rss_2.0Tatra Mountains Mathematical Publications FeedSciendo RSS Feed for Tatra Mountains Mathematical Publicationshttps://sciendo.com/journal/TMMPhttps://www.sciendo.comTatra Mountains Mathematical Publications Feedhttps://sciendo-parsed.s3.eu-central-1.amazonaws.com/647395c94e662f30ba542f63/cover-image.jpghttps://sciendo.com/journal/TMMP140216One Formalized Approach to Truncated Differential Cryptanalysis of Block Ciphershttps://sciendo.com/article/10.2478/tmmp-2024-0022<abstract> <title style='display:none'>Abstract</title> <p>We propose a formalized approach to truncated differential cryptanalysis based on ternary masks that separately account for unchanged, obligatorily changed, and unknown bits in differences. We introduce a security parameter for S-boxes and encryption mappings, which bounds the probability of truncated differentials from below, and examine its basic algebraic properties. Our approach enables the adaptation of existing techniques used in classical differential attacks to truncated differential cryptanalysis, allowing us to extract more information from the encryption process and evaluate the complexity of truncated differential attacks.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00222024-11-09T00:00:00.000+00:00Solving Elliptic Curve Discrete Logarithm Problem on Twisted Edwards Curves Using Quantum Annealing and Index Calculus Methodhttps://sciendo.com/article/10.2478/tmmp-2024-0021<abstract> <title style='display:none'>Abstract</title> <p>This paper presents an approach to solving the elliptic curve discrete logarithm problem on alternative curve models over prime fields using a quantum annealing and index calculus method. Part of the algorithm, relation searching, is transformed into the Quadratic Unconstrained Boolean Optimization (QUBO) problem and then is efficiently solved using the D-Wave computer by quantum annealing. As Faugère et al. showed, twisted Edwards curves, because of their symmetric shape, allow us to obtain solutions of relations searching step using Groebner basis faster than in the case of Weierstrass curves. Because of symmetries, a system of equations of relations searching step for twisted Edwards curves has many symmetric solutions. Using the Groebner basis and having many system solutions makes it easier to find any of them. The same is true using quantum annealing - it is easier to find any solution to the QUBO problem if many are correct. In this paper, we used this observation to find out that a properly constructed QUBO problem for the relations searching step for twisted Edwards curves allows us to find a solution faster for the same size of the base field than in the case of Weierstrass curves. Using the presented approach, we solved the discrete logarithm problem using quantum annealing and index calculus method for elliptic curve discrete logarithm problem defined on twisted Edwards curve over a field 𝔽<sub>1021</sub> with order equal to 4 · 241. It is now the biggest field and size of the group, where the elliptic curve discrete logarithm problem was solved using quantum methods.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00212024-10-29T00:00:00.000+00:00A Fixed Point Approach to the Stability of a Quadratic Functional Equation in Modular Spaces Without Δ-Conditionshttps://sciendo.com/article/10.2478/tmmp-2024-0016<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we investigate the Hyers-Ulam-Rassias stability property of a quadratic functional equation. The even and odd cases for the corresponding function are treated separately before combining them into a single stability result. The study is undertaken in a relatively new structure of modular spaces. The theorems are deduced without using the familiar Δ<sub>2</sub>-property of that space. This complicated the proofs. In the proofs, a fixed point methodology is used for which a modular space version of Banach contraction mapping principle is utilized. Several corollaries and an illustrative example are provided.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00162024-10-20T00:00:00.000+00:00Some Inequalities Involving Weighted Power Meanhttps://sciendo.com/article/10.2478/tmmp-2024-0019<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we first show some inequalities on weighted power mean. When <italic>a, b &gt;</italic> 0, <italic>p</italic> ≥ 1and 0 <italic>&lt;v</italic> ≤ <italic>τ&lt;</italic> 1, we have <disp-formula> <alternatives> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2024-0019_eq_001.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mfrac><mml:mi>v</mml:mi><mml:mi>τ</mml:mi></mml:mfrac><mml:mo>≤</mml:mo><mml:mfrac><mml:mrow><mml:mi>a</mml:mi><mml:msub><mml:mrow><mml:mo>♯</mml:mo></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:msub><mml:mi>b</mml:mi><mml:mo>-</mml:mo><mml:mi>a</mml:mi><mml:msub><mml:mrow><mml:mo>♯</mml:mo></mml:mrow><mml:mi>v</mml:mi></mml:msub><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:msub><mml:mrow><mml:mo>♯</mml:mo></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi></mml:mrow></mml:msub><mml:mi>b</mml:mi><mml:mo>-</mml:mo><mml:mi>a</mml:mi><mml:msub><mml:mrow><mml:mo>♯</mml:mo></mml:mrow><mml:mi>τ</mml:mi></mml:msub><mml:mi>b</mml:mi></mml:mrow></mml:mfrac><mml:mo>≤</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>τ</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:math> <tex-math>\frac{v}{\tau } \le \frac{{a{\sharp _{p,v}}b - a{\sharp _v}b}}{{a{\sharp _{p,\tau }}b - a{\sharp _\tau }b}} \le \frac{{1 - v}}{{1 - \tau }}</tex-math> </alternatives> </disp-formula> and <disp-formula> <alternatives> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2024-0019_eq_002.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mfrac><mml:mi>v</mml:mi><mml:mi>τ</mml:mi></mml:mfrac><mml:mo>≤</mml:mo><mml:mfrac><mml:mrow><mml:mi>a</mml:mi><mml:msub><mml:mrow><mml:mo>♯</mml:mo></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:msub><mml:mi>b</mml:mi><mml:mo>-</mml:mo><mml:mi>a</mml:mi><mml:msub><mml:mrow><mml:mo>!</mml:mo></mml:mrow><mml:mi>v</mml:mi></mml:msub><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:msub><mml:mrow><mml:mo>♯</mml:mo></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi></mml:mrow></mml:msub><mml:mi>b</mml:mi><mml:mo>-</mml:mo><mml:mi>a</mml:mi><mml:msub><mml:mrow><mml:mo>!</mml:mo></mml:mrow><mml:mi>τ</mml:mi></mml:msub><mml:mi>b</mml:mi></mml:mrow></mml:mfrac><mml:mo>≤</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>τ</mml:mi></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:math> <tex-math>\frac{v}{\tau } \le \frac{{a{\sharp _{p,v}}b - a{!_v}b}}{{a{\sharp _{p,\tau }}b - a{!_\tau }b}} \le \frac{{1 - v}}{{1 - \tau }}.</tex-math> </alternatives> </disp-formula> Further, we obtain the range of corresponding inequalities involving the <italic>m</italic> power form of weighted power mean in the same form as above for <italic>m</italic> ∈ ℕ<sup>+</sup> or <italic>p</italic> ≥ <italic>m&gt;</italic> 0, <italic>m</italic> ≤ <italic>p&lt;</italic> 0. As further applications, we provide some inequalities about matrices and determinants, respectively.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00192024-10-20T00:00:00.000+00:00On -Statistically Cauchy Sequences & the Class of -Statistically Concurrent Sequenceshttps://sciendo.com/article/10.2478/tmmp-2024-0013<abstract> <title style='display:none'>Abstract</title> <p>In this article, we present the ideas of <italic>I</italic>-statistically Cauchy criteria and <italic>I</italic><italic><sup>∗</sup></italic>-statistically Cauchy criteria, which are the generalizations of <italic>I</italic>-Cauchy and <italic>I</italic><italic><sup>∗</sup></italic>-Cauchy criterion, respectively. To grasp the differences, we compare this <italic>I</italic>-statistically Cauchy criterion with a few other Cauchy criteria. Also, we investigate a few characteristics of <italic>I</italic><italic><sup>∗</sup></italic>-statistically Cauchy sequences and <italic>I</italic>-statistically Cauchy sequences and demonstrate their equivalence under the condition that the ideal <italic>I</italic> satisfies the property (AP). Furthermore, a relation is defined on the set <italic>S<sub>X </sub></italic>of all sequences in a metric space, which comes out to be an equivalence relation. Finally, we show that if two sequences belong to the same equivalence class, then either both of them are <italic>I</italic>-statistically Cauchy or none of them are.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00132024-10-20T00:00:00.000+00:00Star γ Covers and Their Applications to Selection Principleshttps://sciendo.com/article/10.2478/tmmp-2024-0018<abstract> <title style='display:none'>Abstract</title> <p>By imposing the star operator on <italic>γ</italic> covers, we establish the notion of st-<italic>γ</italic> covers. The interrelationships among many recent variants of <italic>γ</italic> covers have been identified, and various counterexamples have been shown to distinguish them. An investigation has been conducted on the basic topological attributes of these covers. We have also examined the implications between various selection principles and topological games in the context of the class of st-<italic>γ</italic> coverings.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00182024-10-20T00:00:00.000+00:00On O’malley Lower Porouscontinuous Functionshttps://sciendo.com/article/10.2478/tmmp-2024-0014<abstract> <title style='display:none'>Abstract</title> <p>In 2014, J. Borsík and J. Holos defined porouscontinuous functions. Using the notion of density in the sense of O’Malley, new definitions of porouscontinuity were introduced in 2021, namely ℳ𝒪<italic><sub>r</sub></italic> and 𝒮𝒪<italic><sub>r</sub></italic>-continuity. These kinds of porouscontinuity used upper porosity. We consider lower porous-continuity in the sense of O’Malley, where lower porosity is used instead of standard (upper) porosity.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00142024-10-20T00:00:00.000+00:00Sharp and Weighted Boundedness for Multilinear Integral Operatorshttps://sciendo.com/article/10.2478/tmmp-2024-0017<abstract> <title style='display:none'>Abstract</title> <p>In this paper, the weighted boundedness for some multilinear operators related to the Littlewood-Paley operator and Marcinkiewicz operator on the generalized Morrey spaces are obtained by using the sharp estimates of the multilinear operators.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00172024-10-20T00:00:00.000+00:00Neutrosophic Fuzzy Tribonacci ℐ-Lacunary Statistical Convergent Sequence Spaceshttps://sciendo.com/article/10.2478/tmmp-2024-0015<abstract> <title style='display:none'>Abstract</title> <p>The aim of this paper is to introduce and investigate some neutrosophic fuzzy tribonacci ℐ-lacunary statistical convergent sequence spaces by utilizing the domain of regular tribonacci matrix <italic>A</italic> = (<italic>a<sub>jk</sub></italic>). Morever, we also put forward various algebraic and topological features of these convergent sequence spaces and establish several interesting inclusion relations.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00152024-10-20T00:00:00.000+00:00Nonlinear Elliptic Equations with Variable Exponents Anisotropic Sobolev Weights and Natural Growth Termshttps://sciendo.com/article/10.2478/tmmp-2024-0020<abstract> <title style='display:none'>Abstract</title> <p>The purpose of our paper is to prove the existence of the distributional solutions for anisotropic nonlinear elliptic equations with variable exponents, which contain lower order terms dependent on the gradient of the solution and on the solution itself. The terms are weighted, and the main results rely on the possibility of comparing the weights with each other, where the right-hand side is a sum of the natural growth term and the datum <italic>f ∈ L</italic><sup>1</sup>(Ω). Furthermore the weight function <italic>θ</italic>(·) is in <italic>W̊</italic><sup>1,<italic>p→</italic>(·) (Ω)</sup>, with <italic>θ</italic>(·) &gt; 0 and connected with the coefficient <italic>b</italic>(·) <italic>∈ L</italic><sup>1</sup>(Ω) of the lower order term.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00202024-10-01T00:00:00.000+00:00Stability and Hopf Bifurcation in a Modified Sprott Systemhttps://sciendo.com/article/10.2478/tmmp-2024-0012<abstract> <title style='display:none'>Abstract</title> <p>In this article, a Modified Sprott <italic>C</italic> system is considered. The stability of equilibrium points and the occurrence of Hopf bifurcation in the system are investigated. It has been proved that the system displays a Hopf bifurcation at <italic>α</italic> = 0. Additionally, by applying normal form theory, the stability, direction and increase (or decrease) of the period of bifurcating periodic solutions of the system are illustrated. It has been shown that the solutions of bifurcating periodic solutions at the bifurcation value <italic>α</italic> = 0 are unstable. The type of Hopf bifurcation is subcritical and the periods of the bifurcating periodic solutions increase.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00122024-07-19T00:00:00.000+00:00An Application of -Metric and Related Best Proximity Point Results Generalizing Wardowski’s Fixed Point Theoremhttps://sciendo.com/article/10.2478/tmmp-2024-0011<abstract> <title style='display:none'>Abstract</title> <p>In this article, an application of <italic>φ</italic>-metric is given via a geometrical example to show how it can help to measure distance for non-planar surfaces where the classical metric becomes incapable. We also introduce the concept of best proximity point and proximal contraction for a class of mappings in a <italic>φ</italic>-metric space and prove a best proximity point theorem for such class of contraction mappings from which the famous “Wardowski’s fixed point theorem” can be deduced as a particular case. To support our theorem, we provide an example in which Wardowski’s metric fixed point theorem cannot be applied.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00112024-06-18T00:00:00.000+00:00New Fixed Point Theorems on Complete -Metric Space by Using Rus Contraction Mappinghttps://sciendo.com/article/10.2478/tmmp-2024-0010<abstract> <title style='display:none'>Abstract</title> <p>This paper investigates a fixed point over a complete b-metric space for a family of contractive mappings. In this paper, we have discovered new results in the direction of the complete b-metric space by using Rus contraction. Furthermore, we establish a common fixed point theorem between two mappings over complete b-metric space. We also provide some non-trivial examples to display the authenticity of our established results.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00102024-05-29T00:00:00.000+00:00On a Non-Convex Lagrange Optimal Control Problemhttps://sciendo.com/article/10.2478/tmmp-2024-0008<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we are concerned with an iterative differential inclusion governed by the time-dependent maximal monotone operator with perturbation. The approach to solve our problem is based on the Yosida approximation technique. The theoretical result is applied to prove an existence result for a Lagrange optimal control problem without assumptions concerning convexity.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00082024-05-29T00:00:00.000+00:00Cryptanalysis of Strumok Cipher Initializationhttps://sciendo.com/article/10.2478/tmmp-2024-0009<abstract> <title style='display:none'>Abstract</title> <p>The initialization of the <italic>Strumok</italic> cipher is analyzed in this paper. A modified cipher initialization procedure is constructed for the <italic>Strumok-512</italic> cipher, which allows to show that the cipher initialization is not injective. Security against slide attacks is proved. Furthermore, the Guess-and-Determine attack on the <italic>Strumok-512</italic> cipher is constructed with complexity 2<sup>448</sup>.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00092024-05-29T00:00:00.000+00:00Algebraic Cryptanalysis of Ascon Using MRHS Equationshttps://sciendo.com/article/10.2478/tmmp-2024-0007<abstract> <title style='display:none'>Abstract</title> <p>Ascon is a family of lightweight authenticated encryption and hashing algorithms, which is a finalist in the NIST Lightweight Cryptography competition. We study the Ascon algorithm from the perspective of algebraic cryptanalysis based on the MRHS representation of the cipher. We call such an approach an MRHS cryptanalysis.</p> <p>We represent the system on the gate level (focusing on individual AND-gates) and the S-box level (basing MRHS equations on 5-bit S-boxes). We compare the results from the application of two custom MRHS solvers. The RZ solver is based on linear algebra and exhaustive search. The HC solver is based on adaptive bit-flipping with restarts.</p> <p>We show that both the choice of the solver and the choice of the system representation influence the total complexity of the attack. On the other hand, these choices do not change the fundamental properties of the attack, such as scaling with the amount of information the attacker possesses. A similar assessment holds for using a scaled-down version of Ascon for the experiments. Our method can be used for the experimental evaluation of cipher designs against algebraic attacks.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00072024-04-13T00:00:00.000+00:00Existence Result for a Stochastic Functional Differential System Driven by G-Brownian Motion with Infinite Delayhttps://sciendo.com/article/10.2478/tmmp-2024-0005<abstract> <title style='display:none'>Abstract</title> <p>In this article, we are interested in the study of a class of stochastic functional differential systems driven by <italic>G</italic>-Brownian motion with infinite delay. We prove the existence and uniqueness of the solutions when two basic conditions are met: the linear growth condition and the Lipschitz condition.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00052024-04-13T00:00:00.000+00:00Variational Mcshane and Pettis Integrals of Multifunctionshttps://sciendo.com/article/10.2478/tmmp-2024-0004<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we present full characterizations of variationally McShane and Pettis integrable multifunctions in terms of the cubic derivative and the variational McShane measure of additive interval multifunctions.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00042024-04-13T00:00:00.000+00:00The Nemytskiĭ Operator and Vector Measure Solutions for Non-Linear Initial Value Problemshttps://sciendo.com/article/10.2478/tmmp-2024-0006<abstract> <title style='display:none'>Abstract</title> <p>We define and study a Banach-space-valued Nemytskiĭ operator and we find vector measure solutions for associated non-linear initial value problems.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00062024-04-13T00:00:00.000+00:00Some Alternative Interpretations of Strongly Star Semi-Rothberger and Related Spaceshttps://sciendo.com/article/10.2478/tmmp-2024-0003<abstract> <title style='display:none'>Abstract</title> <p>In this article, some under-appreciated characteristics of strongly star-Rothberger subsets are explored. Additionally, with the aid of the SSI property and MSSI property, semi-Rothberger and star semi-Rothberger spaces are represented by families of closed sets. Finally, several selection principle like attributes are produced that can reflect the previously mentioned sequential covering features in an inverted form.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/tmmp-2024-00032024-04-13T00:00:00.000+00:00en-us-1