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arXiv · 1706.07373

Testing isomorphism of lattices over CM-orders

Hendrik W. Lenstra Jr.·Alice Silverberg
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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.

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Hendrik W. Lenstra Jr., Alice Silverberg. 2017-06-22. Testing isomorphism of lattices over CM-orders. https://arxiv.org/abs/1706.07373

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