Fast LLL-type lattice reduction

We modify the concept of LLL-reduction of lattice bases in the sense of Lenstra, Lenstra, Lovász [LLL82] towards a faster reduction algorithm. We organize LLL-reduction in segments of the basis. Our SLLL-bases approximate the successive minima of the lattice in nearly the same way as LLL-bases. For integer lattices of dimension n given by a basis of length 2, SLLL-reduction runs in O(n) bit operations for every ε > 0, compared to O(n) for the original LLL and to O(n) for the LLL-algorithms of Schnorr (1988) and Storjohann (1996). We present an even faster algorithm for SLLL-reduction via iterated subsegments running in O(n log n) arithmetic steps.