Lattice Sparsification: Efficient Sublattice Construction

• Lattice sparsification is a method that uses modular restrictions to create a sparser sublattice while maintaining an additive metric guarantee.
• It employs both randomized and deterministic constructions to enable efficient enumeration for approximate closest vector problems.
• The approach balances the trade-off between enumeration complexity and additive distortion, crucial for practical algorithms in lattice geometry.

Lattice sparsification is a technique for constructing a sparser sublattice of a given lattice such that the metric structure is preserved up to an additive distortion. The primary application is to accelerate algorithms for finding approximate closest lattice vectors in arbitrary norms, in particular enabling deterministic and efficient solutions to the $(1+\epsilon)$-approximate Closest Vector Problem (CVP) with both randomized and derandomized variants. The method is based on modular restrictions and random sublattice selection, yielding sublattices that are sparse but remain within an additive distance of the original lattice, facilitating efficient enumeration for approximate CVP and other computational problems in the geometry of numbers (Dadush et al., 2012).

1. Preliminaries and Definitions

Let $L \subset \mathbb{R}^n$ denote an $n$-dimensional lattice, defined as all integer combinations of a full-rank basis $B \in \mathbb{Q}^{n \times n}$, i.e.,

$L = \{Bz : z \in \mathbb{Z}^n\}.$

Given a symmetric convex body $K = -K \subset \mathbb{R}^n$, containing the origin in its interior, a norm is induced by

$\|x\|_K = \inf \{s \ge 0 : x \in sK \}.$

For a target $y \in \mathbb{R}^n$, the lattice distance is

$d_K(L, y) = \min_{x \in L} \|x - y\|_K.$

The enumeration parameter

$$G(tK, L) = \max_{z \in \mathbb{R}^n} |(z + tK) \cap L|$$

bounds the maximal number of lattice points in a translate of $L \subset \mathbb{R}^n$0.

A sublattice $L \subset \mathbb{R}^n$1 is called a $L \subset \mathbb{R}^n$2–sparsifier if for every $L \subset \mathbb{R}^n$3, $L \subset \mathbb{R}^n$4, and $L \subset \mathbb{R}^n$5. Thus, $L \subset \mathbb{R}^n$6 is significantly sparser than $L \subset \mathbb{R}^n$7 over $L \subset \mathbb{R}^n$8-balls, while all points in $L \subset \mathbb{R}^n$9 remain within additive distance $n$0 to $n$1 compared to $n$2 (Dadush et al., 2012).

2. Sparsification Construction via Modular Restrictions

The core construction uses random or carefully derandomized modular restrictions. For a fixed radius $n$3 and a convex body $n$4, set $n$5 and $n$6. Select a prime $n$7 such that $n$8 and select a (random or deterministic) vector $n$9. Define the sublattice

$B \in \mathbb{Q}^{n \times n}$0

This modular restriction discards the majority of lattice points but preserves a coarse covering of space. Key results rely on the properties of $B \in \mathbb{Q}^{n \times n}$1 reduced modulo $B \in \mathbb{Q}^{n \times n}$2 and probabilistic guarantees derived from uniform choices of $B \in \mathbb{Q}^{n \times n}$3.

Key Random-Sublattice Lemma

Given any $B \in \mathbb{Q}^{n \times n}$4 of size $B \in \mathbb{Q}^{n \times n}$5 with $B \in \mathbb{Q}^{n \times n}$6, a random $B \in \mathbb{Q}^{n \times n}$7 satisfies with constant probability:

1. At most $B \in \mathbb{Q}^{n \times n}$8 points of $B \in \mathbb{Q}^{n \times n}$9 are mapped to zero ($L = \{Bz : z \in \mathbb{Z}^n\}.$0).
2. The set of nonzero images $L = \{Bz : z \in \mathbb{Z}^n\}.$1 has size at least $L = \{Bz : z \in \mathbb{Z}^n\}.$2.

These properties guarantee simultaneously that $L = \{Bz : z \in \mathbb{Z}^n\}.$3 is sparse and that the additive metric distortion is controlled (Dadush et al., 2012).

3. Algorithmic Constructions

Two main approaches—randomized and deterministic—are detailed.

Randomized Construction

• Input: basis $L = \{Bz : z \in \mathbb{Z}^n\}.$4 for $L = \{Bz : z \in \mathbb{Z}^n\}.$5, convex body $L = \{Bz : z \in \mathbb{Z}^n\}.$6, radius $L = \{Bz : z \in \mathbb{Z}^n\}.$7.
• Enumerate $L = \{Bz : z \in \mathbb{Z}^n\}.$8, $L = \{Bz : z \in \mathbb{Z}^n\}.$9.
• If $K = -K \subset \mathbb{R}^n$0, set $K = -K \subset \mathbb{R}^n$1.
• Otherwise, pick $K = -K \subset \mathbb{R}^n$2 and random $K = -K \subset \mathbb{R}^n$3.
• Output $K = -K \subset \mathbb{R}^n$4.

With constant probability, $K = -K \subset \mathbb{R}^n$5 is a $K = -K \subset \mathbb{R}^n$6–sparsifier due to the random-sublattice lemma.

Deterministic Construction

Derandomization proceeds by searching for a suitable $K = -K \subset \mathbb{R}^n$7 via a projection argument:

• Project $K = -K \subset \mathbb{R}^n$8 to $K = -K \subset \mathbb{R}^n$9 through a sequence of injective linear maps.
• In dimension $\|x\|_K = \inf \{s \ge 0 : x \in sK \}.$0, exhaustively enumerate all $\|x\|_K = \inf \{s \ge 0 : x \in sK \}.$1 lines for the small-zero/large-image criteria.
• Lift a successful line to a vector $\|x\|_K = \inf \{s \ge 0 : x \in sK \}.$2.
• Output $\|x\|_K = \inf \{s \ge 0 : x \in sK \}.$3 as above.

All steps run in time $\|x\|_K = \inf \{s \ge 0 : x \in sK \}.$4 and space $\|x\|_K = \inf \{s \ge 0 : x \in sK \}.$5 (Dadush et al., 2012).

4. Structural Properties and Guarantees

A $\|x\|_K = \inf \{s \ge 0 : x \in sK \}.$6–sparsifier $\|x\|_K = \inf \{s \ge 0 : x \in sK \}.$7 produced by this method satisfies the following:

• Distance preservation: For every $\|x\|_K = \inf \{s \ge 0 : x \in sK \}.$8, $\|x\|_K = \inf \{s \ge 0 : x \in sK \}.$9 (randomized) or $y \in \mathbb{R}^n$0 (construction in Theorem 4.2; rescaling $y \in \mathbb{R}^n$1 gives $y \in \mathbb{R}^n$2-additive error).
• Sparsity: For every translate, the number of points in $y \in \mathbb{R}^n$3 is at most $y \in \mathbb{R}^n$4. Specifically, $y \in \mathbb{R}^n$5.
• Trade-off: Choice of prime $y \in \mathbb{R}^n$6 plays a key role in balancing enumeration cost (number of points to check) and additive distortion.

The proof uses properties of sumsets and bounds from additive combinatorics (e.g., Cauchy–Davenport theorem) to ensure metric coverage under repeated sum decompositions (Dadush et al., 2012).

5. Complexity and Limitations

• Randomized algorithm complexity: Time $y \in \mathbb{R}^n$7; space $y \in \mathbb{R}^n$8.
• Deterministic algorithm: Matches above asymptotics by replacing probabilistic steps with explicit search.

A table of principal resource requirements:

Algorithm Type	Time Complexity	Space Complexity
Randomized	$y \in \mathbb{R}^n$9	$d_K(L, y) = \min_{x \in L} \|x - y\|_K.$0
Deterministic	$d_K(L, y) = \min_{x \in L} \|x - y\|_K.$1	$d_K(L, y) = \min_{x \in L} \|x - y\|_K.$2

The constant in $d_K(L, y) = \min_{x \in L} \|x - y\|_K.$3 for general norms arises from enumeration subroutines and remains a subject for further optimization.

Several limitations are identified:

• The $d_K(L, y) = \min_{x \in L} \|x - y\|_K.$4 additive distortion is currently inherent due to the three-sumset argument.
• Further reductions might require more advanced combinatorial or sumset techniques.
• For $d_K(L, y) = \min_{x \in L} \|x - y\|_K.$5 norms, leveraging improved exact CVP solvers can reduce space to $d_K(L, y) = \min_{x \in L} \|x - y\|_K.$6 with the same time asymptotics (Dadush et al., 2012).

6. Integration into $d_K(L, y) = \min_{x \in L} \|x - y\|_K.$7–Approximate CVP

Lattice sparsification is instrumental in accelerating the $d_K(L, y) = \min_{x \in L} \|x - y\|_K.$8–CVP algorithm:

1. Guess a rough upper bound $d_K(L, y) = \min_{x \in L} \|x - y\|_K.$9 on $$G(tK, L) = \max_{z \in \mathbb{R}^n} |(z + tK) \cap L|$$0.
2. Apply sparsification to obtain $$G(tK, L) = \max_{z \in \mathbb{R}^n} |(z + tK) \cap L|$$1 with parameter $$G(tK, L) = \max_{z \in \mathbb{R}^n} |(z + tK) \cap L|$$2. Then $$G(tK, L) = \max_{z \in \mathbb{R}^n} |(z + tK) \cap L|$$3.
3. Enumerate the $$G(tK, L) = \max_{z \in \mathbb{R}^n} |(z + tK) \cap L|$$4-ball in $$G(tK, L) = \max_{z \in \mathbb{R}^n} |(z + tK) \cap L|$$5, whose size is $$G(tK, L) = \max_{z \in \mathbb{R}^n} |(z + tK) \cap L|$$6.
4. Output the closest vector found.

Enumeration over $$G(tK, L) = \max_{z \in \mathbb{R}^n} |(z + tK) \cap L|$$7 now becomes tractable for arbitrary norms, with run time and space matching the enumeration bounds provided by sparsification. This deterministic approach offers an alternative to the AKS Sieve-based algorithms, allowing polynomial space if improved exact $$G(tK, L) = \max_{z \in \mathbb{R}^n} |(z + tK) \cap L|$$8-CVP solvers exist (Dadush et al., 2012).

7. Extensions and Open Questions

Potential extensions include:

• Optimizing the constant factor in $$G(tK, L) = \max_{z \in \mathbb{R}^n} |(z + tK) \cap L|$$9 for general norms.
• Reducing or eliminating the additive distortion via combinatorial innovation (possibly higher-order sumsets).
• Space reduction through hybrid use with exact CVP solvers under $L \subset \mathbb{R}^n$00 for polynomial space and $L \subset \mathbb{R}^n$01 time.

These advances suggest a rich interplay between additive combinatorics, algorithm design, and convex geometry, with lattice sparsification providing a critical bridge for efficient algorithmic solutions in the geometry of numbers (Dadush et al., 2012).