Reduced Ideals in Function Fields

Let F denote a function eld of transcendence degree one over a nite eld k. We assume that the eld is tamely rami ed at in nity, that the valuations at in nity of a set of fundamental units are known and we have gcd(f1; : : : ; fs) = 1, where fi denotes the degree of a place at in nity. In such a situation we describe a simple arithmetic in the divisor class group. One draw back of this arithmetic is that we do not obtain a unique representative for each divisor class. The method makes use of multiplication and reduction of reduced fractional ideals. In this paper we present a notion of reduced ideals for function elds of arbitrary degree, thus generalizing the work of Buchmann, Scheidler, Stein, Thiel and Williams, see [6], [26], [27] and [30]. We show how the notion of reduced ideals can be used to de ne an e cient arithmetic in the divisor class group of certain function elds. We assume throughout that the functions elds are de ned over a nite eld of constants. The ability to e ciently compute in the divisor class group of a curve is required in generalizations of discrete logarithm based cryptosystems, coding theory based on algebraic geometric curves, primality testing and is of independent theoretical interest in its own right. The paper is organized as follows: In section 1 we present the arithmetic of function elds, this is given in enough detail so that the whole paper is mostly self contained. Following this in section 2 we shall cover the basic ideas from the `geometry of numbers' in the Puiseux expansion elds which we shall require. Much of these rst two sections can be found in [29] and [25]. However we di er slightly in some of the notation and this all needs to be xed for the following sections. Indeed there is some disagreement in the literature about certain de nitions, so it is worth while spending some time xing notation. In section 3 we present the basics on reduced ideals in function elds that we will require. In section 4 we explain the reduction algorithm, analyze its complexity and give an example of how an ideal is reduced. In section 5 we show how to use this to perform e cient arithmetic in the divisor class group of certain function elds. Finally in section 6 we outline how some of the ideas in this paper can be used to solve other problems in function elds. Before we begin we comment on some historical notes. The lattice reduction algorithm we shall use is essentially the method of [25], which is itself closely related to the method of [16] and [21]. We shall give a complexity estimate for the number of bit operations of the lattice reduction procedure for lattices in a vector space of Puiseux expansions generated by an ideal of a function eld. 1991 Mathematics Subject Classi cation. Primary: 11G20, 14Q05. Secondary: 11Y16, 14H05.