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On the Lattice Isomorphism Problem (1311.0366v1)
Published 2 Nov 2013 in cs.DS, cs.CC, cs.DM, and math.CO
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 Zn and might be useful elsewhere. We also prove that LIP lies in the complexity class SZK.