Obtuse Lattice Bases

09/01/2020

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by   Kanav Gupta, et al.

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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. We define a class of bases called obtuse bases and show that any lattice basis can be transformed to an obtuse basis. A shortest vector 𝐬 can be written as 𝐬=v_1𝐛_1+…+v_n𝐛_n where 𝐛_1,…,𝐛_n are the input basis vectors and v_1,…,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 obtuse bases makes the lattice enumeration algorithm for finding a shortest vector exponentially faster. We have implemented the algorithm for making bases obtuse, and tested it some small bases.