Subfield Attacks on HSVP in Ideal Lattices.

In this paper, we propose two subfield attacks on HSVP in ideal lattices in number fields (not necessarily Galois fields), both under the canonical embedding and the coefficient embedding. These two attacks use the intersection method and the norm method, respectively. In addition,the reduction steps in our attacks are all in polynomial time. By contrast, the former method reduces HSVP in ideal lattices to HSVP in ideal lattices in subfields with larger approximate factor than the latter one, while the latter method has a wider range of applications. Moreover, in Galois number fields, the intersection method performs better in attacking prime ideals with small decomposition fields, whilst the norm method works better in prime ideals with large decomposition fields. Besides, ideals with small norms are vulnerable by both of these two attacks.