Faster LLL-type Reduction of Lattice Bases

Abstract

We describe an asymptotically fast variant of the LLL lattice reduction algorithm. It takes as input a basis B ∈ Zn x n and returns a (reduced) basis C of the Euclidean lattice L spanned by B, whose first vector satisfies |c1| ≤ (1+c) (4/3)(n-1)/4 (det L)1/n for any fixed c>0. It terminates within O(n4+ε β1+ε) bit operations for any ε >0, with β = log maxi |bi|. It does rely on fast integer arithmetic but does not make use of fast matrix multiplication.