Deterministic Construction of an Approximate M-Ellipsoid and its Application to Derandomizing Lattice Algorithms

We give a deterministic O(log n)^n algorithm for the {\em Shortest Vector Problem (SVP)} of a lattice under {\em any} norm, improving on the previous best deterministic bound of n^O(n) for general norms and nearly matching the bound of 2^O(n) for the standard Euclidean norm established by Micciancio and Voulgaris (STOC 2010). Our algorithm can be viewed as a derandomization of the AKS randomized sieve algorithm, which can be used to solve SVP for any norm in 2^O(n) time with high probability. We use the technique of covering a convex body by ellipsoids, as introduced for lattice problems in (Dadush et al., FOCS 2011). Our main contribution is a deterministic approximation of an M-ellipsoid of any convex body. We achieve this via a convex programming formulation of the optimal ellipsoid with the objective function being an n-dimensional integral that we show can be approximated deterministically, a technique that appears to be of independent interest.

💡 Research Summary

The paper presents a deterministic algorithm for solving the Shortest Vector Problem (SVP) in any norm with a running time of O((log n)^n)·poly(n), thereby removing the randomness that has been inherent in the best known algorithms for general norms. The key technical contribution is a deterministic construction of an approximate M‑ellipsoid for an arbitrary convex body K, which serves as the geometric tool that enables the derandomization of the AKS sieve and the subsequent reduction of SVP in an arbitrary norm to SVP in the Euclidean norm.

An M‑ellipsoid E of a convex body K is defined by the covering numbers N(K,E) and N(E,K), both of which must be bounded by 2^{O(n)}. While Milman proved the existence of such ellipsoids, prior work (Dadush‑Peikert‑Vempala 2011) relied on random sampling to actually construct them, which prevented a fully deterministic algorithm. The authors instead use Pisier’s ℓ‑position, a linear transformation that simultaneously controls the Gaussian average norm ℓ(K) = ∫‖x‖_K γ_n(x)dx and its dual ℓ(K*) . Pisier showed that there exists a transformation T with ℓ(TK)·ℓ(TK*) ≤ O(n log n). This transformation yields an ellipsoid with good covering properties, but the transformation itself must be found algorithmically.

The authors formulate a convex program (CP) that minimizes the functional f(A)=∫‖A x‖K γ_n(x)dx over positive‑definite matrices A with det(A)≥1. The optimal solution corresponds to the ℓ‑position. The main obstacle is that f(A) is defined by a high‑dimensional Gaussian integral. To evaluate it deterministically, the paper introduces a discretization scheme: choose a scaling factor s = Θ(√log n) and define a lattice D = (1/s)ℤ^n intersected with a ball of radius 3√n. The size of D is O((log n)^{n/2}), which is subexponential. For each lattice point x∈D, the authors compute a weight p_x = ∫{C_s}γ_n(x+y)dy, where C_s =