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 E. In analogy with the number field case [Q. Yue, The generalized Reclei-matrix, Math. Z. 261 (2009) 23-37], we define the generalized Reclei-matrix R-E/F of local Hilbert symbols with coefficients in F-l. Using this generalized Reclei-matrix we give an analogue of the Redei-Reichardt formula for E. Furthermore, we explicitly determine the generalized Reclei-matrices for Kummer extensions, biquadratic extensions and Artin-Schreier extensions of F-q(T). Finally, using the generalized Reclei-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 Well descent on elliptic curves, J. Cryptology 15 (2002) 19-46] to perform the Well descent, which may lead to a possible method of attack against the ECDLP, so-called GHS attack. (C) 2012 Elsevier Inc. All rights reserved.