# Testing isomorphism of lattices over CM-orders

## TL;DR

This paper introduces a deterministic polynomial-time algorithm to test isomorphism of lattices over CM-orders, leveraging cryptographic techniques and number theory to efficiently decide ideal class equivalence.

## Contribution

It presents a novel polynomial-time algorithm for lattice isomorphism testing over CM-orders, using lattices instead of ideals to avoid coefficient blow-up.

## Key findings

- Algorithm operates in polynomial time
- Uses a novel existence theorem for auxiliary ideals
- Applies cryptographic techniques to lattice problems

## Abstract

A CM-order is a reduced order equipped with an involution that mimics complex conjugation. The Witt-Picard group of such an order is a certain group of ideal classes that is closely related to the "minus part" of the class group. We present a deterministic polynomial-time algorithm for the following problem, which may be viewed as a special case of the principal ideal testing problem: given a CM-order, decide whether two given elements of its Witt-Picard group are equal. In order to prevent coefficient blow-up, the algorithm operates with lattices rather than with ideals. An important ingredient is a technique introduced by Gentry and Szydlo in a cryptographic context. Our application of it to lattices over CM-orders hinges upon a novel existence theorem for auxiliary ideals, which we deduce from a result of Konyagin and Pomerance in elementary number theory.