Computing a Lattice Basis Revisited

Abstract

Given (a, b) ∈ Z 2 , Euclid's algorithm outputs the generator gcd(a, b) of the ideal aZ + bZ. Computing a lattice basis is a high-dimensional generalization: given a 1 ,. .. , a n ∈ Z m , find a Z-basis of the lattice L = { n i=1 x i a i , x i ∈ Z} generated by the a i 's. The fastest algorithms known are HNF algorithms, but are not adapted to all applications, such as when the output should not be much longer than the input. We present an algorithm which extracts such a short basis within the same time as an HNF, by reduction to HNF. We also present an HNF-less algorithm, which reduces to Euclid's extended algorithm and can be generalized to quadratic forms. Both algorithms can extend primitive sets into bases.

Computing a Basis for an Integer Lattice

“…By first producing a basis B ∈ Z 𝑟 ×𝑚 for A ∈ Z 𝑛×𝑚 , then producing a basis B ∈ Z 𝑟 ×𝑟 for Transpose(B) ∈ Z 𝑚×𝑟 , we can apply the Smith form algorithm with the nonsingular input B to obtain the Smith form of A. We refer to Li and Nguyen [6] for other applications of computing a lattice basis with entries not much larger than those of the input matrix, and for a survey of previous approaches for solving this problem.…”

“…In this paper, we consider the special case 𝑚 = 𝑛 − 1 = 𝑟 , that is, the input is a full column rank matrix A ∈ Z 𝑛×(𝑛 −1) . To the best of our knowledge, the fastest previous algorithm to compute a basis B with log ||B|| ∈ 𝑂 (log 𝑛 + log ||A||) is that of Li and Nguyen [6,Theorem 3.8], which solves the problem in the general case, that is, for 𝑛, 𝑚 and 𝑟 arbitrary. Applied to the special case 𝑚 = 𝑛 − 1 = 𝑟 that we consider here, their algorithm has cost bounded by 𝑂 (𝑛 𝜔 B(𝑛𝑑)) bit operations, where 𝑑 = log 𝑛 + log ||A|| and 𝜔 is the exponent of matrix multiplication.…”