Asymptotics of Random-Lattice Enumeration: Beyond HKZ via Dense Sublattices

Abstract

To bridge the gap between worst-case and heuristic analyses of lattice enumeration, we study the sizes of (unpruned) enumeration trees on $n$-dimensional Haar-random lattices. Writing $n^{cn+o(n)}$ for the tree size at the shortest vector radius, we investigate the asymptotic scaling of $c$, and which bases minimize it. For quasi-HKZ bases, we rigorously analyze the random-lattice HKZ geometry and establish $c=1/(2e)\approx 0.1839$. This matches the worst-case enumeration exponent of Hanrot-Stehlé and confirms earlier predictions for the geometry of HKZ bases on random lattices. Searching beyond quasi-HKZ bases, we prove that with high probability, the minimum achievable tree size exponent $c$ over all bases of a random lattice satisfies: \begin{align} 0.1464 \approx \frac{2-\sqrt{2}}{4}\leq c<0.1474. \end{align} For the lower bound, we use Kim's mean-value formula for nested chains of primitive sublattices to rule out bases with small prefix determinants. This strengthens the earlier Rankin-based heuristic lower bound exponent $c \geq 1/8$, discussed by Nguyen and Hanrot-Stehlé, and rules out the existence of random-lattice bases with trees as small as predicted by the geometric series assumption. For the upper bound, we show that with high probability a dense primitive half-rank sublattice exists with determinant at most $n^{n/8 + o(n)}$. Combining dual HKZ on this sublattice with HKZ on its quotient gives a basis with $c \leq 1/(4\sqrt{e}) \approx 0.1516$. Recursively splitting each half lowers the upper bound below $0.1474$, and under a natural heuristic assumption we prove that the lower bound is tight. Combined with the collision search of Feng-Gao-Liu, our two-block construction gives a polynomial-space algorithm solving the closest vector problem with preprocessing (CVPP) on random lattices in query time $n^{0.0892 n + o(n)}$ for arbitrary targets.