Korkine–Zolotarev lattice basis reduction algorithm

The Korkine–Zolotarev (KZ) lattice basis reduction algorithm is a lattice reduction algorithm invented by A. Korkine and G. Zolotareff in 1877.

Although the KZ reduction has exponential complexity versus the polynomial complexity of the LLL reduction algorithm, it is preferred for solving sequences of Closest Vector Problems (CVPs) in a lattice, where it may be more efficient.

Definition

A KZ-reduced basis for a lattice is defined as follows:

Given a basis

$\mathbf {B} =\{\mathbf {b} _{1},\mathbf {b} _{2},\dots ,\mathbf {b} _{n}\},$

define its Gram–Schmidt process orthogonal basis

$\mathbf {B} ^{*}=\{\mathbf {b} _{1}^{*},\mathbf {b} _{2}^{*},\dots ,\mathbf {b} _{n}^{*}\},$

and the Gram-Schmidt coefficients

$\mu _{i,j}={\frac {\langle \mathbf {b} _{i},\mathbf {b} _{j}^{*}\rangle }{\langle \mathbf {b} _{j}^{*},\mathbf {b} _{j}^{*}\rangle }}$, for any $1\leq j<i\leq n$.

Also define projection functions

$\pi _{i}(\mathbf {x} )=\sum _{j\geq i}{\frac {\langle \mathbf {x} ,\mathbf {b} _{j}^{*}\rangle }{\langle \mathbf {b} _{j}^{*},\mathbf {b} _{j}^{*}\rangle }}\mathbf {b} _{j}^{*}$

which project $\mathbf {x}$ orthogonally onto the span of $b_{i}^{*},\cdots ,b_{n}^{*}$.

Then the basis $B$ is KZ-reduced if the following holds:

1. $\mathbf {b} _{i}^{*}$ is the shortest nonzero vector in $\pi _{i}({\mathcal {L}}(\mathbf {B} ))$

1. For all $j<i$, $\left|\mu _{i,j}\right|\leq 1/2$

References

• Korkine, A.; Zolotareff, G. (1877). "Sur les formes quadratiques positives". {{cite journal}}: Cite journal requires |journal= (help)

• Lyu, Shanxiang; Ling, Cong (2017). "Boosted KZ and LLL Algorithms" (PDF). {{cite journal}}: Cite journal requires |journal= (help)

• Wen, Jinming; Chang, Xiao-Wen (2018). "On the KZ Reduction" (PDF). {{cite journal}}: Cite journal requires |journal= (help)

• Micciancio, Daniele; Goldwasser, Shafi (2002). Complexity of Lattice Problems. pp. 131–136.