A New Estimation for Eigenvalues of Matrix Power Functions

The semigroup associated with the Dirichlet heat kernel is said to be intrinsic ultracontractive if (i) the Dirichlet realization of the associated self adjoint operator has the first positive eigenvalue with positive L2 eigenfunction; (ii) the heat kernel is bounded above and below by the product of the eigenfunctions with positive multiplicative constants depending on time. We give an upper and lower estimate of the first eigenvalue in terms of capacitary width, which yields a sharp sufficient condition for (i). Our parabolic argument also yields an exponential decay property of a certain caloric measure. Then, the caloric measure is controlled by the elliptic Green function with the aid of a parabolic box argument. This is a crucial step for (ii). We give a sharp sufficient integral condition for intrinsic ultracontractivity in terms of capacitary width. A similar integral condition for the boundary Harnack principle is also obtained. Under geometric specifications, these integral conditions generalize known results and give more precise conditions. Sharpness is examined by an infinite funnel, for which we obtain a complete characterization of intrinsic ultracontractivity. Our method is purely analytic and elementary; it enables us to dispense with logarithmic Sobolev inequalities.

In this paper, we mainly consider the existence of homoclinic orbits for the following second-order Hamiltonian systems $$\begin{aligned} \ddot{u}(t)-L(t)u(t)+\nabla W\bigl (t,u(t)\bigr )=f(t), \end{aligned}$$where L(t) is a positive definite and symmetric matrix for all $$t\in \mathbb {R}$$ and the potential function W(t, u) is locally subquadratic. Here, the coefficient of the upper bound for W is a positive constant, whereas in the previous literature the corresponding coefficient need to be some integrable functions a(t) on $$\mathbb {R}$$.

Computer Solution of Large Linear Systems

Recent works show conflicting results: network capacity may increase or decrease with higher transmission power under different scenarios. In this work, we want to understand this paradox. Specifically, we address the following questions: (1)Theoretically, should we increase or decrease transmission power to maximize network capacity? (2) Theoretically, how much network capacity gain can we achieve by power control? (3) Under realistic situations, how do power control, link scheduling and routing interact with each other? Under which scenarios can we expect a large capacity gain by using higher transmission power? To answer these questions, firstly, we prove that the optimal network capacity is a non-decreasing function of transmission power. Secondly, we prove that the optimal network capacity can be increased unlimitedly by higher transmission power in some network configurations. However, when nodes are distributed uniformly, the gain of optimal network capacity by higher transmission power is upper-bounded by a positive constant. Thirdly, we discuss why network capacity in practice may increase or decrease with higher transmission power under different scenarios using carrier sensing and the minimum hop-count routing. Extensive simulations are carried out to verify our analysis.

In one-dimensional unbounded domains, we prove the global existence of strong solutions to the compressible Navier–Stokes system for a viscous radiative gas, when the viscosity μ is a constant and the heat conductivity κ is a power function of the temperature θ according to κ(θ)=κ̃θβ, with β ≥ 0 and κ̃>0. Our result generalizes Zhao and Liao’s result [Y. K. Liao and H. J. Zhao, J. Differ. Equations 265, 2076–2120 (2018)] to the degenerate and nonlinear heat conductivity. In particular, the constant coefficients’ case (μ and κ are positive constants) is also covered in our theorem.

Free semidefinite representation of matrix power functions

An Expectation Formula for the Multivariate Dirichlet Distribution

Shor’s quantum algorithm establishes a polynomial time attack on the discrete logarithm problem in any group. Recent announcements of progress in building quantum computers highlight the need for new concepts to create cryptosystems that are resistant to quantum attacks. In this paper, we present а new message encryption scheme. To enhance the security of the scheme, we suggest double key-exchange protocol (KEP). The first stage of the key exchange uses a matrix power function (MPF) in a tropical semiring. These functions are based on the action of a matrix semiring acting on some matrix set. MPFs can be considered as one-way functions because they are based on some generalized satisfiability problems that are potentially NP-complete. The obtained shared secret key at the first stage of the key exchange serves as an input for the second stage. The security of the second phase relies on the difficulty of the semiring action problem. In our protocol, we suggest using left or right action of the tropical semiring (which can be both min-plus and max-plus) on the group of commutative matrices (circulant matrices, in our case). The fact that the key-exchange protocol works in two phases contributes to its security, since an attacker needs to solve two difficult problems in order to break it. The main advantages of the presented protocol are: the increased efficiency and improved security.

Since the launch of the post-quantum cryptography standardization project by NIST, post-quantum cryptography has become a prominent research area. Non-commutative cryptography constructed using NP-complete problems is widely regarded resistant to quantum computing attacks, so it has become an important branch of post-quantum cryptography. Nonetheless, the existing non-commutative cryptographic protocol still exhibits certain shortcomings. In this paper, a non-commutative combinatorial semigroup with matrix power function is constructed from a modified medial semigroup by the method of semidirect product, and then a key exchange protocol is developed on it to provide a kind of novel non-commutative cryptographic protocol. Due to the non-commutativity of cryptographic platform, the proposed protocol could perform well in respect of antiquantum computing attack, which is superior to traditional cryptographic protocol. In addition, the security analysis shows that the protocol has significant advantages in resisting algebraic and brute force attacks, as well as aganist quantum cryptanalysis; the complexity analysis demonstrates that computation and storage complexities are of polynomial order, ensuring efficient operation even for large matrix sizes.

This paper is devoted to the numerical computation of algebraic linear systems involving several matrix power functions; that is finding x solution to \(\sum _{\alpha \in \mathbb {R}}A^{\alpha }x=b\). These systems will be referred to as Generalized Fractional Algebraic Linear Systems (GFALS). In this paper, we derive several gradient methods for solving these very computationally complex problems, which themselves require the solution to intermiediate standard Fractional Algebraic Linear Systems (FALS) \(A^{\alpha }x=b\), with \(\alpha \in \mathbb {R_+}\). The latter usually require the solution to many classical linear systems \(Ax=b\). We also show that in some cases, the solution to a GFALS problem can be obtained as the solution to a first-order hyperbolic system of conservation laws. We also discuss the connections between this PDE-approach and gradient-type methods. The convergence analysis is addressed and some numerical experiments are proposed to illustrate and compare the methods which are proposed in this paper.

This review explores the current state of public key cryptography based on idempotent semirings, with an emphasis on tropical semirings. It examines key hard problems, such as the tropical discrete logarithm problem, semidirect tropical product problem, the factorization of tropical polynomials, and the matrix power function, that underpin the security of these protocols. Given the significant number of compromised protocols based on tropical semirings, most of which are variations of the Stickel protocol, we present three algorithms and classify schemes of this type. The analysis is further illustrated with a figure that maps the relationships between tropical Stickel’s-like protocols and the attacks targeting them. Additionally, the review provides an in-depth exploration of the vulnerabilities that have led to many tropical semiring-based cryptosystems being compromised. To address these challenges, the review highlights promising alternative approaches, including non-tropical idempotent platforms and non-idempotent options, such as supertropical semirings, which offer potential solutions to overcome known limitations. Furthermore, a discussion on the interplay between tropical cryptography and post-quantum cryptography is presented, raising the following question: what is the future of tropical cryptography?

New NP-complete problem, named as multivariate quadratic power (MQP) problem, is presented. It is based on solution of multivariate quadratic power system of equations over the semigroup Zn, denoted by MQP(Zn), where n is positive integer. Two sequential polynomial-time reductions from known NP-complete multivariate quadratic (MQ) problem over the field Z2, i.e. MQ(Z2) to MQP(Zn) are constructed. It is proved that certain restricted MQP(Zn) problem over some subgroup of Zn is equivalent to MQ(Z2) problem. This allow us to prove that MQP(Zn) is NP-complete also.MQP problem is linked to some author’s previously declared matrix power function (MPF) used for several cryptographic protocols construction. Obtained NP-complete problem will be used to create new candidate one-way function (OWF) based on MPF for new cryptographic primitives’ construction.DOI: http://dx.doi.org/10.5755/j01.itc.41.1.821

This paper continues our research of the so-called matrix power function and its application to cryptography. We consider the simplest case of the improved matrix power asymmetric cypher (MPAC). We show that our protocol is resistant to linear algebra attack, which can be applied to the initial version of MPAC to break it in polynomial time. Our main goal is to present the general idea for the choice of public parameters of MPAC protocol to avoid this attack while also avoiding exhaustive search attack.

Post-quantum cryptography is essential for securing digital communications against threats posed by quantum computers. Researchers have focused on developing algorithms that can withstand attacks from both classical and quantum computers, thereby ensuring the security of data transmissions over public networks. A critical component of this security is the key agreement protocol, which allows two parties to establish a shared secret key over an insecure channel. This paper introduces two novel post-quantum key agreement protocols that can be easily implemented on standard computers using rectangular or rank-deficient matrices, exploiting the generalizations of the matrix power function, which is a generator of NP-hard problems. We provide basic concepts and proofs, pseudocodes, and examples, along with a discussion of complexity.

Dynamic crack propagation simulation with scaled boundary polygon elements and automatic remeshing technique