Lattices that admit logarithmic worst-case to average-case connection factors

We demonstrate an average-case problem that is as hard as finding (n)-approximate shortest vectors in certain n-dimensional lattices in the worst case, where (n) = O( log n). The previously best known factor for any non-trivial class of lattices was (n) = ~O(n). Our results apply to families of lattices having special algebraic structure. Specifically, we consider lattices that correspond to ideals in the ring of integers of an algebraic number field. The worst-case problem we rely on is to find approximate shortest vectors in these lattices, under an appropriate form of preprocessing of the number field. For the connection factors (n) we achieve, the corresponding decision problems on ideal lattices are not known to be NP-hard; in fact, they are in P. However, the search approximation problems still appear to be very hard. Indeed, ideal lattices are well-studied objects in computational number theory, and the best known algorithms for them seem to perform no better than the best kno...

• Type: Conference
• Year: 2007
• Where: STOC
• Authors: Chris Peikert, Alon Rosen
• Keywords: Algorithms, Ideal Lattices, Root Discriminant, Special Algebraic Structure, STOC 2007