Torsieve

Torsieve is a term that refers to a method or algorithm used in number theory, particularly in the context of elliptic curves and their associated torsion subgroups. The core idea behind torsieve is to efficiently determine the structure of these torsion subgroups, which are finite sets of points on an elliptic curve that satisfy certain additive properties. Understanding the torsion subgroup is crucial for many applications of elliptic curves, such as in cryptography.

The torsieve algorithm aims to find generators for the torsion subgroup and to determine its order (the number of elements it contains). This process often involves studying the reduction of the elliptic curve modulo various primes. By examining how the curve and its points behave in different modular arithmetic systems, information about the global torsion subgroup can be deduced.

One common approach within torsieve involves sieving techniques. These techniques typically iterate through a range of numbers, performing local checks or computations. For torsion subgroups, this might involve checking for points of a specific order (like order p for a prime p) on the curve modulo several small primes. If such points exist and satisfy certain conditions, it provides evidence for the existence of a corresponding element in the global torsion subgroup.

The complexity and efficiency of torsieve algorithms are important considerations. Developing faster and more robust methods for determining torsion subgroup structure is an active area of research in computational number theory. The effectiveness of torsieve can depend on the specific elliptic curve being studied and the range of primes being considered.