Closest Vector Problem (CVP) – Best Software Packages

algorithmslatticesmathematical-software

Let $A$ be a positive definite, real $n \times n$ matrix. This defines a norm on $\mathbb{R}^n$. Now I have a given point $p \in \mathbb{R}^n$ and I want to find the lattice point $x \in \mathbb{Z}^n$ that is closest to $p$ with respect to this norm. This is commonly known as the Closest Vector Problem (CVP) and seems to be very important. So I guess algorithms to solve this should be implemented somewhere. However, I was not able to find something.

Is this there a package for solving this problem for some common mathematical software like Maple, Mathematica, Sage, etc?

Note that I am not interested in some approximation using LLL or so, I want really the (or one) closest vector!

Best Answer

fplll (available here: https://github.com/fplll/fplll), a C++ implementation of a selection of lattice algorithms, has a CVP solver:

It also includes a floating-point implementation of the Kannan-Fincke-Pohst algorithm [K83,FP85] that finds a shortest non-zero lattice vector. For the same task, the GaussSieve algorithm [MV10] is also available in fplll. Finally, it contains a variant of the enumeration algorithm that computes a lattice vector closest to a given vector belonging to the real span of the lattice.

Magma has a CVP solver. Details here: https://magma.maths.usyd.edu.au/magma/overview/2/17/9/

Magma includes a highly optimized algorithm for enumerating all short vectors in a lattice with given norm. This algorithm, developed by Damien Stehlë, is used for computing the minimum, the shortest vectors, short vectors in a given range, and vectors close to or closest to a given vector (possibly) outside the lattice.

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