Hardness of Lattice Problems

Katharina Kreuzer 📧

February 2, 2023

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.


This article formalizes the NP-hardness proofs of the Closest Vector Problem (CVP) and the Shortest Vector Problem (SVP) in maximum norm as well as the CVP in any p-norm for p>=1. CVP and SVP are two fundamental problems in lattice theory. Lattices are a discrete, additive subgroup of R^n and are used for lattice-based cryptography. The CVP asks to find the nearest lattice vector to a target. The SVP asks to find the shortest non-zero lattice vector. This entry formalizes the basic properties of lattices, the reduction from CVP to Subset Sum in both maximum and p-norm for a finite p with 1<= p and the reduction of SVP to Partition using the Bounded Homogeneous Linear Equations problem (BHLE) as an intermediate step. The formalization uncovered a number of problems with the existing proofs in the literature.


BSD License


Session CVP_Hardness