The Korkine–Zolotarev (KZ) lattice basis reduction algorithm or Hermite–Korkine–Zolotarev (HKZ) algorithm is a lattice reduction algorithm.
it yields a lattice basis with orthogonality defect at most
bound of the LLL reduction.
[1] KZ has exponential complexity versus the polynomial complexity of the LLL reduction algorithm, however it may still be preferred for solving multiple closest vector problems (CVPs) in the same lattice, where it can be more efficient.
The definition of a KZ-reduced basis was given by Aleksandr Korkin and Yegor Ivanovich Zolotarev in 1877, a strengthened version of Hermite reduction.
The first algorithm for constructing a KZ-reduced basis was given in 1983 by Kannan.
[2] The block Korkine-Zolotarev (BKZ) algorithm was introduced in 1987.
[3] A KZ-reduced basis for a lattice is defined as follows:[4] Given a basis define its Gram–Schmidt process orthogonal basis and the Gram-Schmidt coefficients Also define projection functions which project
is KZ-reduced if the following holds: Note that the first condition can be reformulated recursively as stating that
is a shortest vector in the lattice, and
is a KZ-reduced basis for the lattice
Also note that the second condition guarantees that the reduced basis is length-reduced (adding an integer multiple of one basis vector to another will not decrease its length); the same condition is used in the LLL reduction.