Search Results (25)

In 1970, Dana Scott proposed his highly influential “mathematical theory of computation” to define the relationship between the text of a program and what the program computes (or denotes)—the “semantics” of the program. Scott used a complete lattice based on the “information ordering”, with the bottom element representing undefined—a program failing to terminate normally, thus producing no information. The top element, however, was unused. Hence most subsequent applications of denotational semantics have used mathematical structures that avoid top elements. We suggest that the information ordering is relevant not only to semanticists but also to working programmers as a basis for determining if a program component or a computation is correct according to their intentions. We also suggest that a return to the use of complete lattices is called for, to broaden formal semantics and allow it to encompass programmer intentions. That is because often those intentions permit more than one runtime behaviour for a given input. In this paper we explore the connections between the information ordering, correctness of computations and programs, and debugging. We present a general theory and describe several instances where the intention for what our logic/functional code computes plus what it actually computes can be described by elements in a complete lattice. For correct code, the information order relates (1) what is intended and what is computed, (2) successive states of a computation, and (3) the left and right sides of program component definitions. For bugs, the information order is violated. The technical results are a reasonably straightforward extension to previous denotational semantics work, but the scheme aligns much better with practical programming and software tools. The theory extends both the theoretical basis and practical flexibility of declarative debugging and reasoning about partial correctness and gives an attractive mathematical framework that encompasses our intentions, our programs and what they compute. Full article

The article investigates the dielectric and conductive properties of the polar α-phase of Na₃Fe₂(PO₄)₃ polycrystals synthesized by solid-phase (sample type 1), melt (type 2), and melt-quenching (type 3) methods. To enable a rapid assessment of the dielectric properties of the polar α-phase of Na₃Fe₂(PO₄)₃, the thermo-polarization mobility parameter μ_Tp(T, E(ω)) was introduced. By studying the dielectric properties, it was concluded that the polar α-phase of type 1 samples consists of large and small dipoles and ordered sodium cations, which possess low values of μ_Tp(T, E(ω)), indicating the presence of strong interaction forces between the crystal lattice and the cationic part of the polycrystal. Additional studies of the samples’ conductivity confirm this conclusion. Studies of the polar α-phase of Na₃Fe₂(PO₄)₃ in type 2 samples have established that their structure contains dipoles and sodium cations with higher values of μ_Tr(T, E(ω)), and also exhibits higher conductivity than Type 1 samples. These data indicate a weakening of the interaction forces between the cationic and anionic components in type 2 polycrystals due to a partial increase in crystal symmetry. The results of studies of the polar α-phase of type 3 samples show that their structure contains dipoles and sodium cations with higher values of μ_Tr(T, E(ω)), and also exhibits higher conductivity than type 2 samples. It is concluded that the structure of type 3 samples is characterized by weak interaction forces between the cationic and anionic parts as a result of an increase in the symmetry of the polar α-phase of Na₃Fe₂(PO₄)₃, caused by sharply graded temperature conditions during the synthesis of polycrystals. By studying the dielectric properties of cathode materials, it is possible to obtain information on the extent of interactions between the cationic and anionic components in polycrystals. It is, therefore, appropriate to use this approach when investigating a wide range of new dielectric and ion-conducting materials. Full article

In this paper, we present a novel method to solve trivariate polynomial modular equations of the form $x(y^2+Ay+B)+z\equiv 0 (mod e).$ Our approach integrates Coppersmith’s method with lattice basis reduction to efficiently solve the former equation. Several variants of RSA are based on the cubic Pell equation $x^3+f{y}^3+f^2{z}^3-3fxyz\equiv 1 (mod N)$, where f is a cubic nonresidue modulus $N=pq$. In these variants, the public exponent e and the private exponent d satisfy $ed\equiv 1 (mod \psi (N\left)\right)$ with $\psi \left(N\right)=\left(p^2+p+1\right)\left(q^2+q+1\right)$. Moreover, d can be written in the form $d\equiv {\displaystyle \frac{v_0}{z_0}} (mod \psi \left(N\right))$ with any $z_0$ satisfying $gcd(z_0,\psi \left(N\right))=1$. In this paper, we apply our method to attack the variants when $d\equiv {\displaystyle \frac{v_0}{z_0}} (mod \psi \left(N\right))$ and when $|z_0|$ and $|v_0|$ are suitably small. We also show that our method significantly improves the bounds of the private exponents d of the previous attacks on the variants, particularly in the scenario of small private exponents and in the scenarios where partial information about the primes is available. Full article

SnSe₂, as a prominent member of the post-transition metal dichalcogenides, exhibits many intriguing physical phenomena and excellent thermoelectric properties, calling for both fundamental study and potential application in two-dimensional (2D) devices. In this article, we realized the molecular beam epitaxial growth of SnSe₂ films on a $\sqrt{3}\times \sqrt{3}$-Sn reconstructed Si(111) surface. The analysis of reflection high-energy electron diffraction reveals the in-plane lattice orientation as SnSe₂[110]//$\sqrt{3}$-Sn [112]//Si [110]. In addition, the flat morphology of SnSe₂ film was identified by scanning tunneling microscopy (STM), implying the relatively strong adsorption effect of $\sqrt{3}$-Sn/Si(111) substrate to the SnSe₂ adsorbates. Subsequently, the interfacial charge transfer was observed by X-ray photoemission spectroscopy. Afterwards, the direct characterization of electronic structures was obtained via angle-resolved photoemission spectroscopy. In addition to proving the presence of interfacial charge transfer again, a new relatively flat in-gap band was found in monolayer and few-layer SnSe₂, which disappeared in multi-layer SnSe₂. The interface strain-induced partial structural phase transition of thin SnSe₂ films is presumed to be the reason. Our results provide important information on the characterization and effective modulation of electronic structures of SnSe₂ grown on $\sqrt{3}$-Sn/Si(111), paving the way for the further study and application of SnSe₂ in 2D electronic devices. Full article

Several RSA variants enforce a constraint between their public and private keys through the relation $ed\equiv 1(mod(p^2-1)(q^2-1))$, where p and q are the prime factors of their RSA modulus $N=pq$. In this paper, we introduce a novel attack on RSA variant schemes where the public exponent satisfies an equation of the form $eu\equiv z(mod(p^2-1)(q^2-1))$, with sufficiently small $\left|z\right|$, $\left|u\right|$, in a scenario where the attacker has access to an approximation of one of the prime factors. Our new attack utilizes Coppersmith’s method, combined with lattice basis reduction techniques, to efficiently recover the prime factors of the RSA modulus in these scenarios. This method offers a significant improvement over prior attacks on RSA variants with small private exponents or partial prime information. Full article

Silicon carbide (SiC) has significant potential as a third-generation semiconductor material due to its exceptional thermal and electronic properties, yet its high hardness and brittleness make processing costly and complex. This study introduces ultraviolet laser ablation as a method for direct SiC material removal, investigating the effects of varying scanning speeds on surface composition, hardness, and ablation depth. The results indicate optimal processing speeds for the Si and C faces at 200 mm/s and 100 mm/s, respectively. Ablation depth is linearly correlated with laser repetitions, achieving a 25% improvement in removal efficiency at 100 mm/s on the C face compared to higher speeds. A composition analysis shows that the Si and C faces of SiC exhibit consistent ratios of Si, O, and C both before and after ablation. Post-ablation, the proportion of Si and C decreases with an increased presence of oxygen. At scanning speeds below 200 mm/s, the variation in speed has minimal effect on the compositional ratios, indicating a stable elemental distribution across the surface despite differences in processing speed. Hardness testing indicates an initial hardness of 13,896 MPa for the C face, higher than that of the Si face, with both surfaces experiencing a drop to less than 1% of their original hardness (below 50 MPa) after ablation. Lattice structure analysis shows Moissanite-5H SiC and cubic silicon formation on the Si face, while the C face retains partial SiC structure. This study found that when laser parameters are used to process SiC, the processing parameters required on both sides are different and provide important reference information for future industrial processing applications to shorten the time and process cost of SiC surface thinning. Full article

Partial information decompositions (PIDs) aim to categorize how a set of source variables provides information about a target variable redundantly, uniquely, or synergetically. The original proposal for such an analysis used a lattice-based approach and gained significant attention. However, finding a suitable underlying decomposition measure is still an open research question at an arbitrary number of discrete random variables. This work proposes a solution with a non-negative PID that satisfies an inclusion–exclusion relation for any f-information measure. The decomposition is constructed from a pointwise perspective of the target variable to take advantage of the equivalence between the Blackwell and zonogon order in this setting. Zonogons are the Neyman–Pearson region for an indicator variable of each target state, and f-information is the expected value of quantifying its boundary. We prove that the proposed decomposition satisfies the desired axioms and guarantees non-negative partial information results. Moreover, we demonstrate how the obtained decomposition can be transformed between different decomposition lattices and that it directly provides a non-negative decomposition of Rényi-information at a transformed inclusion–exclusion relation. Finally, we highlight that the decomposition behaves differently depending on the information measure used and how it can be used for tracing partial information flows through Markov chains. Full article

Based on the NTRU trapdoor used in NIST’s Falcon, a signcryption scheme following the sign-then-encrypt paradigm is constructed. The existing partitioning technique based on Waters hash over the lattice can not complete the security reduction in the standard model for the signature part due to the “partiality” of the pre-image generated with the NTRU trapdoor. To address this, a variant of Waters hash over small integers is proposed and, the probability of the successful reduction is analyzed. The resulting signcryption achieves existential unforgeability under the adaptive chosen-message attacks. By utilizing the uniqueness of the secret and the noise in an NTRU instance, the tag used in encryption is eliminated. Furthermore, a method to construct tamper-sensitive lattice public key encryption is proposed. This approach implants the ciphertext-sensitive information into the lattice public key encryption and binds it to the encrypted information. The malleability to the public key ciphertext triggers the change of the message–signature pair so that the IND-CCA2 security of the entire ciphertext can be guaranteed by the signature for the message. Thanks to the rational design and the efficiency of the NTRU trapdoor, the computational overhead of the proposed scheme is reduced significantly compared to the existing lattice-based signcryption scheme, reaching orders of magnitude improvement in efficiency. The experiment shows that the proposed scheme is efficient. Full article

Physics-Informed Neural Networks (PINNs) improve the efficiency of data utilization by combining physical principles with neural network algorithms and thus ensure that their predictions are consistent and stable with the physical laws. PINNs open up a new approach to address inverse problems in fluid mechanics. Based on the single-relaxation-time lattice Boltzmann method (SRT-LBM) with the Bhatnagar–Gross–Krook (BGK) collision operator, the PINN-SRT-LBM model is proposed in this paper for solving the inverse problem in fluid mechanics. The PINN-SRT-LBM model consists of three components. The first component involves a deep neural network that predicts equilibrium control equations in different discrete velocity directions within the SRT-LBM. The second component employs another deep neural network to predict non-equilibrium control equations, enabling the inference of the fluid’s non-equilibrium characteristics. The third component, a physics-informed function, translates the outputs of the first two networks into physical information. By minimizing the residuals of the physical partial differential equations (PDEs), the physics-informed function infers relevant macroscopic quantities of the flow. The model evolves two sub-models that are applicable to different dimensions, named the PINN-SRT-LBM-I and PINN-SRT-LBM-II models according to the construction of the physics-informed function. The innovation of this work is the introduction of SRT-LBM and discrete velocity models as physical drivers into a neural network through the interpretation function. Therefore, the PINN-SRT-LBM allows a given neural network to handle inverse problems of various dimensions and focus on problem-specific solving. Our experimental results confirm the accurate prediction by this model of flow information at different Reynolds numbers within the computational domain. Relying on the PINN-SRT-LBM models, inverse problems in fluid mechanics can be solved efficiently. Full article

The idea of a partial information decomposition (PID) gained significant attention for attributing the components of mutual information from multiple variables about a target to being unique, redundant/shared or synergetic. Since the original measure for this analysis was criticized, several alternatives have been proposed but have failed to satisfy the desired axioms, an inclusion–exclusion principle or have resulted in negative partial information components. For constructing a measure, we interpret the achievable type I/II error pairs for predicting each state of a target variable (reachable decision regions) as notions of pointwise uncertainty. For this representation of uncertainty, we construct a distributive lattice with mutual information as consistent valuation and obtain an algebra for the constructed measure. The resulting definition satisfies the original axioms, an inclusion–exclusion principle and provides a non-negative decomposition for an arbitrary number of variables. We demonstrate practical applications of this approach by tracing the flow of information through Markov chains. This can be used to model and analyze the flow of information in communication networks or data processing systems. Full article

The bounded distance decoding (BDD) is a fundamental problem in lattice-based cryptography which is derived from the closest vector problem (CVP). In this paper, we adapt the lattice enumeration with discrete pruning, a burgeoning method for the shortest lattice vector problem (SVP), to solve BDD in various cryptanalysis scenarios using direct method. We first transfer the basic definition involved in discrete pruning technique from SVP to CVP, prove corresponding properties and give the specific procedures of the algorithm. Additionally, we use the discrete pruning technique to interpret the classical CVP algorithms, including Babai’s nearest plane and Lindner–Peikert nearest planes, which can be regarded as discrete pruned enumeration on some special pruning sets. We propose three probability models in the runtime analysis to accurately estimate the cost of our algorithm in different application scenarios. We study the application of discrete pruned enumeration for BDD mainly on LWE-based cryptosystem and DSA with partially known nonces. The experimental results show that our new algorithm has higher efficiency than the previous algorithms which directly solve BDD, including the nearest plane(s) algorithms and the lattice enumeration with classical pruning strategies, and we are able to recover the DSA secret with less leaked information than the previous works. Full article

Recently, the notion of a three-way fuzzy set is presented, inspired by the basic ideas of three-way decision and various generalized fuzzy sets, including lattice-valued fuzzy sets, partial fuzzy sets, intuitionistic fuzzy sets, etc. As the new theory of uncertainty, it has been used in attribute reduction and as a new control method for the water level. However, as an extension of a three-way decision, this new theory has not been used in multi-criteria decision making (MCDM for short). Based on the previous work, in this paper, we present rough set models based on three-way fuzzy sets, which extend the existing fuzzy rough set models in both complete and incomplete information systems. Furthermore, the new models are used to solve the issue of MCDM. Firstly, three-way fuzzy relation rough set and three-way fuzzy covering rough set models are presented for complete and incomplete information systems. Because almost all existing fuzzy rough set models are proposed under complete information, the new proposed models can be seen as a supplement to these existing models. Then, a relationship between the three-way fuzzy relation rough set and the three-way fuzzy covering rough set is presented. Finally, a novel method for the issue of MCDM is presented under the novel three-way fuzzy rough set models, which is used in paper defect diagnosis. Full article

The Rivest–Shamir–Adleman (RSA) cryptosystem is currently the most influential and commonly used algorithm in public-key cryptography. Whether the security of RSA is equivalent to the intractability of the integer factorization problem is an interesting issue in mathematics and cryptography. Coron and May solved the above most fundamental problem and proved the polynomial-time equivalence of computing the RSA secret key and factoring. They demonstrated that the RSA modulus $N=pq$ can be factored in polynomial time when given RSA key information $(N,e,d)$. The CRT-RSA variant is a fast technical implementation of RSA using the Chinese Remainder Theorem (CRT), which aims to speed up the decryption process. We focus on the polynomial-time equivalence of computing the CRT-RSA secret key and factoring in this paper. With the help of the latest partial key exposure attack on CRT-RSA, we demonstrate that there exists a polynomial-time algorithm outputting the factorization of $N=pq$ for $e{d}_p,e{d}_q<N^{3/2}$ when given the CRT-RSA key information $(N,e,d_p,d_q)$. We apply Coppersmith’s lattice-based method as a basic mathematical tool for finding the small root solutions of modular polynomial equations. Furthermore, we provide validation experiments to illustrate the correctness of the CRT-RSA modulus factorization algorithm, and show that computing the CRT-RSA secret key and factoring its modulus is polynomial-time equivalent by using concrete numerical examples. Full article

Thin films of colloidal CZTS nanocrystals (NCs) synthesized using a “green” approach in water with a variation of the copper-to-tin ratio are investigated by Raman scattering, mid-infrared (molecular vibrations) and near-infrared (free carrier) absorption, X-ray photoemission spectroscopy (XPS), electrical conductivity, and conductive atomic force microscopy (cAFM). We determined the effect of the actual Cu content on the phonon spectra, electrical conductivity, and spectral parameters of the plasmon band. An increase in the electrical conductivity of the NC films upon annealing at 220 °C is explained by three factors: formation of a Cu_xS nanophase at the CZTS NC surface, partial removal of ligands, and improved structural perfection. The presence of the Cu_xS phase is concluded to be the determinant factor for the CZTS NC film conductivity. Cu_xS can be reliably detected based on the analysis of the modified Auger parameter of copper, derived from XPS data and corroborated by Raman spectroscopy data. Partial removal of the ligand is concluded from the agreement of the core-level XPS and vibrational IR spectra. The degree of lattice perfection can be conveniently assessed from the Raman data as well. Further important information derived from a combination of photoelectron and optical data is the work function, ionization potential, and electron affinity of the NC films. Full article

Based on the conceptual basis of information theory, we propose a novel mutual information measure—‘path-based mutual information’. This information measure results from the representation of a set of random variables as a probabilistic graphical model. The edges in this graph are modeled as discrete memoryless communication channels, that is, the underlying data is ergodic, stationary, and the Markov condition is assumed to be applicable. The associated multilinear stochastic maps, tensors, transform source probability mass functions into destination probability mass functions. This allows for an exact expression of the resulting tensor of a cascade of discrete memoryless communication channels in terms of the tensors of the constituting communication channels in the paths. The resulting path-based information measure gives rise to intuitive, non-negative, and additive path-based information components—redundant, unique, and synergistic information—as proposed by Williams and Beer. The path-based redundancy satisfies the axioms postulated by Williams and Beer, the identity axiom postulated by Harder, and the left monotonicity axiom postulated Bertschinger. The ordering relations between redundancies of different joint collections of sources, as captured in the redundancy lattices of Williams and Beer, follow from the data processing inequality. Although negative information components can arise, we speculate that these either result from unobserved variables, or from adding additional sources that are statistically independent from all other sources to a system containing only non-negative information components. This path-based approach illustrates that information theory provides the concepts and measures for a partial information decomposition. Full article