Faster Lattice Enumeration

2019/12/03 by Mithilesh Kumar, Kumar, Mithilesh
Computer Science · #Cryptography and Security (cs.CR) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #cs.CR #cs.DS

paper · pdf · doi:10.48550/arxiv.1912.01781

12 pages, 1 figure

arxiv created 2019/12/03 · arxiv updated 2019/12/05

Abstract

A lattice reduction is an algorithm that transforms the given basis of the lattice to another lattice basis such that problems like finding a shortest vector and closest vector become easier to solve. Some of the famous lattice reduction algorithms are LLL and BKZ reductions. We define a class of bases called obtuse bases and show that any lattice basis can be transformed to an obtuse basis in O(n⁴) time. A shortest vector s can be written as v₁b₁+⋯+v_nb_n where b₁,…,b_n are the input basis vectors and v₁,…,v_n are integers. When the input basis is obtuse, all these integers can be chosen to be positive for a shortest vector. This property of the obtuse basis makes lattice enumeration algorithm for finding a shortest vector exponentially faster. Moreover, extreme pruning, the current fastest algorithm for lattice enumeration, can be run on an obtuse basis.

Related

• Obtuse Lattice Bases
• LLL Algorithm for Lattice Basis Reduction
• A hierarchy of polynomial time lattice basis reduction algorithms
• On the Shortest Lattice Vector vs. the Shortest Basis
• A faster method for computing Gama-Nguyen-Regev's extreme pruning coefficients