Hwanyup Jung

The generalized Redei-matrix for function fields

Abstract Let F be a finite geometric separable extension of the rational function field F q ( 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 O F of F q [ T ] in F is not divisible by l. In analogy with the number field case [Q. Yue, The generalized Redei-matrix, Math. Z. 261 (2009) 23–37], we define the generalized Redei-matrix R E / F of local Hilbert symbols with coefficients in F l . Using this generalized Redei-matrix we give an analogue of the Redei–Reichardt formula for E. Furthermore, we explicitly determine the generalized Redei-matrices for Kummer extensions, biquadratic extensions and Artin–Schreier extensions of F q ( T ) . Finally, using the generalized Redei-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.

CLASS NUMBER FORMULAE IN THE FORM OF A PRODUCT OF DETERMINANTS IN FUNCTION FIELDS

In this paper, we generalize the Kuceras group-determinant formulae to obtain the real and relative class number formulae of any subfield of cyclotomic function fields with arbitrary conductor in the form of a product of determinants. From these formulae, we generalize the relative class number formula of Rosen and Bae-Kang and obtain analogous results of Tsumura and Hirabayashi for an intermediate field in the tower of cyclotomic function fields with prime power conductor.

Determinant formulas for class numbers in function fields

In this paper, by extending Kuceras idea to the function field case, we obtain several determinant formulas involving the real class number and the relative class number of any subfield of cyclotomic function fields. We also provide several examples using these determinant formulas.