Tame Kernels and Further 4-rank Densities

A Note on 4-rank Densities

For certain real quadratic number fields, we prove density results concerning 4-ranks of tame kernels. We also discuss a relationship between 4-ranks of tame kernels and 4-class ranks of narrow ideal class groups. Additionally, we give a product formula for a local Hilbert symbol.

The 3-adic regulators and wild kernels

For any number field, J.-F. Jaulent introduced a new invariant called the group of logarithmic classes in 1994. This invariant is proved to be closely related to the wild kernels of number fields. In this paper, we show how to compute the kernel of the natural homomorphism from the group of logarithmic classes to the group of p-ideal classes by computing the p-adic regulator which is a classica...

Tame kernels of cubic cyclic fields

There are many results describing the structure of the tame kernels of algebraic number fields and relating them to the class numbers of appropriate fields. In the present paper we give some explicit results on tame kernels of cubic cyclic fields. Table 1 collects the results of computations of the structure of the tame kernel for all cubic fields with only one ramified prime p, 7 ≤ p < 5, 000....

TTS: Rank Attacks in Tame-Like Multivariate PKCs

We herein discuss two modes of attack on multivariate public-key cryptosystems. A 2000 Goubin-Courtois article applied these techniques against a special class of multivariate PKC’s called “Triangular-Plus-Minus” (TPM), and may explain in part the present dearth of research on “true” multivariates – multivariate PKC’s in which the middle map is not really taken in a much larger field. These att...

Table of Tame and Wild Kernels of Quadratic Imaginary Number Elds of Discriminants > {5000 (conjectural Values)