an:00176556
Zbl 0788.14024
Okamoto, Tatsuaki; Sakurai, Kouichi
Efficient algorithms for the construction of hyperelliptic cryptosystems
EN
Advances in cryptology, Proc. Conf., CRYPTO '91, Santa Barbara/CA (USA) 1991, Lect. Notes Comput. Sci. 576, 267-278 (1992).
1992
a
14H40 14Q05 94A60 68Q25 14G15 14H52 68Q15
Jacobian of a hyperelliptic curve; group of rational points; finite field; Weil pairing
Let \(g>0\) be a fixed integer. The authors prove that the problem of computing the structure of the group of rational points \(J(\mathbb{F}_ q)\) on the Jacobian \(J\) of a hyperelliptic curve \(X\) over a finite field \(\mathbb{F}_ q\) of cardinality \(q\) is in \(NP\cap co-NP\). The certificate is a set of independent generators of the group of prime power order. The independence is checked by means of the Weil pairing. To prove that the points generate the entire group, the authors employ Pila's algorithm [\textit{J. Pila}, Mth. Comput. 55, No. 192, 745-763 (1990; Zbl 0724.11070)] to compute the number of points in \(J(\mathbb{F}_ q)\). This last part is not really necessary; it suffices to use the fact that \((\sqrt q- 1)^{2g}\leq\#J(\mathbb{F}_ q)\leq(\sqrt q+1)^{2g}\).
For the entire collection see [Zbl 0753.00024].
R.Schoof (Povo)
Zbl 0724.11070