Density of Ideal Lattices

The security of many efficient cryptographic constructions, e.g. collision-resistant hash functions, digital signatures, identification schemes, and more recently public-key encryption has been proven assuming the hardness of worst-case computational problems in ideal lattices. These lattices correspond to ideals in the ring Z[ζ], where ζ is some fixed algebraic integer. Under the assumption that this ring Z[ζ] is the maximal order of the number field Q(ζ), we show that the density of n-dimensional ideal lattices with determinant ≤ b among all lattices under the same bound is in O(b1−n) as b grows. So, for lattices of dimension > 1 with bounded determinant, the subclass of ideal lattices is always vanishingly small. Our assumption, though not valid for all algebraic integers ζ is certainly holds for all ζ that have been suggested for practical use.