On the Lattice Isomorphism Problem

We study the Lattice Isomorphism Problem (LIP), in which given two lattices L_1 and L_2 the goal is to decide whether there exists an orthogonal linear transformation mapping L_1 to L_2. Our main result is an algorithm for this problem running in time n^{O(n)} times a polynomial in the input size, where n is the rank of the input lattices. A crucial component is a new generalized isolation lemma, which can isolate n linearly independent vectors in a given subset of Z^n and might be useful elsewhere. We also prove that LIP lies in the complexity class SZK.Comment: 23 pages, SODA 201

On the Lattice Isomorphism Problem

Abstract We study the Lattice Isomorphism Problem (LIP), in which given two lattices L 1 and L 2 the goal is to decide whether there exists an orthogonal linear transformation mapping L 1 to L 2 . Our main result is an algorithm for this problem running in time n O(n) times a polynomial in the input size, where n is the rank of the input lattices. A crucial component is a new generalized isolation lemma, which can isolate n linearly independent vectors in a given subset of Z n and might be useful elsewhere. We also prove that LIP lies in the complexity class SZK