The generalized Rédei-matrix for function fields

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