulnerabilities are introduced when “clever” low cost (CPU cycles) are implemented. The case in point showed methods for signature identification, upon which tailored targeted attacks could be launched against infrastruture FIPS140-2 devices, such as cryptographic routers. These sorts of attacks can be deployed in polynomial time using sequential programing techniques. Section 20, Overmars shows how factorization can be implemented using parellel processing techniques.
There is still much to be done and areas of further interest are a better understanding of the structure of primes. This will lead to faster prime number generating algorithms and hence faster solutions to the factorization problem. This will also lead to the generation of more robust primes that are less susceptible to factorization methods. An example of this is the use of non-Pythagorean primes. Section 5 showed how Euler’s factorization could be used to attack such composite numbers. Hence a simple method to thwart this would be to use a mix of Pythagorean and Gaussian primes. Section 6 showed how small d values in the RSA private key PR=Nd could be attacked using Wiener’s method. Small e values in the public key PU=Ne can be attacked using a combination of LLL, Coppersmith and Pohlig-Hellman (Sections 15–17). All of these attacks can be mitigated by choosing dande carefully and ensuring that both are sufficiently large.
Development of quantum computing is continuing at break-neck speed, however useful machines are yet to appear. Parallel computing however is here and now and whilst factorizing RSA keys is not achievable on conventional computers in polynomial time, parallel computing has allowed for multiple solutions to be tested simultaneously. This is an area where research continues and new algorithms such as shown in Sections 20 and 14 lend themselves well to GPU parallel processing systems.
“There are known knowns. These are things we know that we know. There are known unknowns. That is to say, there are things that we know we don\'t know. But there are also unknown unknowns. There are things we don\'t know we don\'t know” [25].
\n',keywords:"survey, public keys, vulnerability",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/67685.pdf",chapterXML:"https://mts.intechopen.com/source/xml/67685.xml",downloadPdfUrl:"/chapter/pdf-download/67685",previewPdfUrl:"/chapter/pdf-preview/67685",totalDownloads:164,totalViews:0,totalCrossrefCites:0,dateSubmitted:"November 16th 2018",dateReviewed:"January 31st 2019",datePrePublished:"June 17th 2019",datePublished:"November 27th 2019",readingETA:"0",abstract:"Rivest et al. patented (US) RSA. RSA forms the basis of most public encryption systems. It describes a public key encryption algorithm and certification process, which protects user data over networks. The patent expired in September 2000 and now is available for general use. According to Marketsandmarkets.com, the global network encryption market size is expected to grow from USD 2.9 billion in 2018 to USD 4.6 billion by 2023, at a compound annual growth rate (CAGR) of 9.8%. Major growth drivers for the market include increasing adoption of optical transmission, an increasing demand to meet various regulatory compliances and a growing focus on shielding organizations from network security breaches. In short, RSA forms the basis of almost all public encryption systems. This, however, is not without risk. This chapter explores some of these vulnerabilities in a mathematical context and provides the reader with an appreciation of the strength of RSA.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/67685",risUrl:"/chapter/ris/67685",signatures:"Anthony Overmars",book:{id:"8140",title:"Modern Cryptography",subtitle:"Current Challenges and Solutions",fullTitle:"Modern Cryptography - Current Challenges and Solutions",slug:"modern-cryptography-current-challenges-and-solutions",publishedDate:"November 27th 2019",bookSignature:"Menachem Domb",coverURL:"https://cdn.intechopen.com/books/images_new/8140.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"222778",title:"Prof.",name:"Menachem",middleName:null,surname:"Domb",slug:"menachem-domb",fullName:"Menachem Domb"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:[{id:"285451",title:"Dr.",name:"Anthony",middleName:null,surname:"Overmars",fullName:"Anthony Overmars",slug:"anthony-overmars",email:"3@crykey.com",position:null,institution:null}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Structure of RSA numbers",level:"1"},{id:"sec_3",title:"3. A simple RSA encryption/decryption example",level:"1"},{id:"sec_4",title:"4. If the composite N and the totient φn are known, the original primes can be recovered",level:"1"},{id:"sec_5",title:"5. Fermat’s factorization method",level:"1"},{id:"sec_6",title:"6. Euler’s factorization method",level:"1"},{id:"sec_7",title:"7. Wiener attack",level:"1"},{id:"sec_8",title:"8. Sum of squares",level:"1"},{id:"sec_9",title:"9. Gaussian and Pythagorean primes",level:"1"},{id:"sec_10",title:"10. Overmars factorization method",level:"1"},{id:"sec_11",title:"11. Extensions of the Overmars factorization method",level:"1"},{id:"sec_12",title:"12. Overmars factorization using smooth factors",level:"1"},{id:"sec_13",title:"13. Primes",level:"1"},{id:"sec_14",title:"14. Number systems",level:"1"},{id:"sec_15",title:"15. RSA100 factorization using primorials",level:"1"},{id:"sec_16",title:"16. Lenstra-Lenstra-Lavász lattice reduction (LLL)",level:"1"},{id:"sec_17",title:"17. Coppersmith attack",level:"1"},{id:"sec_18",title:"18. Pohlig-Hellman",level:"1"},{id:"sec_19",title:"19. Shor’s algorithm",level:"1"},{id:"sec_20",title:"20. Attacking public key infrastructure",level:"1"},{id:"sec_21",title:"21. Overmars factorization, bringing it together",level:"1"},{id:"sec_22",title:"22. Conclusion",level:"1"}],chapterReferences:[{id:"B1",body:'Available from: https://www.marketsandmarkets.com/Market-Reports/network-encryption-market-187543224.html [Retrieved: 17 January 2019]'},{id:"B2",body:'Boneh D. Twenty years of attacks on the RS cryptosystems. Notices for the American Mathematical Society (AMS). 1999;46(2)'},{id:"B3",body:'Overmars A, Ntogramatzidis L, Venkatraman S. A new approach to generate all pythagorean triples. AIMS Mathematics. 2019;4(2):242-253. DOI: 10.3934/math.2019.2.242'},{id:"B4",body:'Overmars A, Venkatraman S. 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TBA; 2019'},{id:"B13",body:'Lenstra AK, Lenstra HW Jr, Lovász L. Factoring polynomials with rational coefficients. Mathematische Annalen. 1982;261(4):515-534'},{id:"B14",body:'Bosma W. Chapter 4 LLL. In: Lecture Notes. 2010. pp. 86-109. Available from: http://www.math.ru.nl/∼bosma/onderwijs/voorjaar07/compalg7.pdf [Retrieved: 17 January 2019]'},{id:"B15",body:'Coppersmith D. Finding a small root of a bivariate integer equation; factoring with high bits known. Lecture Notes in Computer Science. 1996;1070:178-189'},{id:"B16",body:'Salah IK, Darwish A, Oqeili S. Mathematical Attacks on RSA cryptosystems. Journal of Computer Science. 2006;2(8):665-671'},{id:"B17",body:'Pohlig SC, Hellman ME. An improved algorithm for computing logarithms over GF(p) and its cryptographic significance. IEEE Transactions on Information Theory. 1978;IT-24(1):106-110'},{id:"B18",body:'Shor PW. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. 1995. 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Finding small roots of univariate modular equations revisited. In: Proceedings of the 6th IMA International Conference on Cryptography and Coding. Springer-Verlag; 1997. pp. 131-142'},{id:"B24",body:'Schumaker R. The formulas for the distribution of the 3-smooth, 5-smooth, 7-smooth and all other smooth numbers. 2016. Available from: https://arxiv.org/abs/1608.06928'},{id:"B25",body:'Rumsfeld D. 2002. Available from: https://en.wikipedia.org/wiki/There_are_known_knowns'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Anthony Overmars",address:"3@crykey.com",affiliation:'
Curtin University, Perth, Australia
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Open Access publication costs can often be designated directly in the grants or in specific budgets allocated for that purpose. Many of the most important funding organisations encourage, and even request, that the projects they fund are made available at no cost to the wider public. IntechOpen strives to maintain excellent relationships with these funders and ensures compliance with mandates.
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In order to help Authors identify appropriate funding agencies and institutions, we have created a list, based on extensive research on various OA resources (including ROARMAP and SHERPA/JULIET) of organizations that have funds available. Before consulting our list we encourage you to petition your own institution or organization for Open Access funds or check the specifications of your grant with your funder to ascertain if publication costs are included. Where you are in receipt of a grant you should clarify:
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Please note that this list is not a definitive one and is updated regularly. To suggest possible modifications or the inclusion of your institution/funder, please contact us at oapf@intechopen.com
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Please be aware that you must be a member, or grantee, of the institutions/funders listed in order to apply for their Open Access publication funds.
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\n\n
In order to help Authors identify appropriate funding agencies and institutions, we have created a list, based on extensive research on various OA resources (including ROARMAP and SHERPA/JULIET) of organizations that have funds available. Before consulting our list we encourage you to petition your own institution or organization for Open Access funds or check the specifications of your grant with your funder to ascertain if publication costs are included. Where you are in receipt of a grant you should clarify:
\n\n
\n\t
Does your institution already have a budget for covering Open Access publication costs?
\n\t
Does your grant list Open Access publication fees as legitimate direct/indirect costs?
\n
\n\n
If you are associated with any of the institutions in our list below, you can apply to receive OA publication funds by following the instructions provided in the links. Please consult the Open Access policies or grant Terms and Conditions of any institution with which you are linked to explore ways to cover your publication costs (also accessible by clicking on the link in their title).
\n\n
Please note that this list is not a definitive one and is updated regularly. To suggest possible modifications or the inclusion of your institution/funder, please contact us at oapf@intechopen.com
\n\n
Please be aware that you must be a member, or grantee, of the institutions/funders listed in order to apply for their Open Access publication funds.
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