Sieve Algorithm Research Articles

Integer factorization and discrete logarithm problems

These are notes for a lecture given at CIRM in 2014, for the Journees Nationales du Calcul Formel. We explain the basic algorithms based on combining congruences for solving the integer factorization and the discrete logarithm problems. We highlight two particular situations where the interaction with symbolic computation is visible: the use of Grobner basis in Joux's algorithm for discrete logarithm in nite eld of small characteristic, and the exact sparse linear algebra tools that occur in the Number Field Sieve algorithm for discrete logarithm in large characteristic.

A parallel line sieve for the GNFS Algorithm

RSA is one of the most important public key cryptosystems for information security. The security of RSA depends on Integer factorization problem, it relies on the difficulty of factoring large integers. Much research has gone into problem of factoring a large number. Due to advances in factoring algorithms and advances in computing hardware the size of the number that can be factorized increases exponentially year by year. The General Number Field Sieve algorithm (GNFS) is currently the best known method for factoring large numbers over than 110 digits. In this paper, a parallel GNFS implementation on a BA-cluster is presented. This study begins with a discussion of the serial algorithm in general and covers the five steps of the algorithm. Moreover, this approach discusses the parallel algorithm for the sieving step. The experimental results have shown that the algorithm has achieved a good speedup and can be used for factoring a large integers.

Sparklab2: A Statistically Based Program to Detect 2D Sparks. Application to Smooth Muscle

We improved the statistical sieve algorithm developed by Izu et al. (Biophys. J. 92(12):4458-65) and simplified the user interface. In this new version, the normalized image (Norm) and background fluorescence is automatically calculated by the program. After reading the stack of images for a given cell, a template is created by averaging all the images. This template will constitute the norm against which each individual frame will be normalized. The cell border is detected from this template, and fluorescence outside this area is considered the background that is subtracted from all images. Then each frame is normalized by the Norm image. The next step highlights events as potential sparks, if fluorescence is ≥ 3 standard deviations from the Norm value. A final statistical test in the time domain is applied at each potential event. If the amplitude of the event is statistically different from the mean value of the previous 10 frames, the event is considered a spark. A table with the amplitude, FWHMx, FHHMy, Tau, Frame and location (x,y) is generated. Additionally the user can inspect each of the sparks in a 3D plot or its fitting profiles in x,y and t axis. A user friendly interface implemented in LabVIEW with intuitive menus guide the user through the whole process. We tested the functionality of the program by analyzing Ca2+ sparks from vascular smooth muscle. SparkLAB2 presents the advantages of being a statistically based spark detector with minimum intervention of the user thus reducing the possibility of human bias.

Improved techniques for computing the ideal class group and a system of fundamental units in number fields

We describe a new algorithm for computing the ideal class group, the regulator and a system of fundamental units in number fields under the generalized Riemann hypothesis. We use sieving techniques adapted from the number field sieve algorithm to derive relations between elements of the ideal class group, and p-adic approximations to manage the loss of precision during the computation of units. This new algorithm is particularily efficient for number fields of small degree for which a speed-up of an order of magnitude is achieved with respect to the standard methods.

Modifications in RSA through Positive Justification of Random Components

There is a surge of interest today in Wireless Networks both for organizations and individuals. It’s emerging as general consensus that the current generation as well as next generation will have more of wireless connectivity. However, wireless networks have many security problems. Effective Wireless Security policies are required as wireless networks have been found to be relatively easy to break in by the hackers. The various techniques are being used for counteracting security risks. The most prevalent algorithm warranting security is RSA .RSA, a public - key cryptography algorithm is based on the factorization of large prime numbers. Presently the use of 512 bits keys are considered as insecure keys after the successful attack and by implementing the General Number Field Sieve (GNFS) algorithm .It has still got gaps of security that needs to be handled. In this, modifications has been proposed in RSA 512 bit and modified RSA through positive justification of alphabets through appending in the plain text and prime numbers in cipher text through multiplication to make the communication more secure. As the large prime numbers are not easily perishable, this modification will provide reliability over the network .This paper proposes a modification in classical RSA and modified RSA in which random numbers are appended in cipher text leading to improvement in previous results. The

Coreference analysis in clinical notes: a multi-pass sieve with alternate anaphora resolution modules

This paper describes the coreference resolution system submitted by Mayo Clinic for the 2011 i2b2/VA/Cincinnati shared task Track 1C. The goal of the task was to construct a system that links the markables corresponding to the same entity. The task organizers provided progress notes and discharge summaries that were annotated with the markables of treatment, problem, test, person, and pronoun. We used a multi-pass sieve algorithm that applies deterministic rules in the order of preciseness and simultaneously gathers information about the entities in the documents. Our system, MedCoref, also uses a state-of-the-art machine learning framework as an alternative to the final, rule-based pronoun resolution sieve. The best system that uses a multi-pass sieve has an overall score of 0.836 (average of B(3), MUC, Blanc, and CEAF F score) for the training set and 0.843 for the test set. A supervised machine learning system that typically uses a single function to find coreferents cannot accommodate irregularities encountered in data especially given the insufficient number of examples. On the other hand, a completely deterministic system could lead to a decrease in recall (sensitivity) when the rules are not exhaustive. The sieve-based framework allows one to combine reliable machine learning components with rules designed by experts. Using relatively simple rules, part-of-speech information, and semantic type properties, an effective coreference resolution system could be designed. The source code of the system described is available at https://sourceforge.net/projects/ohnlp/files/MedCoref.

Impact of decreasing polynomial degree in time needed to factor a 100 digits integer by General Number field sieve algorithm

Impact of decreasing polynomial degree in time needed to factor a 100 digits integer by General Number field sieve algorithm

Fast and fully automatic phalanx segmentation using a grayscale-histogram morphology algorithm

Bone age assessment is a common radiological examination used in pediatrics to diagnose the discrepancy between the skeletal and chronological age of a child; therefore, it is beneficial to develop a computer-based bone age assessment to help junior pediatricians estimate bone age easily. Unfortunately, the phalanx on radiograms is not easily separated from the background and soft tissue. Therefore, we proposed a new method, called the grayscale-histogram morphology algorithm, to segment the phalanges fast and precisely. The algorithm includes three parts: a tri-stage sieve algorithm used to eliminate the background of hand radiograms, a centroid-edge dual scanning algorithm to frame the phalanx region, and finally a segmentation algorithm based on disk traverse-subtraction filter to segment the phalanx. Moreover, two more segmentation methods: adaptive two-mean and adaptive two-mean clustering were performed, and their results were compared with the segmentation algorithm based on disk traverse-subtraction filter using five indices comprising misclassification error, relative foreground area error, modified Hausdorff distances, edge mismatch, and region nonuniformity. In addition, the CPU time of the three segmentation methods was discussed. The result showed that our method had a better performance than the other two methods. Furthermore, satisfactory segmentation results were obtained with a low standard error.

On the use of the lattice sieve in the 3D NFS

ABSTRACT An adaptation of the Number Field Sieve (NFS) algorithm to solve a discrete logarithm problem in degree 6 finite fields (DLP6) requires a modified sieving procedure to find smooth elements of the three dimensional sieve space. In our successful solution [P. Zajac: Discrete Logarithms and Degree Six NumbereField Sieve: A practical Approach. VDM Verlag Dr. M¨uller, Saarbr¨ucken, 2009] we have used a modified line sieving to process a box-shaped region using a large factor base. In this contribution, we compare the results with an alternative approach based on the lattice sieving, which was used in most of the classical factorization and DLP record solutions. Results indicate that this approach does not scale to the 3D-case, making DLP6 more difficult in practice than comparable classical DLP cases.

The Mordell–Weil sieve: proving non-existence of rational points on curves

AbstractWe discuss the Mordell–Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod pinformation at primes of good reduction. We describe our implementation of the Mordell–Weil sieve algorithm and discuss its efficiency.Supplementary materials are available with this article.

An integrated parallel GNFS algorithm for integer factorization based on Linbox Montgomery block Lanczos method over GF(2)

Integer factorization is known to be one of the most important and useful methods in number theory and arithmetic. It also has a very close relationship to some algorithms in cryptography such as RSA algorithm. The RSA cryptosystem is one of the most popular and attractive public-key cryptosystems in the world today. Its security is based on the difficulty of integer factorization. Solving a large and sparse linear system over GF(2) is one of the most time consuming steps in most modern integer factorization algorithms including the fastest one, GNFS algorithm. The Montgomery block Lanczos method from Linbox [13] is for solving large and sparse linear systems over finite fields and it can be integrated into the general number field sieve (GNFS) algorithm which is the best known algorithm for factoring large integers over 110 digits. This paper will present an improved Montgomery block Lanczos method integrated with parallel GNFS algorithm. The experimental results show that the improved Montgomery block Lanczos method has a better performance compared with the original method. It can find more solutions or dependencies than the original method with less time complexities. Implementation details and experimental results are provided in this paper as well.

On the Impossibility of a Quantum Sieve Algorithm for Graph Isomorphism

It is known that any quantum algorithm for graph isomorphism that works within the framework of the hidden subgroup problem (HSP) must perform highly entangled measurements across $\Omega(n\log n)$ coset states. One of the only known models for how such a measurement could be carried out efficiently is Kuperberg's algorithm for the HSP in the dihedral group, in which quantum states are adaptively combined and measured according to the decomposition of tensor products into irreducible representations. This “quantum sieve” starts with coset states and works its way down toward representations whose probabilities differ depending on, for example, whether the hidden subgroup is trivial or nontrivial. In this paper we show that no such approach can produce a polynomial-time quantum algorithm for graph isomorphism. Specifically, we consider the natural reduction of graph isomorphism to the HSP over the wreath product $S_n\wr\mathbb{Z}_2$. Using a recently proved bound on the irreducible characters of $S_n$, we show that no algorithm in this family can solve graph isomorphism in less than $\mathrm{e}^{\Omega(\sqrt{n})}$ time, no matter what adaptive rule it uses to select and combine quantum states. In particular, algorithms of this type can offer essentially no improvement over the best known classical algorithms, which run in time $\mathrm{e}^{O(\sqrt{n\log n})}$.

Learning unions of [formula omitted]-dimensional rectangles

We consider the problem of learning unions of rectangles over the domain [ b ] n , in the uniform distribution membership query learning setting, where both b and n are “large”. We obtain poly ( n , log b ) -time algorithms for the following classes: • poly ( n log b ) -way Majority of O ( log ( n log b ) log log ( n log b ) ) -dimensional rectangles. • Union of poly ( log ( n log b ) ) many O ( log 2 ( n log b ) ( log log ( n log b ) log log log ( n log b ) ) 2 ) -dimensional rectangles. • poly ( n log b ) -way Majority of poly ( n log b ) - Or of disjoint O ( log ( n log b ) log log ( n log b ) ) dimensional rectangles. Our main algorithmic tool is an extension of Jackson’s boosting- and Fourier-based Harmonic Sieve algorithm [J.C. Jackson, An efficient membership-query algorithm for learning DNF with respect to the uniform distribution, Journal of Computer and System Sciences 55 (3) (1997) 414–440] to the domain [ b ] n , building on work of Akavia et al. [A. Akavia, S. Goldwasser, S. Safra, Proving hard core predicates using list decoding, in: Proc. of the 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS ’03, 2003, pp. 146–156]. Other ingredients used to obtain the results stated above are techniques from exact learning [A. Beimel, E. Kushilevitz, Learning boxes in high dimension, Algorithmica 22 (1/2) (1998) 76–90] and ideas from recent work on learning augmented AC 0 circuits [J.C. Jackson, A.R. Klivans, R.A. Servedio, Learnability beyond AC 0 , in: Proc. of the 34th Annual ACM Symposium on Theory of Computing, STOC ’02, 2002, pp. 776–784] and on representing Boolean functions as thresholds of parities [A.R. Klivans, R.A. Servedio, Learning DNF in time 2 O ̃ ( n 1 / 3 ) , Journal of Computer and System Sciences 68 (2) (2004) 303–318].

Sieve algorithms for the shortest vector problem are practical

The most famous lattice problem is the Shortest Vector Problem (SVP), which has many applications in cryptology. The best approximation algorithms known for SVP in high dimension rely on a subroutine for exact SVP in low dimension. In this paper, we assess the practicality of the best (theoretical) algorithm known for exact SVP in low dimension: the sieve algorithm proposed by Ajtai, Kumar and Sivakumar (AKS) in 2001. AKS is a randomized algorithm of time and space complexity 2 O ( n ) , which is theoretically much lower than the super-exponential complexity of all alternative SVP algorithms. Surprisingly, no implementation and no practical analysis of AKS has ever been reported. It was in fact widely believed that AKS was impractical: for instance, Schnorr claimed in 2003 that the constant hidden in the 2 O ( n ) complexity was at least 30. In this paper, we show that AKS can actually be made practical: we present a heuristic variant of AKS whose running time is polynomial-time operations, and whose space requirement is polynomially many bits. Our implementation can experimentally find shortest lattice vectors up to dimension 50, but is slower than classical alternative SVP algorithms in these dimensions.

The 1.5D sieve algorithm

The sieve is a morphological scale-space operator that filters an input signal by removing intensity extrema at a specific scale. In images, this processing can be carried out along a path – the 1D sieve – or over a connected graph – the 2D sieve. The 2D version of the sieve generally performs better; it is however much more complex to implement. In this paper we present the 1.5D sieve, a Hamiltonian path-based version of the sieve algorithm that behaves “in between” the 1D or 2D sieve algorithms, depending on the number of paths used. Experiments show that its robustness to the presence of noise and its performance in texture classification are similar to the original 2D sieve formulation, while being much faster and simpler to implement.

Unconditional lower bounds for learning intersections of halfspaces

We prove new lower bounds for learning intersections of halfspaces, one of the most important concept classes in computational learning theory. Our main result is that any statistical-query algorithm for learning the intersection of $\sqrt{n}$ halfspaces in n dimensions must make $2^{\varOmega (\sqrt{n})}$ queries. This is the first non-trivial lower bound on the statistical query dimension for this concept class (the previous best lower bound was n ?(log?n)). Our lower bound holds even for intersections of low-weight halfspaces. In the latter case, it is nearly tight. We also show that the intersection of two majorities (low-weight halfspaces) cannot be computed by a polynomial threshold function (PTF) with fewer than n ?(log?n/log?log?n) monomials. This is the first super-polynomial lower bound on the PTF length of this concept class, and is nearly optimal. For intersections of k=?(log?n) low-weight halfspaces, we improve our lower bound to $\min\{2^{\varOmega (\sqrt{n})},n^{\varOmega (k/\log k)}\},$ which too is nearly optimal. As a consequence, intersections of even two halfspaces are not computable by polynomial-weight PTFs, the most expressive class of functions known to be efficiently learnable via Jackson's Harmonic Sieve algorithm. Finally, we report our progress on the weak learnability of intersections of halfspaces under the uniform distribution.

New evidence for the saturation of the Froissart bound

Fits to high energy data alone cannot cleanly discriminate between asymptotic $\mathrm{ln}s$ and ${ln}^{2}s$ behavior of total hadronic cross sections. We demonstrate that this is no longer true when we require that these amplitudes also describe, on average, low energy data dominated by resonances. We simultaneously fit real analytic amplitudes to high energy measurements of: (i) the ${\ensuremath{\pi}}^{+}p$ and ${\ensuremath{\pi}}^{\ensuremath{-}}p$ total cross sections and $\ensuremath{\rho}$-values (ratio of the real to the imaginary portion of the forward scattering amplitude), for $\sqrt{s}\ensuremath{\ge}6$ GeV, while requiring that the asymptotic fits smoothly join the ${\ensuremath{\sigma}}_{{\ensuremath{\pi}}^{+}p}$ and ${\ensuremath{\sigma}}_{{\ensuremath{\pi}}^{\ensuremath{-}}p}$ total cross sections at $\sqrt{s}=2.6$ GeV---both in magnitude and slope , and (ii) separately simultaneously fit the $\overline{p}p$ and $pp$ total cross sections and $\ensuremath{\rho}$-values for $\sqrt{s}\ensuremath{\ge}6$ GeV, while requiring that their asymptotic fits smoothly join the the ${\ensuremath{\sigma}}_{\overline{p}p}$ and ${\ensuremath{\sigma}}_{pp}$ total cross sections at $\sqrt{s}=4.0$ GeV---again both in magnitude and slope. In both cases, we have used all of the extensive data of the Particle Data Group [K. Hagiwara et al. (Particle Data Group), Phys. Rev. D 66, 010001 (2002).]. However, we then subject these data to a screening process, the Sieve algorithm [M. M. Block, physics/0506010.], in order to eliminate outliers that can skew a ${\ensuremath{\chi}}^{2}$ fit. With the Sieve algorithm, a robust fit using a Lorentzian distribution is first made to all of the data to sieve out abnormally high $\ensuremath{\Delta}{\ensuremath{\chi}}_{i}^{2}$, the individual i$\mathrm{th}$ point's contribution to the total ${\ensuremath{\chi}}^{2}$. The ${\ensuremath{\chi}}^{2}$ fits are then made to the sieved data. Both the $\ensuremath{\pi}p$ and nucleon-nucleon systems strongly favor a high energy ${ln}^{2}s$ fit of the form: ${\ensuremath{\sigma}}^{\ifmmode\pm\else\textpm\fi{}}={c}_{0}+{c}_{1}\mathrm{ln}(\frac{\ensuremath{\nu}}{m})+{c}_{2}{ln}^{2}(\frac{\ensuremath{\nu}}{m})+{\ensuremath{\beta}}_{{\mathcal{P}}^{\ensuremath{'}}}(\frac{\ensuremath{\nu}}{m}{)}^{\ensuremath{\mu}\ensuremath{-}1}\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}(\frac{\ensuremath{\nu}}{m}{)}^{\ensuremath{\alpha}\ensuremath{-}1}$, basically excluding a $\mathrm{ln}s$ fit of the form: ${\ensuremath{\sigma}}^{\ifmmode\pm\else\textpm\fi{}}={c}_{0}+{c}_{1}\mathrm{ln}(\frac{\ensuremath{\nu}}{m})+{\ensuremath{\beta}}_{{\mathcal{P}}^{\ensuremath{'}}}(\frac{\ensuremath{\nu}}{m}{)}^{\ensuremath{\mu}\ensuremath{-}1}\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}(\frac{\ensuremath{\nu}}{m}{)}^{\ensuremath{\alpha}\ensuremath{-}1}$. The upper sign is for ${\ensuremath{\pi}}^{+}p$ ($pp$) and the lower sign is for ${\ensuremath{\pi}}^{\ensuremath{-}}p$ ($\overline{p}p$) scattering, where $\ensuremath{\nu}$ is the laboratory pion (proton) energy, and $m$ is the pion (proton) mass.

Boosting and Hard-Core Set Construction

This paper connects hard-core set construction, a type of hardness amplification from computational complexity, and boosting, a technique from computational learning theory. Using this connection we give fruitful applications of complexity-theoretic techniques to learning theory and vice versa. We show that the hard-core set construction of Impagliazzo (1995), which establishes the existence of distributions under which boolean functions are highly inapproximable, may be viewed as a boosting algorithm. Using alternate boosting methods we give an improved bound for hard-core set construction which matches known lower bounds from boosting and thus is optimal within this class of techniques. We then show how to apply techniques from Impagliazzo (1995) to give a new version of Jackson's celebrated Harmonic Sieve algorithm for learning DNF formulae under the uniform distribution using membership queries. Our new version has a significant asymptotic improvement in running time. Critical to our arguments is a careful analysis of the distributions which are employed in both boosting and hard-core set constructions.

Computational Micrograph Registration with Sieve Processes

In this paper we present an efficient algorithm for automatically finding the correspondence between pairs of stereo micrographs, the key step in forming a stereo image. The computation burden in this problem is solving for the optimal mapping and transformation between the two micrographs. In this paper, we present a sieve algorithm for efficiently estimating the transformation and correspondence.In a sieve algorithm, a sequence of stages gradually reduce the number of transformations and correspondences that need to be examined, i.e., the analogy of sieving through the set of mappings with gradually finer meshes until the answer is found. The set of sieves is derived from an image model, here a planar graph that encodes the spatial organization of the features. In the sieve algorithm, the graph represents the spatial arrangement of objects in the image. The algorithm for finding the correspondence restricts its attention to the graph, with the correspondence being found by a combination of graph matchings, point set matching and geometric invariants.

A Simple Randomized Sieve Algorithm for the Closest-Pair Problem

We present a linear time randomized sieve algorithm for the closest-pair problem. The algorithm and its analysis are simple. The algorithm is extended to obtain a randomized linear time approximation algorithm for the closest bichromatic pair problem.