Lattices: Reduction

• LLL reduction of lattices, basis matrices and Gram matrices (with numerous parameters)
• Seysen reduction of lattices, basis matrices and Gram matrices (for reducing a lattice and its dual simultaneously)
• Pairwise reduction of lattices, basis matrices and Gram matrices
• Orthogonalization and orthonormalization (Cholesky decomposition) of a lattice
• Testing matrices for positive or negative (semi-)definiteness

The LLL algorithm can operate on either a basis matrix or a Gram matrix (and will use the Gram method even if given a basis matrix and it is deemed appropriate) and can be controlled by many parameters ($\delta$ constant, exact de Weger integral method or Schnorr-Euchner floating point method, step and time limits, selection of methods, etc.). The LLL algorithm can reduce matrices with very large entries as well as matrices having large sizes (e.g., number of rows well over 500).