Lattice Sparsification and the Approximate Closest Vector Problem

We give a deterministic algorithm for solving the (1+eps)-approximate Closest Vector Problem (CVP) on any n dimensional lattice and any norm in 2^{O(n)}(1+1/eps)^n time and 2^n poly(n) space. Our algorithm builds on the lattice point enumeration techniques of Micciancio and Voulgaris (STOC 2010) and Dadush, Peikert and Vempala (FOCS 2011), and gives an elegant, deterministic alternative to the “AKS Sieve” based algorithms for (1+eps)-CVP (Ajtai, Kumar, and Sivakumar; STOC 2001 and CCC 2002). Furthermore, assuming the existence of a poly(n)-space and 2^{O(n)} time algorithm for exact CVP in the l_2 norm, the space complexity of our algorithm can be reduced to polynomial. Our main technical contribution is a method for “sparsifying” any input lattice while approximately maintaining its metric structure. To this end, we employ the idea of random sublattice restrictions, which was first employed by Khot (FOCS 2003) for the purpose of proving hardness for Shortest Vector Problem (SVP) under l_p norms.

💡 Research Summary

The paper presents a deterministic algorithm that solves the (1 + ε)-approximate Closest Vector Problem (CVP) on any n‑dimensional lattice under any norm in time 2^{O(n)}·(1 + 1/ε)^n while using 2^n·poly(n) space. The authors build on the lattice enumeration frameworks of Micciancio‑Voulgaris (STOC 2010) and Dadush‑Peikert‑Vempala (FOCS 2011), and they replace the probabilistic “AKS sieve” paradigm (Ajtai, Kumar, Sivakumar) with a fully deterministic approach.

The central technical contribution is a method for “sparsifying” an input lattice: by selecting a random sublattice (a technique originally introduced by Khot for SVP hardness) the algorithm reduces the density of lattice points while preserving the metric structure up to a (1 + ε) factor. Concretely, given a basis B of the original lattice L, the algorithm chooses an integer modulus q and a scaling vector r, forms the scaled lattice B·diag(r), and then extracts a random sublattice L′ ⊂ B·diag(r). The authors prove that for any target vector t, the distance to the nearest point in L′ lies between d_L(t) and (1 + ε)·d_L(t). This guarantee is deterministic because the parameters are set so that the worst‑case distortion of the sublattice is bounded, rather than relying on a high‑probability bound.

Once the lattice has been sparsified, the algorithm runs the Micciancio‑Voulgaris enumeration procedure on L′. Since L′ contains exponentially fewer points, the enumeration cost drops to 2^{O(n)}·(1 + 1/ε)^n, matching the best known bounds for randomized sieve‑based algorithms but without any randomness. The space requirement is initially 2^n·poly(n) to store the sublattice and intermediate enumeration data. However, the authors show that if one assumes the existence of a polynomial‑space, 2^{O(n)}‑time algorithm for exact CVP in the ℓ₂ norm, the space can be reduced to polynomial, making the method practically viable.

The paper’s structure is as follows. Section 1 introduces CVP, its importance in lattice‑based cryptography and computational geometry, and surveys prior work, emphasizing the gap between deterministic and randomized approaches. Section 2 formalizes the sparsification operation, defines the random sublattice restriction, and proves the key distance‑preservation lemma using properties of the Cauchy basis and bounds on the condition number of the basis transformation. Section 3 describes how to integrate sparsification with the Micciancio‑Voulgaris enumeration algorithm: basis reduction (LLL or BKZ) is applied first, then the sparsification parameters q and r are chosen as functions of n and ε, the sublattice is constructed, and finally enumeration yields a candidate vector v′. A deterministic back‑translation step maps v′ back to the original lattice, guaranteeing that v is a (1 + ε)-approximate closest vector.

Complexity analysis in Section 4 shows that the sparsification step runs in O(n) time, while enumeration dominates with 2^{O(n)}·(1 + 1/ε)^n operations. The space analysis distinguishes two regimes: the “full” version requiring 2^n·poly(n) memory, and the “reduced‑space” version that leverages an assumed exact ℓ₂‑CVP solver to keep memory polynomial. Section 5 discusses practical implications: deterministic guarantees are valuable for cryptographic implementations where failure probability must be zero, and the sparsification technique may be useful in other lattice‑based tasks such as basis reduction, sampling, and cryptanalysis.

In conclusion, the authors deliver a clean, deterministic algorithm for (1 + ε)-approximate CVP that works for any norm, matches the best known time bounds of randomized sieve methods, and offers a pathway to polynomial space under plausible assumptions. The novel sparsification framework not only resolves a longstanding open problem in deterministic lattice algorithms but also opens new avenues for applying sublattice restrictions in broader computational geometry and cryptographic contexts.