In this paper, we present a new algorithm for computing the reduced sum of two divisors of an arbitrary hyperelliptic curve. Our formulas and algorithms are generalizations of Shanks's NUCOMP algorithm, which was suggested earlier for composing and reducing positive definite binary quadratic forms. Our formulation of NUCOMP is derived by approximating the irrational continued fraction expansion used to reduce a divisor by a rational continued fraction expansion, resulting in a relatively simple and efficient presentation of the algorithm as compared to previous versions. We describe a novel, unified framework for divisor reduction on an arbitrary hyperelliptic curve using the theory of continued fractions, and derive our formulation of NUCOMP based on these results. We present numerical data demonstrating that our version of NUCOMP is more efficient than Cantor's algorithm for most hyperelliptic curves, except those of very small genus defined over small finite fields.
In 1976 DiMe and Hellman first introduced their well-known keyexchange protocol which is based on exponentiation in the multiplicative group GF(p)* of integers relatively prime to a large prime p (see I-81). Since then, this scheme has been extended to numerous other finite groups. Recently, Buchmann and Williams [2] introduced a version of the Diffie-Hellman protocol which uses the infrastructure of a real quadratic field. Theirs is the first such system not to require an underlying group structure, but rather a structure which is "almost" like that of a group. We give here a more detailed description of this scheme as well as state the required algorithms and considerations for their implementation.
The first part of this paper classifies all purely cubic function fields over a finite field of characteristic not equal to 3. In the remainder, we describe a method for computing the fundamental unit and regulator of a purely cubic congruence function field of unit rank 1 and characteristic at least 5. The technique is based on Voronoi's algorithm for generating a chain of successive minima in a multiplicative cubic lattice, which is used for calculating the fundamental unit and regulator of a purely cubic number field.
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