Search Graph Formulation and Hasting’s Generalization of Metropolis Algorithm For Solving SVP

Abstract

Shortest Lattice Vector Problem (SVP) has numerous applications spanning from robotics to computational number theory, viz., polynomial factorization. At the same time, SVP is a notoriously hard problem. Not only it is NP-hard, there is not even any polynomial approximation known for the problem that runs in polynomial time. What one normally uses is the LLL algorithm which, although a polynomial time algorithm, may give solutions which are an exponential factor away from the optimum. In this paper, we have defined an appropriate search space for the problem which we use for implementation of the Hasting’s generalization of the Metropolis algorithm. We have defined a suitable neighbourhood structure which makes the diameter of the space polynomially bounded, and we ensure that each search point has only polynomially many neighbours. We also proved that our search space graphs for SVP has magnification greater than half. We have implemented the Metropolis algorithm and Hasting’s generalization of the Metropolis algorithm for the SVP. Our results are quite encouraging in all instances when compared with LLL algorithm.