Smooth Sailing through Spherical Shells: Provable Random-Lattice Sieving in Time 2 0.292n

In an attempt to close the gap between the best provable and heuristic algorithms for hard lattice problems, such as the shortest (SVP) and closest vector problem (CVP), we analyze lattice sieving on Haar-random unimodular lattices. With well-chosen modifications to heuristic sieving, we show that the heuristic assumptions are no longer necessary, and we can provably achieve the same complexities as heuristic sieving on Haar-random lattices for various lattice problems. Concretely, with probability 1 - o(1) over the randomness of the Haar-random lattice and the algorithmic randomness, we show how to: 1. Solve SVP in time 2 0.2924 n + o(n) and space 2 0.2075 n + o(n); 2. Solve CVP for random targets with the same complexities; 3. Produce 2 0.2075 n + o(n) discrete Gaussian samples at any width with these complexities, up to a 2-Ω(n) error in the joint distribution. This improves on the SVP complexities for Haar-random lattices of Pouly-Shen [Eurocrypt, 2026] running in time 2 0.633n + o(n) and space 2 0.5n + o(n), as well as the recent worst-case SVP (and average-case CVP) improvements of Gao-Feng-Hu and Hhan [Cryptology ePrint Archive, 2026], both running in time and space 2 0.5n + o(n) or higher. Similar to standard sieving methods, our approach proceeds through a series of thin spherical shells, starting from a large radius and iteratively combining vectors to obtain vectors from shells with smaller radius. Our main technical contribution is making a series of adjustments to guarantee that for each sieve list generated at each spherical shell, each list vector is independent and uniformly random over all lattice points within this shell. Once this invariant is satisfied, it is a matter of smooth sailing through the spherical shells until we find a solution.