The GHS attack is known to map the discrete logarithm problem(DLP) in the Jacobian of a curve C0 defined over the d degree extension kd of a finite field k to the DLP in the Jacobian of a new curve C over k which is a covering curve of C0, then solve the DLP of curves C/k by variations of index calculus algorithms. It is therefore important to know which curve C0/kd is subjected to the GHS atta...

We study the question of the surjectivity of the Galois correspondence from subHopf algebras to subfields given by the Fundamental Theorem of Galois Theory for abelian Hopf Galois structures on a Galois extension of fields with Galois group Γ, a finite abelian p-group. Applying the connection between regular subgroups of the holomorph of a finite abelian p-group (G,+) and associative, commutati...

In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that ...

In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In case of the smash product for an infinite group our construction specialized for a ring agrees with M. Beatti...

(A) All rings in this announcement are commutative and with 1. For any ring K we denote by W(K) the Witt ring of nondegenerate symmetric bilinear forms over K. DEFINITION 1. A signature o of K is a ring homomorphism from W(K) to Z. REMARK 1. If K is a field, the signatures correspond uniquely with the orderings of K [3], [9]. Thus Theorem 1 below generalizes the main results of Artin-Schreier's...

Two characterizations of an Azumaya Galois extension of a ring are given in terms of the Azumaya skew group ring of the Galois group over the extension and a Galois extension of a ring with a special Galois system is determined by the trace of the Galois group.

The new integrable systems associated to the space of elliptic branched coverings are constructed. The relationship of these systems with elliptic Schlesinger's system Takasaki [1] is described. For the standard twofold elliptic coverings the integrable system is written explicitly. The trigonometric degeneration of our construction is presented.

Strongly-cyclic branched coverings of knots are studied by using their (g, 1)-decompositions. Necessary and sufficient conditions for the existence and uniqueness of such coverings are obtained. It is also shown that their fundamental groups admit geometric g-words cyclic presentations.

We give a decomposition formula for the Bartholdi zeta function of a graph G which is partitioned into some irregular coverings. As a corollary, we obtain a decomposition formula for the Bartholdi zeta function of G which is partitioned into some regular coverings. © 2005 Elsevier B.V. All rights reserved.

We establish automatic realizations of Galois groups among groups M ⋊ G, where G is a cyclic group of order p for a prime p and M is a quotient of the group ring Fp[G]. The fundamental problem in inverse Galois theory is to determine, for a given field F and a given profinite group G, whether there exists a Galois extension K/F such that Gal(K/F ) is isomorphic to G. A natural sort of reduction...