Signature-based Algorithms Research Articles

Signature-based standard basis algorithm under the framework of GVW algorithm

The GVW algorithm, one of the most important so-called signature-based algorithms, is designed to eliminate a large number of useless polynomial reductions from Buchberger's algorithm. The cover theorem serves as the theoretical foundation of the GVW algorithm, and up to now, it applies only to a certain class of monomial orders, namely global orders and a special class of local orders. In this paper we extend this theorem to any semigroup order, which can be either global, local or even mixed. Building upon the pioneering idea of the Mora normal form algorithm, we propose a more comprehensive and general proof for the cover theorem while bypassing the need to choose a minimal element from an infinite set of monomials in all the existing proofs. Therefore, the algorithm for signature-based standard bases is presented for any semigroup order under the framework of the GVW algorithm, and an example is given to provide an illustration of the algorithm.

Abstract B114: Effect of spatial and temporal inter-tumoral heterogeneity on homologous recombination deficiency scores in high grade serous ovarian cancer

Abstract High-grade serous ovarian cancer (HGSC) is the most common and lethal subtype of ovarian cancer. Extensive genomic instability and heterogeneity are characteristics of HGSC, with near-ubiquitous TP53 loss-of-function mutations, defects in homologous recombination (HR) repair, and extensive copy number aberrations contributing to the vast genomic heterogeneity observed. Most patients relapse and acquire resistance to platinum or PARP inhibitor (PARPi)-based therapy. Diverse mechanisms leading to therapy resistance and lack of predictive biomarkers means that matching the best treatment options to patients is difficult. Testing tumors for HR-Deficiency (HRD) status is now a clinical test to predict PARPi sensitivity in HGSC. We aimed to determine the influence of spatial and temporal heterogeneity on HRD status in HGSC. Patients (n=59) at Hammersmith Hospital (HH) underwent maximal-effort upfront cytoreduction for advanced HGSC, had their tumor dissemination mapped and tumor biopsies collected (range 4-15). Paired relapse samples were collected for 10 patients. Tumor DNA was extracted (n=5 tumors per case, plus relapse), Illumina Human OmniExpress genotyping performed for 59 cases and whole genome sequencing (WGS; 30x depth) performed for 21 cases. Allele-specific copy number (CN) was quantified using ASCAT. Multi-site CN data was also accessed from two cohorts (GSE38787, GSE40546). Tumors that failed QC or had an aberrant cell fraction <0.3 were excluded. HR scores were estimated in each cohort using different genomic instability score (GIS) or mutational signature-based algorithms and relevant cut-offs were applied to determine HRD and HR-Proficient (HRP) tumors. Variations in HR scores were observed across the cohorts with a proportion of patients presenting with a mixed HR score, displaying both HRD and HRP scores in their tumors: HH cohort (20%, 11/54 patients); GSE38787 (17%, 4/24 patients); and GSE40546 (28%, 4/14 patients) using a scarHRD algorithm. In the HH cohort (n=45), we examined if mixed HR status was associated with survival and demonstrated that patients with HR-mixed or all HRP tumors had poorer progression-free (p=0.0052) and overall survival (p=0.00092) than patients with all HRD tumors. Applying different approaches to the HH WGS data (scarHRD, CHORD, >63 score cut-off), approximately 20% of patients demonstrated HR-mixed scores in their tumors in each approach. However, depending on the approach chosen to define HRD, different cases showed HR-mixed scores, 2/21 mixed cases overlapped between the different algorithms applied with tumors changing from HRD to HRP and vice versa depending on the approach applied. Differences in HR-deficiency/proficiency scores suggests that HR status may not be consistent across advanced disseminated HGSC, indicating that a single tumor biopsy may not accurately depict spatial HGSC tumor biology. HR score variations in different methods (GIS or mutational signatures) indicates that these tests are not analogous for defining HRD and thus PARPi suitability. Citation Format: Elizabeth L. Christie, Marc Lorentzen, Maiqi Liu, Nikki Burdett, Edward Curry, Chun Hei Kwok, Ahwan Pandey, Katherine Nixon, Jennifer Ploski, James J. Clark, Iain McNeish, David D.L. Bowtell, Jonathan Krell, Christina Fotopoulou, Paula Cunnea. Effect of spatial and temporal inter-tumoral heterogeneity on homologous recombination deficiency scores in high grade serous ovarian cancer [abstract]. In: Proceedings of the AACR Special Conference on Ovarian Cancer; 2023 Oct 5-7; Boston, Massachusetts. Philadelphia (PA): AACR; Cancer Res 2024;84(5 Suppl_2):Abstract nr B114.

Block Chain for IOT Security Using Consensus Algorithms

The first distributed recordkeeping system with a built-in trust structure is the block chain. It creates a dependable architecture for decentralized control through information redundancy across multiple nodes. Based on this, this study suggests a minimal block chain-based IoT information exchange security framework. The framework uses a double-chain approach that combines the data block chain and the transaction block chain. Distributed storage and tamper-proof data are implemented in the data block chain, and the consensus process is improved using the improved practical Byzantine fault-tolerant (PBFT) mechanism. Data registration efficiency, resource and data transfers, and privacy protection are all enhanced by better partial blind signature-based algorithms in the transaction block chain. This article focuses on how well the consensus algorithms employed in a block chain system for the Internet of Things perform (IoT). Such systems' time requirements to accomplish. Consensus ought to be minimal. The three most popular consensus algorithms—modified proof of work, realistic byzantine fault tolerance, and binary consensus—are assessed under various conditions, including mote type, number of participating nodes, and radio propagation model. To enable an IoT node to switch between different consensus algorithms, a comprehensive solution is put forward. The Contiki IoT operating system simulations display strong performance (time to achieve consensus less than seconds)

Signature Gröbner bases, bases of syzygies and cofactor reconstruction in the free algebra

Signature-based algorithms have become a standard approach for computing Gröbner bases in commutative polynomial rings. However, so far, it was not clear how to extend this concept to the setting of noncommutative polynomials in the free algebra. In this paper, we present a signature-based algorithm for computing Gröbner bases in precisely this setting. The algorithm is an adaptation of Buchberger's algorithm including signatures. We prove that our algorithm correctly enumerates a signature Gröbner basis as well as a Gröbner basis of the module generated by the leading terms of the generators' syzygies, and that it terminates whenever the ideal admits a finite signature Gröbner basis. Additionally, we adapt well-known signature-based criteria eliminating redundant reductions, such as the syzygy criterion, the F5 criterion and the singular criterion, to the case of noncommutative polynomials. We also generalise reconstruction methods from the commutative setting that allow to recover, from partial information about signatures, the coordinates of elements of a Gröbner basis in terms of the input polynomials, as well as a basis of the syzygy module of the generators. We have written a toy implementation of all the algorithms in the Mathematica package OperatorGB and we compare our signature-based algorithm to the classical Buchberger algorithm for noncommutative polynomials.

A generic and executable formalization of signature-based Gröbner basis algorithms

We present a generic and executable formalization of signature-based algorithms (such as Faugère's F5) for computing Gröbner bases, as well as their mathematical background, in the Isabelle/HOL proof assistant. Said algorithms are currently the best known algorithms for computing Gröbner bases in terms of computational efficiency. The formal development attempts to be as generic as possible, generalizing most known variants of signature-based algorithms, but at the same time the implemented functions are effectively executable on concrete input for efficiently computing mechanically verified Gröbner bases. Besides correctness the formalization also proves that under certain conditions the algorithms a-priori detect and avoid all useless reductions to zero, and return minimal signature Gröbner bases.To the best of our knowledge, the formalization presented here is the only formalization of signature-based Gröbner basis algorithms in existence so far.

An efficient reduction strategy for signature-based algorithms to compute Gröbner basis

This paper introduces a strategy for signature-based algorithms to compute Gröbner basis. In comparison with Buchberger algorithm, signature-based algorithms generate S-pairs instead of S-polynomials, and operate s -reductions instead of usual reductions which is used in Buchberger algorithm. There are two strategies for s -reductions: one is only-top reduction strategy which is the way that only leading monomials are s -reduced. The other is full reduction strategy which is the way that all monomials are s -reduced. New strategy for s -reduction of S-pairs, which we call selective-full strategy, is introduced in this paper. The idea of the strategy is that when we have computed a signature Gröbner basis, there are unnecessary elements for a minimal Gröbner basis, so candidates of elements included in a minimal Gröbner basis should be sufficiently reduced. According to the experimental result, proposed strategy is efficient for computing the reduced Gröbner basis. For computing a signature Gröbner basis, it's the most efficient or not the worst of the three strategies.

A Signature-Based Algorithm for Computing Gr\xf6bner Bases over Principal Ideal Domains

Signature-based algorithms have become a standard approach for Gröbner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a proof-of-concept signature-based algorithm for computing Gröbner bases over commutative integral domains. It is adapted from a general version of Möller’s algorithm (J Symb Comput 6(2–3), 345–359, 1988) which considers reductions by multiple polynomials at each step. This algorithm performs reductions with non-decreasing signatures, and in particular, signature drops do not occur. When the coefficients are from a principal ideal domain (e.g. the ring of integers or the ring of univariate polynomials over a field), we prove correctness and termination of the algorithm, and we show how to use signature properties to implement classic signature-based criteria to eliminate some redundant reductions. In particular, if the input is a regular sequence, the algorithm operates without any reduction to 0. We have written a toy implementation of the algorithm in Magma. Early experimental results suggest that the algorithm might even be correct and terminate in a more general setting, for polynomials over a unique factorization domain (e.g. the ring of multivariate polynomials over a field or a PID).

Speeding Up the GVW Algorithm via a Substituting Method

The GVW algorithm is an efficient signature-based algorithm for computing Grobner bases. In this paper, the authors consider the implementation of the GVW algorithm by using linear algebra, and speed up GVW via a substituting method. As it is well known that, most of the computing time of a Grobner basis is spent on reductions of polynomials. Thus, linear algebraic techniques, such as matrix operations, have been used extensively to speed up the implementations. Particularly, one-direction (also called signature-safe) reduction is used in signature-based algorithms, because polynomials (or rows in matrices) with larger signatures can only be reduced by polynomials (rows) with smaller signatures. The authors propose a new method to construct sparser matrices for signature-based algorithms via a substituting method. Specifically, instead of only storing the original polynomials in GVW, the authors also record many equivalent but sparser polynomials at the same time. In matrix construction, denser polynomials are substituted by sparser equivalent ones. As the matrices get sparser, they can be eliminated more efficiently. Two specifical algorithms, Block-GVW and LMGVW, are presented, and their combination is the Sub-GVW algorithm. The correctness of the new proposed method is proved, and the experimental results demonstrate the efficiency of this new method.

A survey on signature-based algorithms for computing Gröbner bases

In 1965 Buchberger introduced an algorithmic approach to compute Gröbner bases. Later on, he and many others presented various attempts to improve the computation by removing useless elements a priori. One approach, initiated by Gebauer, Möller, Mora and Traverso in the 1990s, is to keep track of the corresponding syzygies which is related to the topic of this survey: signature-based algorithms for Gröbner bases. This area was initiated by Faugère's F5 algorithm in 2002. The general idea of signatures is to keep track of the history of the computation with a minimal overhead and to exploit this information to detect redundant elements. Here we give a summary of the literature on signature-based algorithms and show how to classify known algorithms by 3 different orderings. For this we give translations between different notations and show the relationships (differences and similarities) among many approaches. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. We hope that this survey would help to outline this field of active research.

Minimal contrast frequent pattern mining for malware detection

Currently, most of widely-used malware detection software products utilize signature-based algorithms to recognize threats. However, this approach is problematic because it relies on file hashes and byte (or instruction) signatures. Consequently, obfuscation techniques are straightforward ways to modify these features syntactically and evade detection. Since it is harder for an attacker to radically change the behavior of a malware than to morph its syntactic structure, behavior-based detection techniques are a promising solution to this problem. However behavior-based techniques can be applied using static analysis or dynamic analysis or hybrid analysis. While dynamic behavior-based detection methods are time consuming and fail to obtain all possible malicious execution traces, most of static behavior-based approaches suffer from a high growth rate in the number of behavioral signatures and suffer from high false positive rates. In this paper, we present a new graph mining method to detect variants of malware using static analysis, while covering the existing defects. We propose a novel algorithm, called minimal contrast frequent subgraph miner algorithm (MCFSM), for extracting minimal discriminative and widely employed malicious behavioral patterns which can identify precisely an entire family of malicious programs, in contrast to another set of benign programs. The proposed method shows high detection rates and low false positive rates and generates a limited number of behavioral malware signatures.

On implementing signature-based Gröbner basis algorithms using linear algebraic routines from M4RI

On implementing signature-based Gröbner basis algorithms using linear algebraic routines from M4RI

A new signature-based algorithms for computing Gröbner bases

Gao, Volny and Wang (2010) gave a simple criterion for signature-based algorithms to compute Grobner bases. It gives a unified frame work for computing Grobner bases for both ideals and syzygies, the latter is very important in free resolutions in homological algebra. Sun and Wang (2011) later generalized the GVW criterion to a more general situation (to include the F5 Algorithm). Signature-based algorithms have become increasingly popular for computing Grobner bases. The current paper introduces a concept of factor pairs that can be used to detect more useless J-pairs than the generalized GVW criterion, thus improving signature-based algorithms.

An analysis of inhomogeneous signature-based Gröbner basis computations

In this paper we give an insight into the behavior of signature-based Gröbner basis algorithms, like F5, G2V or SB, for inhomogeneous input. On the one hand, it seems that the restriction to sig-safe reductions puts a penalty on the performance. The lost connection between polynomial degree and signature degree can disallow lots of reductions and can lead to an overhead in the computations. On the other hand, the way critical pairs are sorted and corresponding s-polynomials are handled in signature-based algorithms is a very efficient one, strongly connected to sorting w.r.t. the well-known sugar degree of polynomials.

Improving incremental signature-based Gröbner basis algorithms

In this paper we describe a combination of ideas to improve incremental signature-based Gröbner basis algorithms which have a big impact on their performance. Besides explaining how to combine already known optimizations to achieve more efficient algorithms, we show how to improve them even further. Although our idea has a postive effect on all kinds of incremental signature-based algorithms, the way this impact is achieved can be quite different. Based on the two best-known algorithms in this area, F5 and G2V, we explain our idea, both from a theoretical and a practical point of view.

Safeguarding information intensive critical infrastructures against novel types of emerging failures

The complexity of information intensive critical infrastructures, like electricity networks, telecommunication networks and public transportation networks is today augmented much more than in the past: such complexity augments the number of possible failures and anomalous working conditions and consequently decreases the survivability of the infrastructures. In this paper, the possibility is investigated to detect early anomalies and failures inside information intensive critical infrastructures by the introduction of anomaly detectors being “self-aware” about the normal working conditions of the infrastructure itself. This approach has the objective to improve the performance of the most popular signature-based algorithms for intrusion detection, and makes use of different classes of time-oriented algorithms based on artificial intelligence paradigm. It has the advantage to work also in presence of unknown and unexpected types of attacks or failures. The tests, to evaluate the performance of the utilised detectors, are executed inside an emulated supervisory control and data acquisition (SCADA) system of an electrical power transmission grid, and a proposal for the future integration inside real SCADA systems is also reported.

Experimental study of mutual coupling compensation in smart antenna applications

This paper investigates the benefit of mutual coupling compensation via a method of moments (MoM) approach in a uniform circular antenna array operating at 1.8 GHz. This mutual coupling compensation technique is applied to a direction of arrival (DOA) study of up to two cochannel mobile users. Field measurements and computer simulations are examined to explore the assumptions of the technique and verify its effect when using the Bartlett and MUSIC DOA algorithms. Computer simulations considering the application of the technique to down-link beamforming are also included. Experimental results show that the mutual coupling compensation technique improves up-link DOA algorithm performance primarily by reducing unwanted sidelobe levels. This reduction in sidelobe levels aids in down-link beamforming weight design. Specifically, simulation results show that use of the compensation technique allows DOA-based down-link beamforming algorithms to perform similarly to spatial signature-based algorithms. All field measurements were made using the smart antenna testbed at the University of Texas at Austin.