On the Complexity of Computing with p-adic Lattices

Abstract

We study the complexity of two fundamental problems attached to p-adic lattices: computing an N-orthogonal basis, and deciding whether a given basis is N-orthogonal. When the norm is supplied through an explicit diagonal representation, both problems are polynomial. However, in the black-box model, where the ultrametric norm N is fixed and only accessible through an evaluation oracle, we prove that both problems require $Ω$(p^{v-1} ) queries in the worst case, where v $\le$ n is the largest multiplicity of a commensurability class of N and n is the dimension of the ambient vector space. We then analyse the optimality of these bounds by providing explicit algorithms. Our results answer formally an open question of Zhang, Deng and Wang and refine the recent results of Duan and Wu by providing pointwise query bounds depending on the norm's structure, in addition to uniform bounds.