Closest Vector Problem (CVP)

• Closest Vector Problem (CVP) is a computational challenge defined on a lattice basis where the goal is to find the lattice vector nearest to a target under a given norm.
• Researchers employ exact and approximate algorithms, including randomized methods, lattice sparsification, and geometric covering techniques, to tackle CVP efficiently.
• CVP’s computational hardness underpins post-quantum cryptographic schemes and influences optimization, integer programming, and coding theory with broad practical applications.

The Closest Vector Problem (CVP) is a foundational computational problem in lattice theory, discrete optimization, and cryptography. Given a full-rank lattice $\Lambda \subset \mathbb{R}^n$, specified by a basis $B \in \mathbb{R}^{n \times n}$, and a target vector $t \in \mathbb{R}^n$, CVP asks for a lattice vector $v \in \Lambda$ that minimizes $\|t - v\|$ with respect to a specified norm (typically $\ell_p$). The intractability of CVP underpins the security of post-quantum cryptographic schemes, and its computational complexity has been the subject of intense research due to its connections with complexity theory, integer programming, coding theory, and geometric algorithms.

1. Formal Definition and Structural Variants

The formal CVP is as follows: For a given lattice $$\Lambda = \text{span}_{\mathbb{Z}}\{b_1, \ldots, b_n\}$$ (with basis $B$), and a target $t \in \mathbb{R}^n$, find $v \in \Lambda$ minimizing $\|t - v\|_p$. Variants include:

• Exact CVP: Output $v$ satisfying $$\|t - v\|_p = d_p(t, \Lambda) := \min_{w \in \Lambda} \|t - w\|_p$$.
• Approximate CVP: Output $v$ with $\|t - v\|_p \leq \gamma(n) \cdot d_p(t, \Lambda)$ for some $\gamma(n) > 1$.
• Preprocessing CVP (CVPP): The lattice is preprocessed arbitrarily (with unbounded space/time), and then, given $t$, answer CVP for that particular target efficiently (Dadush et al., 2014, Bonifas et al., 2014).
• (0,1)-CVP: Restrict lattice vectors to be $\{0,1\}$ combinations of the basis (Abboud et al., 7 Jan 2025).

For $\ell_2$ (Euclidean) norm, CVP is strongest in cryptographic applications, but other norms such as $\ell_p$, especially $\ell_\infty$, are studied for their role in complexity-theoretic analyses (Bennett et al., 2017, Aggarwal et al., 2019).

2. Exact Algorithms and Complexity Barriers

CVP is known to be NP-hard for the Euclidean and most other norms, and remains NP-hard to approximate within some small constant factor (Bennett et al., 2017). The best known exact algorithm (for general lattices in $\ell_2$) is randomized and runs in time $2^{n + o(n)}$ and space $2^{n + o(n)}$ (Aggarwal et al., 2015). Earlier, deterministic Voronoi cell-based methods achieved $2^{O(n)}$ time and $2^{O(n)}$ space (Hunkenschröder et al., 2018, Bonifas et al., 2014).

Barriers and Fine-Grained Complexity: Reductions from $k$-SAT establish no $2^{(1-\varepsilon)n}$-time algorithm exists for CVP$_p$ with odd $p \geq 1$ and $p=\infty$ (under SETH), but these reductions fundamentally fail for $p=2$ (Euclidean norm) and other even $p$ (Bennett et al., 2017, Aggarwal et al., 2019). The inherent rigidity of Euclidean CVP solutions, as analyzed by additive combinatorics (notably through Szemerédi's cube lemma on boolean cubes), restricts the possible structure of solution sets; in particular, closest vector sets cannot simulate all $3$-CNF structure in $\ell_2$ (Aggarwal et al., 2019). This structural limitation is pivotal in explaining the current inability to rule out $2^{(1-\varepsilon)n}$-time algorithms for exact CVP$_2$ via natural reductions.

Problem Variant	Best Known Complexity	Hardness Result
CVP (exact, $\ell_2$)	$2^{n + o(n)}$ randomized	No $2^{(1-\varepsilon)n}$ known (barrier) (Bennett et al., 2017Aggarwal et al., 2019)
CVP, $p$ odd or $\infty$	$2^{n + o(n)}$	SETH-hard for $2^{(1-\varepsilon)n}$ (Bennett et al., 2017Aggarwal et al., 2019)
(0,1)-CVP$_2$	$O(1.7299^n)$	Equivalence to Max-$p$-SAT, clique (Abboud et al., 7 Jan 2025)

Special Cases and Algorithmic Breakthroughs: In the restricted (0,1)–CVP (where lattice vectors are 0–1 combinations of basis vectors), recent (Abboud et al., 7 Jan 2025) results break the $2^n$ barrier, achieving $O(1.7299^n)$ time for exact CVP$_2$ when basis/target coordinates are bounded by $2^{o(n)}$. This exploits reductions to weighted Max-SAT and $k$-clique, achieving computational equivalence (for even $p$) with well-understood problems.

3. Approximation Algorithms: Lattice Sparsification and Geometric Techniques

A core theme in recent work is achieving improved approximations in high-dimensional CVP via lattice sparsification (Dadush et al., 2012, Naszódi et al., 2019). The central idea is to construct a sublattice $L'$ such that $d_K(L', x) \leq d_K(L, x)+t$ (for norm ball $K$), with low local density, reducing enumeration cost. This enables deterministic $(1+\varepsilon)$-approximate CVP solvers in time $2^{O(n)} (1+1/\varepsilon)^n$ and $2^n \operatorname{poly}(n)$ space, improving the dependency on $\varepsilon$ and derandomizing earlier sieve-based approaches.

Geometric covering arguments improve the dependence of $(1+\varepsilon)$-CVP on $\varepsilon$ for $\ell_p$ norms:

• For $p>2$: $2^{O(n)}(1/\varepsilon)^{n/2}$ time;
• For $p \in [1,2]$: $2^{O(n)}(1/\varepsilon)^{n/p}$ (Naszódi et al., 2019).

These improvements leverage coverings of convex bodies with their homothets (parameterized by the modulus of smoothness), boosting coarser solvers to $(1+\varepsilon)$-accuracy.

Heuristic approaches such as nearest-colattice algorithms (Espitau et al., 2020) apply filtration and projection strategies, achieving HSVP-equivalent distance guarantees and efficient batch solving after precomputation.

4. Quantum and Classical Algorithms in Special Regimes

Recent work explores leveraging both deep learning and quantum algorithms for CVP:

• Deep Neural Solvers for Structured Lattices: For root lattices, folding symmetries dramatically reduce the number of decision regions characterizing the CVP, yielding polynomial-time solution by deep ReLU networks, where shallow networks would require exponential width (Corlay et al., 2019).
• QUBO and Optimization Techniques: CVP can be efficiently reduced to quadratic unconstrained binary optimization (QUBO) formulations with $O(n^2(\log n + \log b))$ binary variables, permitting solution via Ising-type quantum annealers and classical QUBO solvers, with polynomial reduction overhead (Canale et al., 2023).
• Quantum Approximate Optimization Algorithm (QAOA): For CVP instances arising in lattice-based sieving for integer factorization (the “prime lattice”), heuristic application of QAOA with pre-trained, fixed angles yields an effective sampling complexity $O(2^{0.225 n})$ (for circuit depth $p=10$), exceeding the quadratic speed-up of Grover’s search in these structured CVP settings (Priestley et al., 11 Mar 2025). Here, the pre-training scheme allows parameter sets to generalize across increasing lattice dimensions, facilitating scalability for sieving in cryptanalytic settings.

5. Structural Rigidity, Reductions, and Hardness

Fine-grained complexity results show exact and approximate CVP in $\ell_p$ with $p \notin 2\mathbb{Z}$ (odd $p$ and $p=\infty$) are SETH-hard, i.e., no $2^{(1-\varepsilon)n}$-time algorithm exists unless SETH fails (Bennett et al., 2017, Aggarwal et al., 2019). For the Euclidean norm ($p=2$), the situation is subtler:

• Structural Rigidity: Closest vector solutions in $\ell_2$ exhibit combinatorial rigidity, forbidding arbitrary expressiveness required to encode generic NP-complete problems (notably $k$-SAT).
• Szemerédi's Cube Lemma: This lemma, a key tool from additive combinatorics, guarantees the existence of affine cubes (i.e., large structured subspaces) in any sufficiently large subset of $\{0,1\}^n$. The inability to realize arbitrary affine cubes among the coordinates (modulo 2) of closest vectors in CVP$_2$ instances shows a fundamental barrier: natural reductions from $k$-SAT to CVP$_2$ lack the flexibility needed, as the set of closest vectors' mod-2 projections cannot reproduce arbitrary $3$-CNF satisfiability structure (Aggarwal et al., 2019). This demonstrates a true boundary for proof methods—faster than $2^n$ exact Euclidean CVP algorithms cannot be ruled out by known (natural) reductions.

Setting	SETH-hardness (for $2^{(1-\varepsilon)n}$)?	Underlying Obstacle
CVP$_p$, $p$ odd/$\infty$	Yes (Bennett et al., 2017Aggarwal et al., 2019)	Reduction via isolating parallelepipeds
CVP$_2$	No (barriers)	Combinatorial rigidity/Szemerédi cube lemma

This combinatorial rigidity blocks complexity-theoretic reductions, motivating research into “unnatural” reductions or alternate problem structures. The structural constraints deepen understanding of the landscape of computational hardness for lattice problems.

6. Applications and Broader Impact

CVP is not only central in theoretical computer science and cryptography but also arises in diverse applications:

• Cryptographic constructions: Lattice-based encryption schemes and signatures rely on the worst-case hardness of CVP for security. Advances or weaknesses in algorithms for CVP or its variants directly impact the parameter choices and security margins in post-quantum cryptography (Laarhoven, 2016, Li et al., 2017).
• Communications and Coding: CVP appears in lattice decoding, e.g., for error-correction in high-dimensional spaces, where efficiency and accuracy of CVP solvers influence practical system performance.
• Optimization: Integer programming and cutting plane methods often reduce, via rounding arguments, to solving CVP in various norms (Dadush et al., 2012).
• Biometrics and Fuzzy Extraction: Efficient linear-time CVP solvers for structured lattices (such as triangular lattices) directly enable fast cryptographic fuzzy extractors and signatures on high-dimensional biometric data (Takahashi et al., 8 Dec 2024).

Further, methods such as lattice sparsification, convex covering, and reductions to well-studied combinatorial optimization problems (Max-SAT, $k$-Clique) highlight the tight links between lattice algorithms and broader discrete optimization paradigms.

7. Future Directions and Open Problems

Several key avenues for ongoing research emerge:

• Exploring Hardness for CVP$_2$: Given that reductions based on affine cube structure are blocked, research may pursue more intricate (“unnatural”) reductions or explore other problem classes for potential connections.
• Algorithmic Improvements Under Structural Constraints: The (0,1)-CVP algorithm breaking the $2^n$ barrier (Abboud et al., 7 Jan 2025) points toward leveraging combinatorial decompositions and problem structure more generally.
• Quantum and Deep Learning Methods: Further refinement of quantum pre-training and neural network folding techniques for CVP-like problems could expand practical capability in large-scale or constrained classes of lattices (Corlay et al., 2019, Priestley et al., 11 Mar 2025).
• Understanding Clustering Phenomena in Gaussian Samplers: Analysis of discrete Gaussian clustering underlying efficient sampling and recursion for exact CVP could lead to tighter bounds or new algorithms (Aggarwal et al., 2015).
• Applications to New Domains: Adapting advanced CVP solvers to emerging settings, such as biometric cryptosystems, coding for next-gen communication, or integer optimization under non-symmetric norms.

The Closest Vector Problem thus serves both as a driver and a barometer for progress in computational geometry, cryptography, optimization, and the fine-grained paper of computational intractability. The interplay between hardness (barriers as formalized through combinatorial rigidity) and algorithmic innovation (as evidenced by breakthroughs in restricted domains and new connections) continues to fuel the development of both theory and practical methods.