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Another look at real quadratic fields of relative class number 1

Debopam Chakraborty, Anupam Saikia (2014)

Acta Arithmetica

The relative class number H d ( f ) of a real quadratic field K = ℚ (√m) of discriminant d is defined to be the ratio of the class numbers of f and K , where K denotes the ring of integers of K and f is the order of conductor f given by + f K . R. Mollin has shown recently that almost all real quadratic fields have relative class number 1 for some conductor. In this paper we give a characterization of real quadratic fields with relative class number 1 through an elementary approach considering the cases when...

Cryptography based on number fields with large regulator

Johannes Buchmann, Markus Maurer, Bodo Möller (2000)

Journal de théorie des nombres de Bordeaux

We explain a variant of the Fiat-Shamir identification and signature protocol that is based on the intractability of computing generators of principal ideals in algebraic number fields. We also show how to use the Cohen-Lenstra-Martinet heuristics for class groups to construct number fields in which computing generators of principal ideals is intractable.

Galois module structure of the rings of integers in wildly ramified extensions

Stephen M. J. Wilson (1989)

Annales de l'institut Fourier

The main results of this paper may be loosely stated as follows.Theorem.— Let N and N ' be sums of Galois algebras with group Γ over algebraic number fields. Suppose that N and N ' have the same dimension and that they are identical at their wildly ramified primes. Then (writing 𝒪 N for the maximal order in N ) 𝒪 N 𝒪 N Γ Γ 𝒪 N ' 𝒪 N ' Γ . In many cases 𝒪 N Γ 𝒪 N ' . The role played by the root numbers of N and N ' at the symplectic characters of Γ in determining the relationship between the Γ -modules 𝒪 N and 𝒪 N ' is described. The theorem includes...

Jacobi symbols, ambiguous ideals, and continued fractions

R. A. Mollin (1998)

Acta Arithmetica

The purpose of this paper is to generalize some seminal results in the literature concerning the interrelationships between Legendre symbols and continued fractions. We introduce the power of ideal theory into the arena. This allows significant improvements over the existing results via the infrastructure of real quadratic fields.

On Bilinear Structures on Divisor Class Groups

Gerhard Frey (2009)

Annales mathématiques Blaise Pascal

It is well known that duality theorems are of utmost importance for the arithmetic of local and global fields and that Brauer groups appear in this context unavoidably. The key word here is class field theory.In this paper we want to make evident that these topics play an important role in public key cryptopgraphy, too. Here the key words are Discrete Logarithm systems with bilinear structures.Almost all public key crypto systems used today based on discrete logarithms use the ideal class groups...

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