Ideal Class Number Research Articles

On the Performance Analysis for CSIDH-Based Cryptosystems

In this paper, we present the performance and security analysis for various commutative SIDH (CSIDH)-based algorithms. As CSIDH offers a smaller key size than SIDH and provides a relatively efficient signature scheme, numerous CSIDH-based key exchange algorithms have been proposed to optimize the CSIDH. In CSIDH, the private key is an ideal class in a class group, which can be represented by an integer vector. As the number of ideal classes represented by these vectors determines the security level of CSIDH, it is important to analyze whether the different vectors induce the same public key. In this regard, we generalize the existence of a collision for a base prime p≡7mod8. Based on our result, we present a new interval for the private key to have a similar security level for the various CSIDH-based algorithms for a fair comparison of the performance. Deduced from the implementation result, we conclude that for a prime p≡7mod8, CSIDH on the surface using the Montgomery curves is the most likely to be efficient. For a prime p≡3mod8, CSIDH on the floor using the hybrid method with Onuki’s collision-free method is the most likely to be efficient and secure.

Maximal Solvable Subgroups of Size 2 Integer Matrices

Studying the solvable subgroups of 2 × 2 matrix groups over Z, we find a maximal finite order primitive solvable subgroup of GL(2, Z) unique up to conjugacy in GL(2, Z). We describe the maximal primitive solvable subgroups whose maximal abelian normal divisor coincides with the group of units of a quadratic ring extension of Z. We prove that every real quadratic ring R determines h classes of conjugacy in GL(2, Z) of maximal primitive solvable subgroups of GL(2, Z), where h is the number of ideal classes in R.

Class number parities of compositum of quadratic function fields

Abstract The parities of ideal class numbers of compositum of quadratic function fields are studied. Especially, the parities of ideal class numbers of F q ( t ) ( P , Q ) and F q ( t ) ( P , Q , R ) $\mathbb{F}_q(t)(\sqrt P,\sqrt Q) \text{ and } \mathbb{F}_q(t)(\sqrt P,\sqrt Q,\sqrt R)$ are completely determined in detail, where P,Q,R are monic irreducible polynomials of even degrees.

On l-class groups of global function fields

Let [Formula: see text] be a finite geometric separable extension of the rational function field [Formula: see text], and let [Formula: see text] be a finite cyclic extension of [Formula: see text] of prime degree [Formula: see text]. Assume that the ideal class number of the integral closure [Formula: see text] of [Formula: see text] in [Formula: see text] is not divisible by [Formula: see text]. Using genus theory and Conner–Hurrelbrink exact hexagon for function fields, we study in this paper the [Formula: see text]-class group of [Formula: see text] (i.e. the Sylow [Formula: see text]-subgroup of the ideal class group of [Formula: see text]) as Galois module, where [Formula: see text] is the integral closure of [Formula: see text] in [Formula: see text]. The resulting conclusion is used to discuss the relations of class numbers for the biquadratic function fields with their quadratic subfields.

Ternary quadratic forms over number fields with small class number

We enumerate all positive definite ternary quadratic forms over number fields with class number at most 2. This is done by constructing all definite quaternion orders of type number at most 2 over number fields. Finally, we list all definite quaternion orders of ideal class number 1 or 2.

Artin–Schreier extensions of the rational function field

In this paper we study the arithmetic of Artin–Schreier extensions of \(\mathbb {F}_{q}(T)\). We determine the integral closure of \(\mathbb {F}_{q}[T]\) in Artin–Schreier extension of \(\mathbb {F}_{q}(T)\). We also investigate the average values of the \(L\)-functions of orders of Artin–Schreier extensions and study the average values of ideal class numbers when \(p=3\) in detail.

The generalized Rédei-matrix for function fields

Let F be a finite geometric separable extension of the rational function field Fq(T). Let E be a finite cyclic extension of F with degree ℓ, where ℓ is a prime number. Assume that the ideal class number of the integral closure OF of Fq[T] in F is not divisible by ℓ. In analogy with the number field case [Q. Yue, The generalized Rédei-matrix, Math. Z. 261 (2009) 23–37], we define the generalized Rédei-matrix RE/F of local Hilbert symbols with coefficients in Fℓ. Using this generalized Rédei-matrix we give an analogue of the Rédei–Reichardt formula for E. Furthermore, we explicitly determine the generalized Rédei-matrices for Kummer extensions, biquadratic extensions and Artin–Schreier extensions of Fq(T). Finally, using the generalized Rédei-matrix given in this paper, we completely determine the 4-ranks of the ideal class groups for a large class of Artin–Schreier extensions. In cryptanalysis, this class of Artin–Schreier extensions has been used in [P. Gaudry, F. Hess, N.P. Smart, Constructive and destructive facets of Weil descent on elliptic curves, J. Cryptology 15 (2002) 19–46] to perform the Weil descent, which may lead to a possible method of attack against the ECDLP, so-called GHS attack.

Real quadratic function fields of Richaud-Degert type with ideal class number one

We determine all real quadratic function fields of Richaud-Degert type with ideal class number one.

THE RAMIFICATION GROUPS AND DIFFERENT OF A COMPOSITUM OF ARTIN–SCHREIER EXTENSIONS

Let K be a function field over a perfect constant field of positive characteristic p, and L the compositum of n (degree p) Artin–Schreier extensions of K. Then much of the behavior of the degree pnextension L/K is determined by the behavior of the degree p intermediate extensions M/K. For example, we prove that a place of K totally ramifies/is inert/splits completely in L if and only if it totally ramifies/is inert/splits completely in every M. Examples are provided to show that all possible decompositions are in fact possible; in particular, a place can be inert in a non-cyclic Galois function field extension, which is impossible in the case of a number field. Moreover, we give an explicit closed form description of all the different exponents in L/K in terms of those in all the M/K. Results of a similar nature are given for the genus, the regulator, the ideal class number and the divisor class number. In addition, for the case n = 2, we provide an explicit description of the ramification group filtration of L/K.

On the independence of Heegner points in the function field case

Let ∞ be a fixed place of a global function field k. Let E be an elliptic curve defined over k which has split multiplicative reduction at ∞ and fix a modular parametrization ΦE:X0(N)→E. Let P1,…,Pr∈E(k¯) be Heegner points associated to the rings of integers of distinct quadratic “imaginary” fields K1,…,Kr over (k,∞). We prove that if the “prime-to-2p” part of the ideal class numbers of ring of integers of K1,…,Kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P1,…,Pr are independent in E(k¯)/Etors. Moreover, when k is rational, we show that there are infinitely many imaginary quadratic fields for which the prime-to-2p part of the class numbers are larger than C.

CM cycles and nonvanishing of class group $L$–functions

Let $K$ be a totally imaginary quadratic extension of a totally real number field $F$, and assume that $F$ has narrow ideal class number 1. Let $\chi$ be a character of the ideal class group $\Cl(K)$ of $K$, and let $L_K(\chi,s)$ be its associated $L$--function. In this paper we prove that for all $\epsilon > 0$, \begin{align*} \# \{\chi \in \textrm{CL}(K)^{\wedge}:~L_K(\chi,\frac{1}{2}) \neq 0\} \gg_{\epsilon, F} d_K^{\frac{1}{100}-\epsilon} \end{align*} as the absolute discriminant $d_K \rightarrow \infty$.

Global Units Modulo Circular Units: Descent Without Iwasawa’s Main Conjecture

Abstract.Iwasawa's classical asymptotical formula relates the orders of the p-parts Xn of the ideal class groups along a ℤp-extension F∞/F of a number field F to Iwasawa structural invariants ƛ and μ attached to the inverse limit X∞ = . It relies on “good” descent properties satisfied by Xn. If F is abelian and F∞ is cyclotomic, it is known that the p-parts of the orders of the global units modulo circular units Un/Cn are asymptotically equivalent to the p-parts of the ideal class numbers. This suggests that these quotientsUn/Cn, so to speak unit class groups, also satisfy good descent properties. We show this directly, i.e., without using Iwasawa's Main Conjecture.

On the L -functions of the curves y 2 = x ℓ + A

Let > 3 be an odd prime and let A be an integer not divisible by . Let CA be the non-singular projective model over of the affine curve CA: y2 = x + A. In this paper, we use the theory of Hilbert modular Eisenstein series to obtain a formula for the central value of the Hasse�Weil L-function L(CA, s) of the curve CA when the cyclotomic field () has ideal class number 1

On some questions related to the Gauss conjecture for function fields

We show that, for any finite field F q , there exist infinitely many real quadratic function fields over F q such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field F q , of infinitely many real function fields over F q with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over F q such that the numerator of their zeta function is an irreducible polynomial.

Indice des unités elliptiques dans les $\mathbb{Z}_p$-extensions

We compare the behavior in ℤ p -extensions of the ideal class number with the behavior of the index of Rubin’s group of elliptic units.

Class groups of imaginary function fields: The inert case

Let F \mathbb {F} be a finite field and T T a transcendental element over F \mathbb {F} . An imaginary function field is defined to be a function field such that the prime at infinity is inert or totally ramified. For the totally imaginary case, in a recent paper the second author constructed infinitely many function fields of any fixed degree over F ( T ) \mathbb {F}(T) in which the prime at infinity is totally ramified and with ideal class numbers divisible by any given positive integer greater than 1. In this paper, we complete the imaginary case by proving the corresponding result for function fields in which the prime at infinity is inert. Specifically, we show that for relatively prime integers m m and n n , there are infinitely many function fields K K of fixed degree m m such that the class group of K K contains a subgroup isomorphic to ( Z / n Z ) m − 1 (\mathbb {Z}/n\mathbb {Z})^{m-1} and the prime at infinity is inert.

Abelian subgroups of any order in class groups of global function fields

Let F be a finite field with q elements, and T a transcendental element over F . In this paper, we construct infinitely many real function fields of any fixed degree over F(T) with ideal class numbers divisible by any given positive integer greater than 1. For imaginary function fields, we obtain a stronger result which shows that for any relatively prime integers m and n with m, n>1 and relatively prime to the characteristic of F , there are infinitely many imaginary fields of fixed degree m such that the class group contains a subgroup isomorphic to ( Z/n Z) m−1 .

On CM abelian varieties over imaginary quadratic fields

In this paper, we associate canonically to every imaginary quadratic field $K=\Bbb Q(\sqrt{-D})$ one or two isogenous classes of CM abelian varieties over $K$, depending on whether $D$ is odd or even ($D \ne 4$). These abelian varieties are characterized as of smallest dimension and smallest conductor, and such that the abelian varieties themselves descend to $\Bbb Q$. When $D$ is odd or divisible by 8, they are the `canonical' ones first studied by Gross and Rohrlich. We prove that these abelian varieties have the striking property that the vanishing order of their $L$-function at the center is dictated by the root number of the associated Hecke character. We also prove that the smallest dimension of a CM abelian variety over $K$ is exactly the ideal class number of $K$ and classify when a CM abelian variety over $K$ has the smallest dimension.

Bounds of the Ideal Class Numbers of Real Quadratic Function Fields

The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields $$ K = k{\left( {{\sqrt D }} \right)} $$ over k = F q (T). For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e. g., if D = F 2 + c, then h(D) ≥ degF/degP; if D = (SG)2 + cS, then h(D) ≥ degS/degP; if D = (A m + a)2 + A, then h(D) ≥ degA/degP, where P is an irreducible polynomial splitting in K, c ∈ F q . In addition, three types of quadratic function fields K are found to have ideal class numbers bigger than one.

Computation of the Fundamental Units and the Regulator of a Cyclic Cubic Function Field

This paper presents algorithms for computing the two fundamental units and the regulator of a cyclic cubic extension of a rational function field over a field of order q ≡ 1 (mod 3). The procedure is based on a method originally due to Voronoi that was recently adapted to purely cubic function fields of unit rank one. Our numerical examples show that the two fundamental units tend to have large degree, and frequently, the extension has a very small ideal class number.