Algebraic Analysis Research Articles

Double laser-faults based PFA on cryptographic circuits with algebraic analysis

Cryptographic algorithms have been employed in a variety of fields as the primary method to protect information security. The security of a cryptographic algorithm is closely related to its operating environment and physical devices. Algebraic Persistent Fault Analysis (APFA) is a new fault analysis method for block ciphers proposed in CHES 2022, which utilizes the fault that persists in encryptions and introduces algebraic analysis in the fault analysis step. In the fault injection step, as the transistors of the integrated circuit are getting smaller and tighter, even high-precision devices may cause more than one fault per injection. However, more faults may lead to a more efficient attack in the fault analysis step. In this paper, APFA for double faults is proposed, which can deal with the double faults model and reduce the number of required ciphertexts. The practicality of our fault injection is validated by laser fault injection experiments on the SRAM embedded in an ATmega163L microcontroller. The effectiveness of our fault analysis is proven by successfully recovering the key of PRESENT-128 and AES-128. The number of ciphertexts needed for key recovery is reduced by 46% compared to PFA with a single fault.

Chemo-рroteomic analysis of the pharmacological properties of vitamin В12 derivatives

Background. Chemical derivatives of vitamin B12 are characterized by a wide range of pharmacological effects. It is important to learn how to establish relationships between changes of the corrin ring structure in vitamin B12 derivatives and changes in pharmacological properties.Objective: to evaluate the interaction of six vitamin B12 derivatives (aquacobalamin, diaquacobinamide, aquacyano-forms of heptaethanolamine, heptaethylenediamine, heptamethyl and heptabutyl cobyrinates) with human proteome proteins.Material and methods. Using the method of chemoinformational (chemoproteomic) analysis, implemented within the framework of algebraic recognition theory and topological data analysis, the constants of half maximal inhibitory concentration (IC50) and half maximal effective concentration (EC50) of human proteome proteins were assessed.Results. Significant differences were found in the interactions of the studied molecules with 1200 proteins. It was shown that the chemoproteomic profiles of each of the compounds form three groups of molecules with similar proteomic properties: (1) aquacobalamin, (2) diaquacobinamide, aquacyano-forms of heptaethanolamine and heptaethylenediamine cobyrinates, (3) aquacyano-forms of heptamethyl and heptabutyl cobyrinates. A more detailed analysis of the chemoproteomic profiles of the studied compounds using the GO (Gene Ontology) nomenclature of biological functions of proteins made it possible to identify functional GO categories indicating differences in the biological effects of the studied compounds: neuroprotective regulation of neurotransmitter activity (serotonin receptor activity, cholinergic synapses, regulation of dopamine secretion, receptor thyroid hormones), reduction of inflammation (inhibition of cytokine biosynthesis, including tumor necrosis factor alpha and interleukin 1 beta, I-kappa-B kinases / nuclear factor kappa В, leukocyte migration), etc.Conclusion. Based on the obtained data conclusions were drawn about the potential effects and safety of the studied substances.

Algebraic surrogate-based flexibility analysis of process units with complicating process constraints

Flexibility analyses are widespread in chemical engineering to quantify allowed deviations from nominal conditions. Standard approaches to perform flexibility analysis can be hard to apply if process constraints are difficult to handle, as it happens in bioprocesses with dynamic constraints. Here, focusing on the computation of the traditional flexibility index in problems with complicating constraints, we apply symbolic regression to build algebraic expressions of the said complicating constraints, simplifying the flexibility analysis of complex process models by enabling the application of state-of-the-art deterministic solvers. Our approach is applied to ethanol production in fed-batch operation mode and a chromatographic process. The performance is assessed in terms of model building time, predictive accuracy of the model, and the time required to solve the flexibility formulations. Overall, our approach, which focuses on computing the original flexibility index proposed in the literature, provides an alternative way to analyse the flexibility of processes entailing complicating constraints.

Verification and identification of the speech signal using algebraic analysis and DWT

We presented a flowchart, of a new approach that relies on the mathematical operations of matrices and to verify and identifies the speaker, this is a new method difficult to detect by attackers, and this is due to the difficulty to determine its base matrix used in the speaker verification and identification process, Although there are many artificial intelligence programs that can imitate voices and generating false conversations, which poses security and hacking problems.

Study of algebra integration and analysis in teaching mathematics and its applications

Study of algebra integration and analysis in teaching mathematics and its applications

Algebraic spectral analysis of the locating degree matrix of a graph

In this article, we present a new graph matrix based on locating degree, we call it the locating degree matrix. The spectrum and energy related to this new matrix are established for certain classes of graphs. Bounds of the new related energy for a special type of regular graph are obtained.

Adversarially Trained Persistent Homology Based Graph Convolutional Network for Disease Identification Using Brain Connectivity.

Brain disease propagation is associated with characteristic alterations in the structural and functional connectivity networks of the brain. To identify disease-specific network representations, graph convolutional networks (GCNs) have been used because of their powerful graph embedding ability to characterize the non-Euclidean structure of brain networks. However, existing GCNs generally focus on learning the discriminative region of interest (ROI) features, often ignoring important topological information that enables the integration of connectome patterns of brain activity. In addition, most methods fail to consider the vulnerability of GCNs to perturbations in network properties of the brain, which considerably degrades the reliability of diagnosis results. In this study, we propose an adversarially trained persistent homology-based graph convolutional network (ATPGCN) to capture disease-specific brain connectome patterns and classify brain diseases. First, the brain functional/structural connectivity is constructed using different neuroimaging modalities. Then, we develop a novel strategy that concatenates the persistent homology features from a brain algebraic topology analysis with readout features of the global pooling layer of a GCN model to collaboratively learn the individual-level representation. Finally, we simulate the adversarial perturbations by targeting the risk ROIs from clinical prior, and incorporate them into a training loop to evaluate the robustness of the model. The experimental results on three independent datasets demonstrate that ATPGCN outperforms existing classification methods in disease identification and is robust to minor perturbations in network architecture. Our code is available at https://github.com/CYB08/ATPGCN.

Understanding Episode Hardness in Few-Shot Learning.

Achieving generalization for deep learning models has usually suffered from the bottleneck of annotated sample scarcity. As a common way of tackling this issue, few-shot learning focuses on "episodes", i.e. sampled tasks that help the model acquire generalizable knowledge onto unseen categories - better the episodes, the higher a model's generalisability. Despite extensive research, the characteristics of episodes and their potential effects are relatively less explored. A recent paper discussed that different episodes exhibit different prediction difficulties, and coined a new metric "hardness" to quantify episodes, which however is too wide-range for an arbitrary dataset and thus remains impractical for realistic applications. In this paper therefore, we for the first time conduct an algebraic analysis of the critical factors influencing episode hardness supported by experimental demonstrations, that reveal episode hardness to largely depend on classes within an episode, and importantly propose an efficient pre-sampling hardness assessment technique named Inverse-Fisher Discriminant Ratio (IFDR). This enables sampling hard episodes at the class level via class-level (cl) sampling scheme that drastically decreases quantification cost. Delving deeper, we also develop a variant called class-pair-level (cpl) sampling, which further reduces the sampling cost while guaranteeing the sampled distribution. Finally, comprehensive experiments conducted on benchmark datasets verify the efficacy of our proposed method. Codes are available at: https://github.com/PRIS-CV/class-level-sampling.

The Archimedean Origin of Modern Positional Number Systems

A symbolic analysis of Archimedes’s periodical number system is developed, from which a natural link emerges with the modern positional number systems with zero. After the publication of Fibonacci’s Liber Abaci, the decimal Indo-Arabic positional system was the basis of the algorithmic and algebraic trend of modern mathematics, but even if zero plays a crucial role in algebra and mathematical analysis, zeroless positional systems show the same capability of producing efficient arithmetical algorithms based on operation tables over digits. The crucial role of digits is assessed, by considering a representation of numbers based on strings in lexicographic order. A new algorithm for the determination of decimal periods is presented by remarking on the cruciality of this topic in number theory. Periods of ordinal numbers and enumerations of recursive enumerability are shortly recalled. Concluding remarks are formulated about the deep relationship between numbers and information, which shed new light on a red line passing through the whole history of mathematics.

Applications of the Spectral Theorem: Utilizing Eigenvalues and Eigenvectors

This paper explores the applications and principles of both the real and complex spectral theorems, cornerstones of linear algebra and functional analysis. We begin with an overview of the spectral theorem and delve into the pivotal roles of eigenvalues and eigenvectors, culminating in a detailed proof of the theorem. A discerning comparison is made between the real and complex versions; notably, the latter seamlessly integrates with the fundamental theorem of algebra, an attribute absent in the former. The ensuing sections illuminate practical applications. The real spectral theorem emerges as instrumental in multivariable calculus, notably in the second derivative test, in data-driven techniques such as Principal Component Analysis (PCA), and in electrical network analyses via Kirchhoff’s matrix. The complex variant takes center stage in quantum mechanics, illuminating the Schrödinger Equation. This paper underscores the spectral theorem’s profound relevance in diverse theoretical and practical arenas.

Algebraic And Geometric Properties Of A Family of Rational Curves

This paper consists of two components - an application part and a theoretical part, where the former targets the applications of computer aided geometric designs in generating parametric curves, and the latter focuses on the algebraic analysis of rational space curves. At the application level, we construct a family of rational space curves via quaternion products of two generating curves. At the theoretical level, we use algebraic methods to extract a $\mu$-basis for this family of curves, and describe a basis for a special submodule of the syzygy module in terms of a $\mu$-basis for the syzygy module of this family of curves. A commutative diagram is provided to summarize these results.

Digitisation of Classical Exercise Practices with STACK: Management of Large Mathematics Classes in Higher Education Institutions

Management of large classes’ tutorials is a known problem in Mathematics. Engineering mathematics classes enrol around 1000 students at the University of Dar es Salaam. For effective learning, each tutorial session should register not more than 40 students. This requires at least 25 sessions of tutorials per week, which may not be feasible due to both scarce human resources and venues. In this work, we developed online Mathematics exercises based on the System for Teaching and Assessment using Computer Algebra Kernel (STACK). About 300 STACK questions in linear algebra, calculus, complex numbers, and numerical analysis were constructed for weekly tutorials and quizzes. Students were given an unlimited number of attempts in tutorials and only one attempt for quizzes. The quality of the questions was analysed by examining their facility indices and discriminative efficiencies. Majority of the questions (87%) were within acceptable region. The questions, therefore, provided reasonable insight as appropriate alternative to classical practice. Competent authoring of STACK questions can improve the quality of teaching and learning of Mathematics and save scarce human and material resources required to serve large classes. This can also address the issue of running online programmes in Mathematics and computational subjects to support distance learning.

Algebraic model analysis of shape phase transitions in odd-odd nuclei

Algebraic model analysis of shape phase transitions in odd-odd nuclei

Construction of algebraic complex 9-bit lookup tables using non-chain-ring and its applications in data security

A substitution box (S-box) is the only nonlinear component of a symmetric key encryption cipher that directly affects an encryption scheme’s performance and security level. So it is appealing to produce S-boxes with good performance and efficiency. Customarily, in algebra, Galois fields are used to generate such structures. It requires the storage memory of size 8×28 bits. Similarly, in GF(29) this figure reaches to 9×29 bits. In this study, a novel construction of lookup tables over the algebraic structure of non-chain ring R=Z8[u]<u3>=Z8+uZ8+u2Z8,(u3=0) of cardinality 512 is presented that holds 9×28 computer memory calls. Moreover, the use of non-chain ring operations in the building of lookup tables results in a more complex S-box. Afterward, we use this nonlinear layer to produce confusion in a novel image-enciphering scheme. A nonlinear encryption scheme such as a non-chain-ring S-box construction is susceptible to adversaries because of its complex connections between key ciphertexts and cryptographic primitives. With non-chain ring encryption, you can benefit from enhanced security, complexity of algorithm, resistance to modern attacks, as well as improved computing performance. Texture, algebraic, and statistical analyses were performed on the suggested S-box encryption technique. A comparison of the encryption data using the proposed S-box with the literature available S-boxes demonstrates that the suggested S-box is superior and can be employed in various ciphers. The results of the statistical, differential, and security analyses also reveal that our approach is more effective for image encryption.

Adjoint and Direct Characteristic Equations for Two-Dimensional Compressible Euler Flows

The method of characteristics is a classical method for gaining understanding in the solution of a partial differential equation. It has recently been applied to the adjoint equations of the 2D steady-state Euler equations and the first goal of this paper is to present a linear algebra analysis that greatly simplifies the discussion of the number of independent characteristic equations satisfied along a family of characteristic curves. This method may be applied for both the direct and the adjoint problem. Our second goal is to directly derive in conservative variables the characteristic equations of 2D compressible inviscid flows. Finally, the theoretical results are assessed for a nozzle flow with a classical scheme and its dual consistent discrete adjoint.

Positive solution for a nonlocal problem with strong singular nonlinearity

Abstract In this article, we consider a nonlocal problem with a strong singular term and a general weight function. By using Ekeland’s variational principle, we prove a necessary and sufficient condition for the existence of a positive solution. Moreover, a method of algebraic analysis is used to deal with the multiplicity of solutions. Compared with the existing literature, our problems and results are novel.

Tutorials in climate modeling

This article describes a learner-centered, tutorial-based approach to climate modeling. Using these tutorials, students in a general education course in global climate change taught in the physics department engaged with compelling, “real-world” questions by drawing on foundational physics concepts along with their prior experiences and conceptual resources in the process of knowledge construction. Students used simulations, energy tracking diagrams, and algebraic analysis to construct climate models that address challenging quantitative questions and reveal key concepts in climate science. Special emphasis was given to the learning goals, pre-requisite knowledge, and conceptual challenges associated with these activities. Instructional materials are provided for educators who wish to adapt these tutorials for use in their own educational context. The supplementary materials are appropriate for both non-science majors and upper-level physics majors.

Revisiting the Computation Analysis against Internal Encodings in White-Box Implementations

White-box implementations aim to prevent the key extraction of the cryptographic algorithm even if the attacker has full access to the execution environment. To obfuscate the round functions, Chow et al. proposed a pivotal principle of white-box implementations to convert the round functions as look-up tables which are encoded by random internal encodings. These encodings consist of a linear mapping and a non-linear nibble permutation. At CHES 2016, Bos et al. introduced differential computation analysis (DCA) to extract the secret key from the runtime information, such as accessed memory and registers. Following this attack, many computation analysis methods were proposed to break the white-box implementations by leveraging some properties of the linear internal encodings, such as Hamming weight and imbalance. Therefore, it becomes an alternative choice to use a non-linear byte encoding to thwart DCA. At CHES 2021, Carlet et al. proposed a structural attack and revealed the weakness of the non-linear byte encodings which are combined with a non-invertible linear mapping. However, such a structural attack requires the details of the implementation, which relies on extra reverse engineering efforts in practice. To the best of our knowledge, it still lacks a thorough investigation of whether the non-linear byte encodings can resist the computation analyses.In this paper, we revisit the proposed computation analyses by investigating their capabilities against internal encodings with different algebraic degrees. Particularly, the algebraic degree of encodings is leveraged to explain the key leakage on the non-linear encodings. Based on this observation, we propose a new algebraic degree computation analysis (ADCA), which targets the mappings from the inputs to each sample of the computation traces. Different from the previous computation analyses, ADCA is a higher-degree attack that can distinguish the correct key by matching the algebraic degrees of the mappings. The experimental results prove that ADCA can break the internal encodings from degree 1 to 6 with the lowest time complexity. nstead of running different computation analyses separately, ADCA can be used as a generic tool to attack the white-box implementations.

SCHEME OF FUNCTION STUDY AND METHODOLOGICAL APPROACH OF SKETCHING GRAPHS OF THE FUNCTIONS IN THE SCHOOL COURSE OF MATHEMATICAL ANALYSIS

The article examines one of the most important topics of the high school course "Elements of Algebra and Mathematical Analysis": scheme of function study and methodological approaches to graph construction. In RA the methods of sketching graphs of function are studied during the high school course "Elements of Algebra and Mathematical Analysis" in two stages: without using the concept of derivative (10th grade) and with the use of the derivative (11th grade) [1, 2]. The derivative is introduced in the 11th grade during the step-by-step study of a function, which facilitates the study of the properties of the function. However, after analyzing the function with the recommended steps, the students sometimes find it difficult to sketch the graph of the function because they encounter a number of difficulties that are insurmountable due to the lack of relevant knowledge. This article recommends introducing the concepts of function convexity (concavity), the inflection point during the teaching of second-order derivatives in the 11th grade in basic mathematics streams of the high school course "Elements of Algebra and Mathematical Analysis", which will significantly simplify the process of sketching the graph of the function. As a result, the course will be more comprehensive, and the outline of the function study and sketching graphs will be more understandable to high school students

Nonlinear computational framework for hybrid ductile glulam joists

This paper starts by presenting a nonlinear algebraic analysis of hybrid glulam sections, including ductile compression-softening constitutive models obtained via regression analysis of material test data, to compute full-range admissible values of moment (M), co-existent curvature (κ) and excluded-area axial force (Fea) for the sections. The (M,κ,Fea) states, double-checked by treating the section alternately as discretised and as a continuum, are clustered into (M,κ) and (M,Fea) data-sets that permit regression-analysis of κ and Fea as polynomial functions of M. For any load on a glulam member the M profile is known, so κ(M) is a more efficient route to calculating deflections than is M(κ). The κ(M) and Fea(M) constitutive functions, which enable assessment of any section state without tedious recalculation, are fused with longitudinal compatibility and equilibrium requirements to predict the joists’ nonlinear responses up to ultimate. Using quartic or Glos compressive constitutive models, spreadsheet-coding of this framework is shown to predict nonlinear local (κ(M)) and global (load–deflection) responses close to test data, also axial and longitudinal-shear stress redistributions mimicking FE predictions for distributed- or point-loaded hybrid glulam joists comprising combinations of poplar, blue-gum, maritime-pine and larch. The results show that post-peak reductions on compressive stress–strain curves cause through-depth reversal of longitudinal-shear at high moments.