Python API · Submodule

alkahest.lattice

alkahest.lattice — LLL lattice reduction (V2-6).

LLL reduces the rows of a matrix (basis given as integer rows, all having the same length). Optional delta = delta_num/delta_den must lie strictly between 1/4 and 1 when both parts are supplied; otherwise the canonical δ = 3/4 is used.

The reduction runs on FLINT's fmpz_lll and the result is then verified against the exact rational LLL inequalities — |μ_ij| ≤ 1/2 and the Lovász condition at the requested δ — before it is returned; a floating-point LLL promises only |μ_ij| ≤ η for some η > 1/2. An LLL-reduced basis is not unique, so assert those properties (and that the lattice is unchanged) rather than an exact matrix.

For the rest of the lattice toolkit — determinants, duals, exact shortest and closest vectors, theta series, kissing numbers and sphere-packing densities — see experimental.Lattice.

Raises

LatticeError Empty basis (E-LAT-001), ragged rows (E-LAT-002), invalid δ (E-LAT-003), or iteration guard (E-LAT-004). OverflowError (subclass PyOverflowError after reduction) A reduced coefficient no longer fits in a signed i64.

function

lattice.lll_reduce_rows(
    rows: list[list[int]],
    *,
    delta_num: int | None=None,
    delta_den: int | None=None
) → list[list[int]]

#source

LLL‑reduce rows with optional rational Lovász parameter delta_num/delta_den.

alkahest.lattice

#Raises

Raises