A. Lauder and Wan. D. Counting points on varieties over nite elds of small characteristic. In J.P. Buhler and P. Stevenhagen, editors, Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography, Mathematical Sciences Research Institute Publications, 2002. To appear.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of small odd characteristic, using the theory of Monsky Washnitzer cohomology. The running time of the algorithm is O(g q) for a hyperelliptic curve of genus g. The algorithm readily generalises to superelliptic curves as shown by Gaudry and Gurel [10] A related approach by Lauder and Wan [15] is based on Dwork s proof of the rationality of the zeta function and leads to a polynomial time algorithm for computing the zeta function of an arbitrary variety over a nite eld. Despite its polynomial complexity, the algorithm in its most general form is not practical. Using Dwork cohomology, ....

A.G.B. Lauder and D. Wan. Counting points on varieties over nite elds of small characteristic. Preprint 2001.

....80 bits, and one needs a lot of tries to nd a curve with a large prime order subgroup. Alternatives are to construct the curve via the CM method (see Weng [37] to choose Koblitz (sub eld) curves (see Lange [19] or to restrict to elds of small characteristic (see Kedlaya [16] Lauder and Wan [22], Vercauteren [36] In this article we propose a further kind of groups suitable for cryptographic applications as the computation of scalar multiples the main operation in the protocols can be carried out eciently, the group order can be determined and there are no known weaknesses. The ....

A. Lauder and D. Wan. Counting points on varieties over nite elds of small characteristic. submitted. 13

....odd characteristic, using the theory of Monsky Washnitzer cohomology. The running time of the algorithm is ) for a hyperelliptic curve of genus g over F p n . The algorithm readily generalizes to superelliptic curves as shown by Gaudry and Gurel [14] A related approach by Lauder and Wan [18] is based on Dwork s proof of the rationality of the zeta function and leads to a polynomial time algorithm for computing the zeta function of an arbitrary variety over a nite eld. Note that Wan [37] suggested the use of p adic methods, including the method of Dwork and Monsky, already several ....

A.G.B. Lauder and D. Wan. Counting points on varieties over nite elds of small characteristic. Preprint 2001.

....of the papers in this area can be found in [3] and more recent work includes [9, 10, 20, 22] Various fast methods have been developed for elliptic curves, but until recently general curves, let al..one higher dimensional varieties, seemed beyond reach both theoretically and practically. In [14] the present authors proved polynomial time computability of the zeta function of an arbitrary variety of xed dimension over a nite eld of small characteristic. This result was based upon Dwork s proof of the rationality of the zeta function of a variety [6] but unfortunately does not lead to ....

....a large class of hyperelliptic curves in characteristic 2, which is the main motivation behind the work. Such curves are of interest in cryptography [12] and the fast computation of their Jacobian orders was a long standing open problem, rst resolved for arbitrary genus in a special case in [14], and with no restrictions in the recent work [4, 5] Unfortunately, we are only able to prove the correctness of our algorithm for the case p 5. We now describe the main theorem. Denote by F q the nite eld with q elements, where q = p and p is prime. Fix F q an algebraic closure of F ....

A.G.B. Lauder and D. Wan, Counting points on varieties over nite elds of small characteristic, to appear in the proceedings of the MSRI workshop in Algorithmic Number Theory Aug-Dec 2000. Available at: http://web.comlab.ox.ac.uk/oucl/work/alan.lauder/

....time complexity to [18] Moreover, it is the rst practical algorithm for hyperelliptic curves in characteristic 2 which has polynomial time growth in both the eld size and genus. The problem of polynomial time computability for arbitrary varieties in small characteristic was already solved in [15], but the general algorithm there is not very practical. As previously mentioned, our algorithm can be extended to arbitrary hyperelliptic curves in characteristic 2, but we focus on the simplest cases in this paper. We refer to the references in [2] for the large literature on point counting, ....

....contains a statement of the algorithm for what we call Type 1 Artin Schreier curves, and Section 7 describes exactly how to perform the main steps. The complexity analysis is tied up in Section 8, and Section 9 discusses the remaining type of Artin Schreier curve in a more condensed fashion. As in [15] we aim to give a largely self contained presentation. 2. L functions and Artin Schreier Curves 2.1. General theory. Let Q denote an algebraic closure of the rationals Q . Let : F p Q be a non trivial additive character, and Tr k : F k F p the absolute trace map. A speci c will ....

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A.G.B. Lauder and D. Wan, Counting points on varieties over nite elds of small characteristic, preprint 2001.

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A. Lauder and Wan. D. Counting points on varieties over nite elds of small characteristic. In J.P. Buhler and P. Stevenhagen, editors, Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography, Mathematical Sciences Research Institute Publications, 2002. To appear.

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A.G.B. Lauder and D. Wan. Counting points on varieties over nite elds of small characteristic. Preprint 2001.

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