Field F2 Research Articles

An Applied Image Cryptosystem on Moore’s Automaton Operating on δ ( q k )/𝔽 2

The volume of multimedia-based image data or video frames in Web 3.0 is constantly increasing, owing to the advancement of real-time data transmission. However, security vulnerabilities frequently impair the performance of real-time applications. Many researchers have recently proposed image encryption schemes based on a high-dimensional chaotic system due to properties such as ergodicity and initial state sensitivity. Nonetheless, most schemes have suffered from excessive computational complexity, low security, and the generation of cryptographically secure random numbers. To overcome these challenges, an efficient and highly secure cryptosystem is necessary for safe multimedia transmission in Web 3.0. This article proposes a novel work on the image cryptosystem based on the Escalation function with a one-time key-oriented Moore’s Automaton over a finite field 𝔽 2 . The Escalation function is a nonlinear scrambling technique for plaintext images that goes through the confusion phase and plays an essential role in row-column permutation. To make the algorithm more secure and robust in the diffusion phase, the proposed Moore’s Automaton produced ciphertext images through a highly random key stream generated by the combination of a logistic map and cyclic group. Specifically, the proposed Moore’s Automaton operates on δ ( q k )/𝔽 2 to render random binary bits into unpredictable sequences to construct ciphertext images. Our new finding quickens the speed and provides adequate key space, and pixel distributions are more uniform, have high entropy value, and are secure against differential and statistical attacks.

Detecting Exoplanets Closer to Stars with Moderate Spectral Resolution Integral-field Spectroscopy

While radial velocity surveys have demonstrated that the population of gas giants peaks around 3 au, the most recent high-contrast imaging surveys have only been sensitive to planets beyond ∼10 au. Sensitivity at small angular separations from stars is currently limited by the variability of the point-spread function. We demonstrate how moderate-resolution integral-field spectrographs can detect planets at smaller separations (≲ 0.3“) by detecting the distinct spectral signature of planets compared to the host star. Using OSIRIS (R ≈ 4000) at the W.M. Keck Observatory, we present the results of a planet search via this methodology around 20 young targets in the Ophiuchus and Taurus star-forming regions. We show that OSIRIS can outperform high-contrast coronagraphic instruments equipped with extreme adaptive optics and non-redundant masking in the 0.05“–0.3“ regime. As a proof of concept, we present the 34σ detection of a high-contrast M dwarf companion at ≈0.1“ with flux ratio of ≈ 0.92% around the field F2 star HD 148352. We developed an open-source Python package, breads, for the analysis of moderate-resolution integral-field spectroscopy data in which the planet and the host star signal are jointly modeled. The diffracted starlight continuum is forward-modeled using a spline model, which removes the need for prior high-pass filtering or continuum normalization. The code allows for analytic marginalization of linear hyperparameters, which simplifies the posterior sampling of other parameters (e.g., radial velocity, effective temperature). This technique could prove very powerful when applied to integral-field spectrographs such as NIRSpec on the JWST and other upcoming first-light instruments on the future Extremely Large Telescopes.

An optimal tester for k-linear

A Boolean function f:{0,1}n→{0,1} is k-linear if it returns the sum (over the binary field F2) of exactly k coordinates of the input. In this paper, we study property testing of the classes k-Linear, the class of all k-linear functions, and k-Linear⁎, the class ∪j=0kj-Linear. We give a non-adaptive distribution-free two-sided ϵ-tester for k-Linear that makesO(klog⁡k+1ϵ) queries. This matches the lower bound known from the literature.We then give a non-adaptive distribution-free one-sided ϵ-tester for k-Linear⁎ that makes the same number of queries and show that any non-adaptive uniform-distribution one-sided ϵ-tester for k-Linear must make at least Ω˜(k)log⁡n+Ω(1/ϵ) queries. The latter bound almost matches the upper bound O(klog⁡n+1/ϵ) known from the literature. We then show that any adaptive uniform-distribution one-sided ϵ-tester for k-Linear must make at least Ω˜(k)log⁡n+Ω(1/ϵ) queries.

A note on the distribution of weights of fixed-rank matrices over the binary field

Let M be a random m×n rank-r matrix over the binary field F2, and let wt(M) be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as m,n→+∞ with r fixed and m/n tending to a constant, we have thatwt(M)−1−2−r2mn2−r(1−2−r)4(m+n)mn converges in distribution to a standard normal random variable.

Monitoring of Soybean Maturity Using UAV Remote Sensing and Deep Learning

Soybean breeders must develop early-maturing, standard, and late-maturing varieties for planting at different latitudes to ensure that soybean plants fully utilize solar radiation. Therefore, timely monitoring of soybean breeding line maturity is crucial for soybean harvesting management and yield measurement. Currently, the widely used deep learning models focus more on extracting deep image features, whereas shallow image feature information is ignored. In this study, we designed a new convolutional neural network (CNN) architecture, called DS-SoybeanNet, to improve the performance of unmanned aerial vehicle (UAV)-based soybean maturity information monitoring. DS-SoybeanNet can extract and utilize both shallow and deep image features. We used a high-definition digital camera on board a UAV to collect high-definition soybean canopy digital images. A total of 2662 soybean canopy digital images were obtained from two soybean breeding fields (fields F1 and F2). We compared the soybean maturity classification accuracies of (i) conventional machine learning methods (support vector machine (SVM) and random forest (RF)), (ii) current deep learning methods (InceptionResNetV2, MobileNetV2, and ResNet50), and (iii) our proposed DS-SoybeanNet method. Our results show the following: (1) The conventional machine learning methods (SVM and RF) had faster calculation times than the deep learning methods (InceptionResNetV2, MobileNetV2, and ResNet50) and our proposed DS-SoybeanNet method. For example, the computation speed of RF was 0.03 s per 1000 images. However, the conventional machine learning methods had lower overall accuracies (field F2: 63.37–65.38%) than the proposed DS-SoybeanNet (Field F2: 86.26%). (2) The performances of the current deep learning and conventional machine learning methods notably decreased when tested on a new dataset. For example, the overall accuracies of MobileNetV2 for fields F1 and F2 were 97.52% and 52.75%, respectively. (3) The proposed DS-SoybeanNet model can provide high-performance soybean maturity classification results. It showed a computation speed of 11.770 s per 1000 images and overall accuracies for fields F1 and F2 of 99.19% and 86.26%, respectively.

A note on the hit problem for the polynomial algebra of six variables and the sixth algebraic transfer

Let A be the classical, singly-graded Steenrod algebra over the prime order field F2 and let P⊗h:=F2[t1,…,th] denote the polynomial algebra on h generators, each of degree 1, viewed as a module over A. Write GLh for the usual general linear group of rank h over F2. As well known, the (mod 2) cohomology groups of the Steenrod algebra, ExtAh,h+⁎(F2,F2) is still largely mysterious for all homological degrees h≥6. The h-th algebraic transferTrhA:(F2⊗GLhAnnA‾[P⊗h]⁎)n⟶ExtAh,h+n(F2,F2), defined by William Singer [28], is a helpful tool to describe that Ext groups. Singer conjectured that this transfer is a monomorphism, but it remains open for any h≥5. There is currently no information on the conjecture for h=6. In this paper, we verify Singer's conjecture for homological degree h=6 and the internal degree of the general form ns:=6(2s−1)+10⋅2s. This result is important, since it tells us that the non-zero elements h22g1=h4Ph2∈ExtA6,6+n1(F2,F2), and D2∈ExtA6,6+n2(F2,F2) are not in the image of the sixth algebraic transfer. This Note is a continuation of our previous one [22], which will refer to as Part I. Our approach is based on explicitly solving the hit problem for the Steenrod algebra in the case h=6 and degree ns. This extends a result in Mothebe, Kaelo and Ramatebele [13]. At the same time, we also use this obtained result to establish the dimension result for the space of the unhit elements, F2⊗AP⊗7 in a certain general degree.

New singly and doubly even binary [72,36,12] self-dual codes from M2(R)G - group matrix rings

In this work, we present a number of generator matrices of the form [I2n|τ2(v)], where I2n is the 2n×2n identity matrix, v is an element in the group matrix ring M2(R)G and where R is a finite commutative Frobenius ring and G is a finite group of order 18. We employ these generator matrices and search for binary [72,36,12] self-dual codes directly over the finite field F2. As a result, we find 134 Type I and 1 Type II codes of this length, with parameters in their weight enumerators that were not known in the literature before. We tabulate all of our findings.

Hamiltonian and pseudo-Hamiltonian cycles and fillings in simplicial complexes

We introduce and study a d-dimensional generalization of graph Hamiltonian cycles. These are the Hamiltonian d-dimensional cycles in Knd (the complete simplicial d-complex over a vertex set of size n). Hamiltonian d-cycles are the simple d-cycles of a complete rank, or, equivalently, of size 1+(n−1d).The discussion is restricted to the fields F2 and Q. For d=2, we characterize the n's for which Hamiltonian 2-cycles exist. For d=3 it is shown that Hamiltonian 3-cycles exist for infinitely many n's. In general, it is shown that there always exist simple d-cycles of size (n−1d)−O(nd−3). All the above results are constructive.Our approach naturally extends to (and in fact, involves) d-fillings, generalizing the notion of T-joins in graphs. Given a (d−1)-cycle Zd−1∈Knd, F is its d-filling if ∂F=Zd−1. We call a d-filling Hamiltonian if it is acyclic and of a complete rank, or, equivalently, is of size (n−1d). If a Hamiltonian d-cycle Z over F2 contains a d-simplex σ, then Z∖σ is a Hamiltonian d-filling of ∂σ (a closely related fact is also true for cycles over Q). Thus, the two notions are closely related.Most of the above results about Hamiltonian d-cycles hold for Hamiltonian d-fillings as well.

The symptom-syndrome analysis of multivariate categorical data based on Zhegalkin polynomials

In this article, we study the distribution, entropy and other informational properties of finite projective subspaces (syndromes) parameterized by impulse sequences with basic elements in the form of symptoms - polynomials over the field F2 which are known as Zhegalkin polynomials. It has been proven that the super syndrome, which is a linear syndrome with basic elements in the form of a multiplicative syndrome, is closed. If in the multiplication of two symptoms one is neutral, then we are talking about its majorization. The ordered by majorization symptoms form a majorized syndrome. Is proved that the majorized syndrome is closed and coincides with the super syndrome. The statements formulated in the first part of the paper are used to justify the convergence of the iterative procedure (PI), in which the most informative symptoms selected from partial super syndromes are again used in the next step. The stationary state of PI is obtained if all elements of the input set belong to either the same partial super syndrome or to the majorized syndrome. Thanks IP it is possible to quickly find the optimal syndrome from a large set of variables. An example from phthisiology shows how the specificity of classification can be improved using symptom analysis.

Networks of Picture Processors with Filtering Based on Evaluation Sets as Solvers for Cryptographic Puzzles Based on Random Multivariate Quadratic Equations

Networks of picture processors is a massively distributed and parallel computational model inspired by the evolutionary cellular processes, which offers efficient solutions for NP-complete problems. This bio-inspired model computes two-dimensional strings (pictures) using simple rewriting rules (evolutionary operations). The functioning of this model mimics a community of cells (pictures) that are evolving according to these bio-operations via a selection process that filters valid surviving cells. In this paper, we propose an extension of this model that empowers it with a flexible method that selects the processed pictures based on a quantitative evaluation of its content. In order to show the versatility of this extension, we introduce a solver for a cryptographic proof-of-work based on the hardness of finding a solution to a set of random quadratic equations over the finite field F2. This problem is demonstrated to be NP-hard, even with quadratic polynomials over the field F2, when the number of equations and the number of variables are of roughly the same size. The proposed solution runs in O(n2) computational steps for any size (n,m) of the input pictures. In this context, this paper opens up a wide field of research that looks for theoretical and practical solutions of cryptographic problems via software/hardware implementations based on bio-inspired computational models.

Digital quantum groups

We find and classify all bialgebras and Hopf algebras or “quantum groups” of dimension ≤4 over the field F2={0,1}. We summarize our results as a quiver, where the vertices are the inequivalent algebras and there is an arrow for each inequivalent bialgebra or Hopf algebra built from the algebra at the source of the arrow and the dual of the algebra at the target of the arrow. There are 314 distinct bialgebras and, among them, 25 Hopf algebras, with at most one of these from one vertex to another. We find a unique smallest noncommutative and noncocommutative one, which is moreover self-dual and resembles a digital version of uq(sl2). We also find a unique self-dual Hopf algebra in one anyonic variable x4 = 0. For all our Hopf algebras, we determine the integral and associated Fourier transform operator, viewed as a representation of the quiver. We also find all quasitriangular or “universal R-matrix” structures on our Hopf algebras. These induce solutions of the Yang–Baxter or braid relations in any representation.

The Effect of the Primitive Irreducible Polynomial on the Quality of Cryptographic Properties of Block Ciphers

Substitution boxes are the only nonlinear component of the symmetric key cryptography and play a key role in the cryptosystem. In block ciphers, the S-boxes create confusion and add valuable strength. The majority of the substitution boxes algorithms focus on bijective Boolean functions and primitive irreducible polynomial that generates the Galois field. For binary field F2, there are exactly 16 primitive irreducible polynomials of degree 8 and it prompts us to construct 16 Galois field extensions of order 256. Conventionally, construction of affine power affine S-box is based on Galois field of order 256, depending on a single degree 8 primitive irreducible polynomial over ℤ2. In this manuscript, we study affine power affine S-boxes for all the 16 distinct degree 8 primitive irreducible polynomials over ℤ2 to propose 16 different 8×8 substitution boxes. To perform this idea, we introduce 16 affine power affine transformations and, for fixed parameters, we obtained 16 distinct S-boxes. Here, we thoroughly study S-boxes with all possible primitive irreducible polynomials and their algebraic properties. All of these boxes are evaluated with the help of nonlinearity test, strict avalanche criterion, bit independent criterion, and linear and differential approximation probability analyses to measure the algebraic and statistical strength of the proposed substitution boxes. Majority logic criterion results indicate that the proposed substitution boxes are well suited for the techniques of secure communication.

12×12 S-box Design and its Application to RGB Image Encryption

Customarily, 8×8 S-box is a 16×16 look up table over Galois field GF28, algebra over the binary field F2. A 12×12 S-box over GF212 is not workable as the computer memory does not manage it. The finite chain ring F2[x]<x12> is fundamentally an algebra over the binary field F2. In this paper we considered the advantage and devise a novel method of 12×12 S-box design over the unit elements of finite chain ring F2[x]<x12>. The newly obtained S-box has greatly higher confusion ability than any of 8×8 S-box. To observe this effect, an RGB image encryption application is given. In the proposed encryption scheme, we doubled the bits of 12×12 S-box from 12 to 24 for creating confusion in the data. Consequently, a new method of color image encryption is designed by which 24 binary bits are utilized at the place of byte. Whereas for diffusion we apply the linear permutation P=i×32 mod 257 and finally bitwise Exclusive-or operation is operationalized. Analysis reveals that the proposed image encryption approach might switch many 8×8 S-box based color image encryption schemes.

Корреляционный анализ пространственных аномалий вычисленных и измеренных полей в системе GIS-ENDDB

The GIS-ENDDB expert-information system includes the global seismic catalogs and measurement data for various geophysical fields. Statistical processing of these data and visualization of the distributing characteristics of the seismic-geodynamic process in the form of isolines make it possible to reveal the spatial relationship between the seismicity distribution and anomalies of geophysical fields. The first attempts to identify this relationship were made with L.S. Sokolova in 2016 based on a visual comparison of maps obtained as a result of applying methods of various data analysis (in particular, the isolines maps of the maximum magnitude of recorded earth-quakes Mmax) for different samples of seismological catalogs on the one hand, and various modifications of the measured geophysical fields (in this case, the regional component of heat flow (HF) field) on the other hand. These results were more likely qualitative, but later a correlation analysis function was implemented into GIS-ENDDB for the quantitative comparison of the fields. The correlation analysis is carried out on a histogram of the distributing number of cells (nodes) NC from the difference ∆F of the reduced values of the fields F1 and F2 (relative to the arithmetic mean Fi.OCP) in the cells that are equal for the compared fields. To identify non-random variations, NC histograms (∆F) are given against the background of the normal distribution curve, which allows not only visually (by the deviation of the first columns from the norm), but also quantitatively to establish the degree of correlation. For the studied territories, histograms have obtained that show: 1. a moderate correlation of HF and Mmax fields in the North American continent (NEIC catalog, M≥5, cell 3×4); 2. no correlation across Asia - Far East region (COMPLEX, M≥4, H≥50 km, cell 6×8); and 3. high correlation in a rectangular region covering the Inside-Asian mountain belt (30-40N; 66-84E., COMPLEX, M≥5, H≥50 km, cell 1×1).

A note on the stability of trinomials over finite fields

A polynomial f(x) over a field K is called stable if all of its iterates are irreducible over K. In this paper, we study the stability of trinomials over finite fields. We show that if f(x) is a trinomial of even degree over the binary field F2, then f(x) is not stable. We prove similar results for some families of monic trinomials over finite fields of odd characteristic. We also study the stability of polynomials of higher weights and prove some results and pose a new conjecture.

The Adjacency Graphs of a Class of LFSRs and Their Applications*

Nonlinear feedback shift registers (NFSRs) are widely used in communication and cryptography. How to construct more NFSRs with maximal periods, which can generate sequences with maximal periods, i.e., de Brujin sequences, is an attractive problem. Recently many results on constructing de Bruijn sequences from adjacency graphs of Linear feedback shift registers (LFSRs) by means of the cycle joining method have been obtained. In this paper we discuss a class of LFSRs with characteristic polynomial p2(x), where p(x) is a primitive polynomial of degree n ≥ 2 over the finite field F2. As results, we determine their cycle structures and adjacency graphs, and further construct a class of new de Bruijn sequences from these LFSRs.

A closed formula for the inverse of a reversible cellular automaton with [formula omitted]-cyclic rule

Reversibility of cellular automata (CA) has been an extensively studied problem from both a theoretical and a practical point of view. It is known when a (2R+1)-cyclic cellular automaton with periodic boundary conditions (p.b.c.) is reversible (see Siap et al., 2013) but, as far as we know, no explicit expression is given for its inverse cellular automaton apart from the case R=1 (see Encinas and del Rey, 2007). In this paper we give a closed formula for the inverse rule of a reversible (2R+1)-cyclic cellular automaton with p.b.c. over the finite field F2 for any value of the neighbourhood radius R. It turns out that the inverse of a reversible (2R+1)-cyclic CA with p.b.c. is again a cyclic CA with p.b.c., but with a different neighbourhood radius, and this radius depends on certain numbers which need to be computed by a new algorithm we introduce. Finally, we apply our results to the case R=1 (which is the ECA with Wolfram rule number 150) to introduce an alternative and improved expression for the inverse transition dipolynomial formulated in Encinas and del Rey (2007). We also illustrate these results by giving explicit computations for the inverse transition dipolynomial of a reversible cellular automaton with penta-cyclic rule.

Symbolic computation on the partition lattice of an n-set

In this paper, an injection from the partition lattice Πn to a zero-dimensional affine algebra over the finite field F2 is constructed. We study some interesting properties of the injection and present two direct sum decompositions of the zero-dimensional affine algebra. Based upon these, some properties of Πn are revealed in an algebraic manner, for instance, the rank-generating function and characteristic polynomial of Πn can be derived using the symbolic method.

On the k‐Error Linear Complexity of Binary Sequences Derived from the Discrete Logarithm in Finite Fields

Let be the finite field with q = pr elements, where p is an odd prime. For the ordered elements , the binary sequence σ = (σ0, σ1, …, σq−1) with period q is defined over the finite field as follows: σn = {0, if n = 0, (1 − χ(ξn))/2, if 1 ≤ n &lt; q}, σn+q = σn, where χ is the quadratic character of . Obviously, σ is the Legendre sequence if r = 1. In this paper, our first contribution is to prove a lower bound on the linear complexity of σ for r ≥ 2, which improves some results of Meidl and Winterhof. Our second contribution is to study the distribution of the k‐error linear complexity of σ for r = 2. Unfortunately, the method presented in this paper seems not suitable for the case r &gt; 2 and we leave it open.

Artin–Schreier extensions of normal bases

In this paper we extend a normal basis of a finite field over its base field to a new basis which permits both computationally inexpensive exponentiation and multiplication. We focus primarily on extensions of the finite field F2. These bases are motivated by Artin–Schreier theory and we conclude that they are particularly useful in Artin–Schreier extensions; that is, extensions F2n of F2 with n a power of two.