Lattices and Quadratic Forms
Magma implements state-of-the-art lattice algorithms - LLL and BKZ reduction, shortest and closest vector algorithms, and sphere enumeration. Quadratic forms over number fields and local fields are fully supported.
Contributors
A Magma package for lattices over number fields was based on code contributed by Gael Collinet (Strasbourg). This package was extended by Markus Kirschmer, David Lorch (both Aachen) and Mark Watkins (Magma).
The construction of the sublattice of an integral lattice is performed by code developed by Markus Kirschmer (Aachen).
A collection of lattices derived from the on-line tables of lattices prepared by Neil Sloane (AT&T Research) and Gabriele Nebe (Aachen) is included in Magma.
The original functions for computing automorphism groups and isometries of integral lattices are based on the AUTO and ISOM programs of Bernd Souvignier (Nijmegen). In V2.16 they are replaced by much faster versions developed by Bill Unger (Magma).
Coppersmith's method (based on LLL) for finding small roots of univariate polynomials modulo an integer has been implemented by Damien Stehlé (ENS Lyon).
Given a quadratic form F in an arbitrary number of variables, Mark Watkins (Bristol) has used Denis Simon's ideas as the basis of an algorithm he has implemented in Magma for finding a large (totally) isotropic subspace of F.