Finite Field Research Articles

Sum representations of Appell–Lauricella functions over finite fields using confluent hypergeometric functions and their applications

Sum representations of Appell–Lauricella functions over finite fields using confluent hypergeometric functions and their applications

Magnetization without spin: Effective Lagrangian of itinerant electrons

In this paper, an effective Lagrangian of an itinerant electron system of finite density at a finite magnetic field is obtained. It includes a Chern-Simons term of electromagnetic potentials of lower-scale dimensions compared to those studied before. This term has an origin in the many-body wave function and a unique topological property that is independent of a spin degree of freedom. The coupling strength is proportional to ρeB, which is singular at B=0 for a constant charge density. The effective Lagrangian at a finite B represents the physical effects at B≠0 properly. A universal shift of the magnetic field known as the Slater-Pauling curve is obtained from the effective Lagrangian.

On the construction of certain odd degree irreducible polynomials over finite fields

On the construction of certain odd degree irreducible polynomials over finite fields

A coupled crystal inelasticity-phase field model for crack growth in polycrystalline nitinol microstructures

Abstract This paper introduces a comprehensive computational framework, comprising a finite deformation crystal inelasticity constitutive model and phase field model, for modeling crack growth in superelastic nitinol polycrystalline microstructures. The crystal inelasticity model represents crystal stretching and lattice rotation from elastic mechanisms, as well as local inelastic deformation due to austenite-martensite phase transformation. The phase field formulation decomposes the Helmholtz free energy density into stored elastic energy, phase transformation energy, and crack surface energy components. The elastic energy accounts for tension-compression asymmetry with the formation of the crack through a spectral decomposition. Kinetic Monte Carlo simulations generate equilibrium area fractions of different surface orientations, which serve as weights for the surface energy. An adaptive wavelet-enhanced hierarchical finite element (FE) model is introduced to alleviate high computational overhead in phase field crack simulations. Simulations with the coupled inelasticity phase field model are conducted under various loading conditions including Mode-I tension, a quasi-static Kalthoff experiment, and cyclic loading of polycrystalline microstructures. Crack propagation is effectively predicted by this model, providing valuable insights into the material mechanical behavior with growing cracks.

On Some Permutation Trinomials in Characteristic Three

In this paper, we determine the permutation properties of the polynomial x^3+x^q+2−x^4q−1 over the finite field Fq2 in characteristic three. Moreover, we consider the trinomials of the form x^4q−1 +x^2q+1 ±x^3. In particular, we first show that x^3 +x^q+2 −x^4q−1 permutes F_q^2 with q = 3^m if and only if m is odd. This enables us to show that the sufficient condition in [35, Theorem 4] is also necessary. Next, we prove that x^4q−1 + x^2q+1 − x^3 permutes F_q^2 with q = 3^m if and only if m is congruent to 0 modulo 4. Consequently, we prove that the sufficient condition in [20, Theorem 3.2] is also necessary. Finally, we investigate the trinomial x^4q−1 +x^2q+1 +x^3 and show that it is never a permutation polynomial of F_q^2 in any characteristic. All the polynomials considered in this work are not quasi-multiplicative equivalent to any known class of permutation trinomials.

Homotopy properties of the complex of frames of a unitary space

AbstractLet be a finite‐dimensional vector space equipped with a nondegenerate Hermitian form over a field . Let be the graph with vertex set the one‐dimensional nondegenerate subspaces of and adjacency relation given by orthogonality. We give a complete description of when is connected in terms of the dimension of and the size of the ground field . Furthermore, we prove that if , then the clique complex of is simply connected. For finite fields , we also compute the eigenvalues of the adjacency matrix of . Then, by Garland's method, we conclude that for all , where is a field of characteristic 0, provided that . Under these assumptions, we deduce that the barycentric subdivision of deformation retracts to the order complex of the certain rank selection of that is Cohen–Macaulay over . Finally, we apply our results to the Quillen poset of elementary abelian ‐subgroups of a finite group and to the study of geometric properties of the poset of nondegenerate subspaces of and the poset of orthogonal decompositions of .

A different base approach for better efficiency on range proofs

Zero-knowledge range proofs (ZKRPs) are commonly used to prove the validation of a secret integer lies in an interval to some other party in a secret way. In many ZKRPs, the secret is represented in binary and then committed via a suitable commitment scheme or represented as an appropriate encryption scheme. This paper is an extended version of the conference paper presented at the 14th IEEE International Conference on Security of Information and Networks. To this end, after summarizing the conference paper, we first analyze the proof proposed by Mao in 1998 in the elliptic-curve setting. Mao’s proof contains a bit commitment scheme with an OR construction as a sub-protocol. We have extended Mao’s range proof to base-u with a modified OR-proof. We investigate and compare the efficiency of different base approaches on Mao’s range proof with both Pedersen commitment and ElGamal encryption. Later, we analyze the range proof proposed by Bootle et al. in both finite fields and elliptic-curve settings. This proof contains polynomial commitment with matrix row operations. We take the number of computations in modulo exponentiation and the cost of the number of exchanged integers between parties. Then, we generalize these costs for u-based construction. We show that compared with the base-2 representation, different base approach provides efficiency in communication cost or computation cost, or both.

The Structure of the Unit Group of $\mathbb{F}_{3^{k}}(C_{3}\times D_{2n})$ and Finite Group Algebras of Groups of Order $42$

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p>0$ with $|\mathbb{F}_{q}|=q=p^{k}$ and $\mathcal{U}(\mathbb{F}_{q}G)$ be the unit group of the group algebra $\mathbb{F}_{q}G$ for some group $G$. There are $6$ groups of order $42$ up to isomorphism. In this paper, we provide a characterization of $\mathcal{U}(\mathbb{F}_{3^{k}}(C_{3}\times D_{2n}))$ and establish the structures of the unit groups of some finite group algebras of groups of order $42$.

The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II

Let E and E′ be 2-isogenous elliptic curves over Q. Following [6], we call a prime of good reduction p anomalous if E(Fp)≃E′(Fp) but E(Fp2)≄E′(Fp2). Our main result is an explicit formula for the proportion of anomalous primes for any such pair of elliptic curves. We consider both the CM case and the non-CM case.

Metrical properties of beta-transformations on power series rings

Let A ∞ be an integer ring of a formal Laurent series field in one variable 1 t over a finite field of order q, with a maximal ideal M ∞ . Then, we introduce beta-transformations U β on A ∞ that are topologically isomorphic and measurably isomorphic to the well-known beta-transformation T β on M ∞ . We prove various metrical properties of U β such as (total) ergodicity and mixing of any order through explicit representations of both U β and its nth iterate U β n with respect to the shift operators. Furthermore, we examine dynamical connections between the measure-preserving property of 1-Lipschitz functions and the locally scaling property of ( q − k , q k ) -Lipschitz functions, in terms of the coefficients of van der Put and Carlitz–Wagner.

Involutions of finite abelian groups with explicit constructions on finite fields

Involutions of finite abelian groups with explicit constructions on finite fields

Effect of Coriolis force on electrical conductivity tensor for the rotating hadron resonance gas

We have investigated the influence of the Coriolis force on the electrical conductivity of hadronic matter formed in relativistic nuclear collisions, employing the hadron resonance gas model. A rotating matter in the peripheral heavy-ion collisions can be expected from the initial stage of quark matter to late-stage hadronic matter. Present work is focused on rotating hadronic matter, whose medium constituents—hadron resonances—can face a nonzero Coriolis force, which can influence the hadronic flow or conductivity. We estimate this conductivity tensor by using the relativistic Boltzmann transport equation. In the absence of Coriolis force, an isotropic conductivity tensor for hadronic matter is expected. However, our study finds that the presence of Coriolis force can generate an anisotropic conductivity tensor with three main conductivity components—parallel, perpendicular, and Hall—similarly to the effect of Lorentz force at a finite magnetic field. Our study has indicated that a noticeable anisotropy of conductivity tensor can be found within the phenomenological range of angular velocity Ω=0.001–0.02 GeV and hadronic scattering radius a=0.2–2 fm. Published by the American Physical Society 2024

Novel Weakness Multivariate Quadratic Structures Detected within Macaulay Matrix

The security of a Multivariate Public-Key Cryptosystem (MPKC) is based on the hard mathematical problem of solving Multivariate Quadratic (MQ) equations over finite fields, also known as the MQ problem. An MPKC has the potential to be a post-quantum cryptosystem. In this paper, we identify new weaknesses in the Macaulay matrix identified via Wang's technique, which was initially designed for solving multivariate quadratic equation systems. This new weakness occurs in the case of random coefficients in any column vector for different variables of monomials and random coefficients are assigned to other monomials. The weakness is exposed through the use of Gaussian elimination to obtain a univariate equation. We illustrate our findings using a random example.

New upper bounds on the number of non-zero weights of constacyclic codes

For any simple-root constacyclic code C over a finite field Fq, as far as we know, the group G generated by the multiplier, the constacyclic shift and the scalar multiplications is the largest subgroup of the automorphism group Aut(C) of C. In this paper, by calculating the number of G-orbits of C﹨{0}, we give an explicit upper bound on the number of non-zero weights of C and present a necessary and sufficient condition for C to meet the upper bound. Some examples in this paper show that our upper bound is tight and better than the upper bounds in Zhang and Cao (2024) [26]. In particular, our main results provide a new method to construct few-weight constacyclic codes. Furthermore, for the constacyclic code C belonging to two special types, we obtain a smaller upper bound on the number of non-zero weights of C by substituting G with a larger subgroup of Aut(C). The results derived in this paper generalize the main results in Chen et al. (2024) [9].

On 2-superirreducible polynomials over finite fields

We investigate k-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most k. Let F be a finite field of characteristic p. We show that no 2-superirreducible polynomials exist in F[t] when p=2 and that no such polynomials of odd degree exist when p is odd. We address the remaining case in which p is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree d. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity.

Separable Polynomials and Separable Extensions

Summary We continue the formalization of field theory in Mizar [2], [3], [4]. We introduce separability of polynomials and field extensions: a polynomial is separable, if it has no multiple roots in its splitting field; an algebraic extension E of F is separable, if the minimal polynomial of each a ∈ E is separable. We prove among others that a polynomial q(X) is separable if and only if the gcd of q(X) and its (formal) derivation equals 1 – and that a irreducible polynomial q(X) is separable if and only if its derivation is not 0 – and that q(X) is separable if and only if the number of q(X)’s roots in some field extension equals the degree of q(X). A field F is called perfect if all irreducible polynomials over F are separable, and as a consequence every algebraic extension of F is separable. Every field with characteristic 0 is perfect [13]. To also consider separability in fields with prime characteristic p we define the rings Rp = { ap | a ∈ R} and the polynomials Xn − a for a ∈ R. Then we show that a field F with prime characteristic p is separable if and only if F = F p and that finite fields are perfect. Finally we prove that for fields F ⊆ K ⊆ E where E is a separable extension of F both E is separable over K and K is separable over F .

Polynomial calculus for optimization

MaxSAT is the problem of finding an assignment satisfying the maximum number of clauses in a CNF formula. We consider a natural generalization of this problem to generic sets of polynomials and propose a weighted version of Polynomial Calculus to address this problem.Weighted Polynomial Calculus is a natural generalization of the systems MaxSAT-Resolution and weighted Resolution. Unlike such systems, weighted Polynomial Calculus manipulates polynomials with coefficients in a finite field and either weights in N or Z. We show the soundness and completeness of weighted Polynomial Calculus via an algorithmic procedure.Weighted Polynomial Calculus, with weights in N and coefficients in F2, is able to prove efficiently that Tseitin formulas on a connected graph are minimally unsatisfiable. Using weights in Z, it also proves efficiently that the Pigeonhole Principle is minimally unsatisfiable.

Research on key technologies of finite element flow field in airflow drying process analysis

Research on key technologies of finite element flow field in airflow drying process analysis

A phase field method for predicting hydrogen-induced cracking on pipelines

An accurate determination of the threshold conditions to initiate cracks on aged hydrogen pipelines is paramount for ensuring energy transport safety. In this work, a finite element-based phase field method was developed to assess the crack initiation on dented pipelines while considering the hydrogen (H) impact. Theoretical and multi-physics numerical formulas were derived for prediction of the elastic-plastic fracture behavior of H-contained steel. A critical phase field parameter, ϕ=0.69, is defined for predicting crack initiation at the dent on pipelines. The presence of H within the steel decreases the threshold dent depth for initiating H-induced cracks. When the initial H concentration increases from 0 to 0.5 wppm, the maximum dent depth for crack initiation reduces from 17.5 mm to 10.7 mm. The maximum dent depth required for crack initiation reduces from 17.5 mm to 7.8 mm when an internal pressure of 8 MPa is applied on the steel pipe. The site with the maximum phase field parameter changes during indentation, implying that the location initiating cracks depends on the dent dimension. The existing criteria in ASME B31.12 standard are not applicable for predicting H-induced crack initiation on dented pipelines. This study proposes a new method to predict hydrogen-induced cracking on aged pipelines when transporting hydrogen.

Detecting arrays for effects of multiple interacting factors

Detecting arrays provide test suites for complex engineered systems in which many factors interact. The determination of which interactions have a significant impact on system behaviour requires not only that each interaction appear in a test, but also that its effect can be distinguished from those of other significant interactions. In this paper, compact representations of detecting arrays using vectors over the finite field are developed. Covering strong separating hash families exploit linear independence over the field, while the weaker elongated covering perfect hash families permit some linear dependence. For both, probabilistic analyses are employed to establish effective upper bounds on the number of tests needed in a detecting array for a wide variety of parameters. The analyses underlie efficient algorithms for the explicit construction of detecting arrays.