Extensions Of Quadratic Fields Research Articles

Paving the Way for SQIsign: Toward Efficient Deployment on 32-bit Embedded Devices

The threat of quantum computing has spurred research into post-quantum cryptography. SQIsign, a candidate submitted to the standardization process of the National Institute of Standards and Technology, is emerging as a promising isogeny-based signature scheme. This work aimed to enhance SQIsign’s practical deployment by optimizing its low-level arithmetic operations. Through hierarchical decomposition and performance profiling, we identified the ideal-to-isogeny translation, primarily involving elliptic curve operations, as the main bottleneck. We developed efficient 32-bit finite field arithmetic for elliptic curves, such as basic operations, like addition with carry, subtraction with borrow, and conditional move. We then implemented arithmetic operations in the Montgomery domain, and extended these to quadratic field extensions. Our implementation offers improved compatibility with 32-bit architectures and enables more fine-grained SIMD acceleration. Performance evaluations demonstrated the practicality in low-level operations. Our work has potential in easing the development of SQIsign in practice, making SQIsign more efficient and practical for real-world post-quantum cryptographic applications.

Regular complete permutation polynomials over quadratic extension fields

Regular complete permutation polynomials over quadratic extension fields

The Pentagon Theorem in Miquelian Möbius Planes

We give an algebraic proof of the Pentagon Theorem. The proof works in all Miquelian Möbius planes obtained from a separable quadratic field extension. In particular, the theorem holds in every finite Miquelian plane. The arguments also reveal that the five concyclic points in the Pentagon Theorem are either pairwise distinct or identical to one single point. In addition we identify five additional quintuples of points in the pentagon configuration which are concyclic.

Structure of 2-class groups in the $${\mathbb {Z}}_{2}$$-extensions of certain real quadratic fields

Structure of 2-class groups in the $${\mathbb {Z}}_{2}$$-extensions of certain real quadratic fields

The Hidden Twin of Morley’s Five Circles Theorem

We give an algebraic proof of a slightly extended version of Morley’s Five Circles Theorem. The theorem holds in all Miquelian Möbius planes obtained from a separable quadratic field extension, in particular in the classical real Möbius plane. Moreover, the calculations bring to light a hidden twin of the Five Circles Theorem that seems to have been overlooked until now.

Optimal spinor selectivity for quaternion orders

Let D be a quaternion algebra over a number field F, and G be an arbitrary genus of OF-orders of full rank in D. Let K be a quadratic field extension of F that embeds into D, and B be an OF-order in K that can be optimally embedded into some member of G. We provide a necessary and sufficient condition for B to be optimally spinor selective for the genus G, which generalizes previous existing optimal selectivity criterions for Eichler orders as given by Arenas, Arenas-Carmona and Contreras, and by Voight independently. This allows us to obtain a refinement of the classical trace formula for optimal embeddings, which will be called the spinor trace formula. When G is a genus of Eichler orders, we extend Maclachlan's relative conductor formula for optimal selectivity from Eichler orders of square-free levels to all Eichler orders.

Efficient Construction of CGL Hash Function Using Legendre Curves

The CGL hash function is a provably secure hash function using walks on isogeny graphs of supersingular elliptic curves. A dominant cost of its computation comes from iterative computations of power roots over quadratic extension fields. In this paper, we reduce the necessary number of power root computations by almost half, by applying and also extending an existing method of efficient isogeny sequence computation on Legendre curves (Hashimoto and Nuida, CASC 2021). We also point out some relationship between 2-isogenies for Legendre curves and those for Edwards curves, which is of independent interests, and develop a method of efficient computation for 2e-th roots in quadratic extension fields.

The equivariant Tamagawa number conjecture for abelian extensions of imaginary quadratic fields

We prove the Iwasawa-theoretic version of a conjecture of Mazur–Rubin and Sano in the case of elliptic units. This allows us to derive the p -part of the equivariant Tamagawa number conjecture at s = 0 for abelian extensions of imaginary quadratic fields in the semi-simple case and, provided that a standard \mu -vanishing hypothesis is satisfied, also in the general case.

Monogenity in totally real extensions of imaginary quadratic fields with an application to simplest quartic fields

We describe an efficient algorithm to calculate generators of power integral bases in composites of totally real fields and imaginary quadratic fields with coprime discriminants. We show that the calculation can be reduced to solving index form equations in the original totally real fields. We illustrate our method by investigating monogenity in the infinite parametric family of imaginary quadratic extensions of the simplest quartic fields.

On In-forms over field extensions

For various types of field extensions [Formula: see text] and values of [Formula: see text] we consider quadratic forms [Formula: see text] that lie in the [Formula: see text]th power [Formula: see text] of the fundamental ideal of [Formula: see text] but are already defined over [Formula: see text]. We then search for some [Formula: see text] of minimal dimension that maps to [Formula: see text] when extending the scalars to [Formula: see text]. This problem can be easily solved completely for finite extensions of odd degree. For quadratic extensions [Formula: see text], the situation is more involved, but solved for [Formula: see text]. For [Formula: see text], we construct quadratic field extensions [Formula: see text] and a form [Formula: see text] such that any form [Formula: see text] that maps to [Formula: see text] when extending the scalars to [Formula: see text] has dimension at least [Formula: see text].

Unramified extensions of quadratic number fields with certain perfect Galois groups

In this paper, we provide infinite families of quadratic number fields with everywhere unramified Galois extensions of Galois group [Formula: see text] and [Formula: see text], respectively. To my knowledge, these are the first instances of infinitely many such realizations for perfect groups which are not generated by involutions, a property which makes them difficult to approach for the problem in question and leads to somewhat delicate local–global problems in inverse Galois theory.

Measured Foliations and Hilbert 12th Problem

Yu. I. Manin conjectured that the maximal abelian extensions of the real quadratic number fields are generated by the pseudo-lattices with real multiplication. We prove this conjecture using theory of measured foliations on the modular curves.

On the solution of a Riesz equilibrium problem and integral identities for special functions

The aim of this note is to provide a full space quadratic external field extension of a classical result of Marcel Riesz for the equilibrium measure on a ball with respect to Riesz s-kernels. We address the case s=d−3 for arbitrary dimension d, in particular the logarithmic kernel in dimension 3. The equilibrium measure for this full space external field problem turns out to be a radial arcsine distribution supported on a ball with a special radius. As a corollary, we obtain new integral identities involving special functions such as elliptic integrals and more generally hypergeometric functions. It seems that these identities are not found in the existing tables for series and integrals, and are not recognized by advanced mathematical software. Among other ingredients, our proofs involve the Euler–Lagrange variational characterization, the Funk–Hecke formula, the Weyl regularity lemma, the maximum principle, and special properties of hypergeometric functions.

Rationalizability of field extensions with a view towards Feynman integrals

In 2021, Marco Besier and the first author introduced the concept of rationalizability of square roots to simplify arguments of Feynman integrals. In this work, we generalize the definition of rationalizability to field extensions. We then show that the rationalizability of a set of quadratic field extensions is equivalent to the rationalizability of the compositum of the field extensions, providing a new strategy to prove rationalizability of sets of square roots of polynomials.

Linear spaces embedded into projective spaces via Baer sublines

Every nontrivial linear space embedded in a Pappian projective space such that the blocks of the linear space are projectively equivalent Baer sublines with respect to a separable quadratic field extension is either a Baer subspace, or a Hermitian unital.Mathematics Subject Classifications: 51A45, 51E10

MULTIPLICITY ONE FOR THE PAIR (GLn(D);GLn(E))

Let F be a local field of characteristic zero. Let D be a quaternion algebra over F. Let E be a quadratic field extension of F. Let {\mu} be a character of GL(1,E). We study the distinction problem for the pair (GL(n,D), GL(n,E)) and we prove that any bi-(GL(n,E), {\mu})-equivariant tempered generalized function on GL(n,D) is invariant with respect to an anti-involution. Then it implies that dimHom({\pi},{\mu}) is at most 1 by the generalized Gelfand-Kazhdan criterion. Thus we give a new proof to the fact that (GL(2n,F),GL(n,E)) is a Gelfand pair when {\mu} is trivial and D splits.

Finite subunitals of the Hermitian unitals

Every finite subunital of any generalized hermitian unital is itself a hermitian unital; the embedding is given by an embedding of quadratic field extensions. In particular, a generalized hermitian unital with a finite subunital is a hermitian one (i.e., it originates from a separable field extension).

Finite submodules of Iwasawa modules for multi-variable extensions

For multi-variable extensions of number fields, we study the maximal finite submodules of classical Iwasawa modules. Our principal objects are the unramified Iwasawa modules for two-variable extensions of imaginary quadratic fields. On the one hand, we obtain a novel way to show a certain upper bound, or even the vanishing, of the maximal finite submodules. On the other hand, we obtain a sufficient condition for the non-vanishing, and then construct numerical examples for which the non-vanishing holds.

An infinitesimal variant of the Guo–Jacquet trace formula, II

We establish an infinitesimal variant of Guo-Jacquet trace formula for the case of a central simple algebra over a number field $F$ containing a quadratic field extension $E/F$. It is an equality between a sum of geometric distributions on the tangent space of some symmetric space and its Fourier transform. To prove this, we need to define an analogue of Arthur's truncation and then use the Poisson summation formula. We describe the terms attached to regular semi-simple orbits as explicit weighted orbital integrals. To compare them to those for another case studied in our previous work, we state and prove the weighted fundamental lemma at the infinitesimal level by using Labesse's work on the base change for $GL_n$.

On metabelian 2-class field towers over -extensions of real quadratic fields

AbstractWe give a family of real quadratic fields such that the 2-class field towers over their cyclotomic $\mathbb Z_2$ -extensions have metabelian Galois groups of abelian invariants $[2,2,2]$ . We also consider the boundedness of the Galois groups in relation to Greenberg’s conjecture, and calculate their class-2 quotients with an explicit example.