Finite Field Extension Research Articles

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.

An Improvement for Error-Correcting Pairs of Some Special MDS Codes

The error-correcting pair is a general algebraic decoding method for linear codes. Since every linear code is contained in an MDS linear code with the same minimum distance over some finite field extensions, we focus on MDS linear codes. Recently, He and Liao showed that for an MDS linear code [Formula: see text] with minimum distance [Formula: see text], if it has an [Formula: see text]-error-correcting pair, then the parameters of the pair have three possibilities. Moreover, for the first case, they gave a necessary condition for an MDS linear code [Formula: see text] with minimum distance [Formula: see text] to have an [Formula: see text]-error-correcting pair, and for the other two cases, they only gave some counterexamples. For the second case, in this paper, we give a necessary condition for an MDS linear code [Formula: see text] with minimum distance [Formula: see text] to have an [Formula: see text]-error-correcting pair, and then basing on the Product Singleton Bound, we prove that there are two cases for such pairs, and then give some counterexamples basing on twisted generalized Reed–Solomon codes for these cases.

On the number of k-normal elements and [formula omitted]-practical numbers

A normal element in a finite field extension Fqn/Fq is characterized by having linearly independent conjugates over Fq. We consider the generalization of normal elements known as k-normal elements, where a subset of the conjugates are required to be linearly independent. In this paper, we provide an explicit combinatorial formula for counting the number of k-normal elements in a finite field extension motivated by an open problem proposed by Huczynska, Mullen, Panario, and Thomson in 2013. Furthermore, we use these results to establish new insights about Fq-practical numbers.

Permutation rational functions over quadratic extensions of finite fields

Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over Fq2, whose numerators are so-called q-quadratic polynomials. To this end, we will first determine the exact number of zeros of a special q-quadratic polynomial in Fq2, by calculating character sums related to quadratic forms of Fq2/Fq. Then given some rational function, we can demonstrate whether it induces a permutation of Fq2.

Spatial diffusion of heavy quarks in a background magnetic field

The ratio of shear viscosity to entropy density shows a valley-shaped pattern well-known in the community of heavy-ion physics. Diffusion coefficients of heavy quark and meson show a similar structure, and both sketches have become quite popular in the community. Present work has attempted a finite magnetic field extension of the diffusion coefficients of heavy quark and meson. Using Einstein's diffusion relation, we calculated heavy quark and heavy meson diffusion by the ratio of conductivity to susceptibility in the kinetic theory framework of relaxation time approximation. The relaxation time of heavy quark and meson are tuned from the knowledge of earlier works on spatial diffusion estimations, and then we have extended the framework for a finite magnetic field, where our outcomes have revealed two aspects—anisotropic and quantum aspects of diffusion with future possibilities of phenomenological signature. Published by the American Physical Society 2024

Algorithmic aspects of elliptic bases in finite field discrete logarithm algorithms

Elliptic bases, introduced by Couveignes and Lercier in 2009, give an elegant way of representing finite field extensions. A natural question which seems to have been considered independently by several groups is to use this representation as a starting point for discrete logarithm algorithms in small characteristic finite fields.This idea has been recently proposed by two groups working on it, in order to achieve provable quasi-polynomial time for discrete logarithms in small characteristic finite fields.In this paper, we do not try to achieve a provable algorithm but, instead, investigate the practicality of heuristic algorithms based on elliptic bases. Our key idea is to use a different model of the elliptic curve used for the elliptic basis that allows for a relatively simple adaptation of the techniques used with former Frobenius representation algorithms.We have not performed any record computation with this new method but our experiments with the field $ {\mathbb F}_{3^{1345}} $ indicate that switching to elliptic representations might be possible with performances comparable to the current best practical methods.

Bounding the Pythagoras number of a field by 2n + 1

Given a positive integer n, a sufficient condition on a field is given for bounding its Pythagoras number by 2n+1. The condition is satisfied for n=1 by function fields of curves over iterated formal power series fields over R, as well as by finite field extensions of R((t0,t1)). In both cases, one retrieves the upper bound 3 on the Pythagoras number. The new method presented here might help to establish more generally 2n+1 as an upper bound for the Pythagoras number of function fields of curves over R((t1,…,tn)) and for finite field extensions of R((t0,…,tn)).

The Ostrowski quotient of an elliptic curve

For [Formula: see text] a finite Galois extension of number fields, the relative Pólya group [Formula: see text] is the subgroup of the ideal class group of [Formula: see text] generated by all the strongly ambiguous ideal classes in [Formula: see text]. The notion of Ostrowski quotient [Formula: see text], as the cokernel of the capitulation map into [Formula: see text], has been recently introduced in [E. Shahoseini, A. Rajaei and A. Maarefparvar, Ostrowski quotients for finite extensions of number fields, Pacific J. Math. 321 (2022) 415–429, doi: 10.2140/pjm.2022.321.415]. In this paper, using some results of [C. D. González-Avilés, Capitulation, ambiguous classes and the cohomology of the units, J. Reine Angew. Math. 613 (2007) 75–97], we find a new approach to define [Formula: see text] and [Formula: see text] which is the main motivation for us to investigate analogous notions in the elliptic curve setting. For [Formula: see text] an elliptic curve defined over [Formula: see text], we define the Ostrowski quotient [Formula: see text] and the coarse Ostrowski quotient [Formula: see text] of [Formula: see text] relative to [Formula: see text], for which in the latter group we do not take into account primes of bad reduction. Our main result is a nontrivial structure theorem for the group [Formula: see text] and we analyze this theorem, in some details, for the class of curves [Formula: see text] over quadratic extensions [Formula: see text].

The subfield metric and its application to quantum error correction

We introduce a new weight and corresponding metric over finite extension fields for asymmetric error correction. The weight distinguishes between elements from the base field and the ones outside of it, which is motivated by asymmetric quantum codes. We set up the theoretic framework for this weight and metric, including upper and lower bounds, asymptotic behavior of random codes, and we show the existence of an optimal family of codes achieving the Singleton-type upper bound.

Character sums over elements of extensions of finite fields with restricted coordinates

We obtain nontrivial bounds for character sums with multiplicative and additive characters over finite fields over elements with restricted coordinate expansion. In particular, we obtain a nontrivial estimate for such a sum over a finite field analogue of the Cantor set.

Skew bracoids

Skew braces are intensively studied owing to their wide ranging connections and applications. We generalize the definition of a skew brace to give a new algebraic object, which we term a skew bracoid. Our construction involves two groups interacting in a manner analogous to the compatibility condition found in the definition of a skew brace. We formulate tools for characterizing and classifying skew bracoids, and study substructures, quotients, homomorphisms, and isomorphisms. As a first application, we prove that finite skew bracoids correspond with Hopf-Galois structures on finite separable extensions of fields, generalizing the existing connection between finite skew braces and Hopf-Galois structures on finite Galois extensions.

Chabauty limits of groups of involutions In SL(2,F) for local fields

We classify Chabauty limits of groups fixed by various (abstract) involutions over SL ( 2 , F ) , where F is a finite field-extension of Q p , with p ≠ 2 . To do so, we first classify abstract involutions over SL ( 2 , F ) with F a quadratic extension of Q p , and prove p-adic polar decompositions with respect to various subgroups of p-adic SL 2 . Then we classify Chabauty limits of: SL ( 2 , F ) ⊂ SL ( 2 , E ) where E is a quadratic extension of F, of SL ( 2 , R ) ⊂ SL ( 2 , C ) , and of H θ ⊂ SL ( 2 , F ) , where H θ is the fixed point group of an F-involution θ over SL ( 2 , F ) .

A Brief Introduction to Splitting Of Primes over Number Fields

The study of Dedekind Zeta Functions over a number field extension uses different aspects of both Algebraic and Analytic Number Theory. In this paper, we shall learn about the structure and different analytic aspects of such functions, namely the domain of its convregence and analyticity at different points of ℂ when the function is defined over any finite field extension K over ℚ . Moreover, given any two Number Fields L and K over ℚ with L being Normal over K, our intention is to classify and study the primes in K which split completely in L. Also, we shall explore some special cases related to this result

Certain diagonal equations and conflict-avoiding codes of prime lengths

We study the construction of optimal conflict-avoiding codes (CAC) from a number theoretical point of view. The determination of the size of optimal CAC of prime length p and weight 3 is formulated in terms of the solvability of certain twisted Fermat equations of the form g2Xℓ+gYℓ+1=0 over the finite field Fp for some primitive root g modulo p. We treat the problem of solving the twisted Fermat equations in a more general situation by allowing the base field to be any finite extension field Fq of Fp. We show that for q greater than a lower bound of the order of magnitude O(ℓ2) there exists a generator g of Fq× such that the equation in question is solvable over Fq. Using our results we are able to contribute new results to the construction of optimal CAC of prime lengths and weight 3.

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

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

On the error-correcting pair for MDS linear codes with even minimum distance

The error-correcting pair is a general algebraic decoding method for linear codes, which exists for many classical linear codes. Since every linear code is contained in an MDS linear code with the same minimum distance over some finite field extension, we focus our study on MDS linear codes. It is well-known that an MDS linear code with minimum distance 2ℓ+1 has an ℓ-error-correcting pair if and only if it is a generalized Reed-Solomon code. In this paper, we show that for an MDS linear code C with minimal distance 2ℓ+2, if it has an ℓ-error-correcting pair, then the parameters of the pair are three cases. For one case, we give a necessary condition that C is a generalized Reed-Solomon code, and then give some counterexamples that C is a non-generalized Reed-Solomon code for the other two cases.

A Hilbert irreducibility theorem for Enriques surfaces

We define the over-exceptional lattice of a minimal algebraic surface of Kodaira dimension 0 0 . Bounding the rank of this object, we prove that a conjecture by Campana [Ann. Inst. Fourier (Grenoble) 54 (2004), pp. 499–630] and Corvaja–Zannier [Math. Z. 286 (2017), pp. 579–602] holds for Enriques surfaces, as well as K3 surfaces of Picard rank ≥ 6 \geq 6 apart from a finite list of geometric Picard lattices. Concretely, we prove that such surfaces over finitely generated fields of characteristic 0 0 satisfy the weak Hilbert Property after a finite field extension of the base field. The degree of the field extension can be uniformly bounded.

On the functional graph of f(X)=c(Xq+1 + aX2) over quadratic extensions of finite fields

Let Fq be the finite field with q elements and char(Fq) odd. In this article we will describe completely the dynamics of the map f(X)=c(Xq+1+aX2) over the finite field Fq2, for a∈{±1} and c∈Fq⁎. We will also give some partial results for a∈Fq⁎∖{±1}.

The equivariant complexity of multiplication in finite field extensions

We study the complexity of multiplication of two elements in a finite field extension given by their coordinates in a normal basis. We show how to control this complexity using the arithmetic and geometry of algebraic curves.

The complexity of solving Weil restriction systems

The solving degree of a system of multivariate polynomial equations provides an upper bound for the complexity of computing the solutions of the system via Gröbner basis methods. In this paper, we consider polynomial systems that are obtained via Weil restriction of scalars. The latter is an arithmetic construction which, given a finite Galois field extension k↪K, associates to a system F defined over K a system Weil(F) defined over k, in such a way that the solutions of F over K and those of Weil(F) over k are in natural bijection. In this paper, we find upper bounds for the complexity of solving a polynomial system Weil(F) obtained via Weil restriction in terms of algebraic invariants of the system F.