Solving the Shortest Vector Problem in $2^{0.6039n}$ Time via Mid-point Hessian

Author: Minki Hhan
arXiv Identifier: 2608.02478
Categories: Data Structures and Algorithms (cs.DS); Cryptography and Security (cs.CR)

📄 Executive Summary

This research introduces novel randomized algorithms designed to tackle the Shortest Vector Problem (SVP) within an $n$ -dimensional lattice $\mathcal{L}$ . By leveraging the properties of the Hessian of periodic Gaussian functions, the author significantly reduces the time complexity compared to previous benchmarks.

The primary breakthrough involves analyzing the Hessian at the half shortest vector. Specifically, if $v$ is a shortest vector in $\mathcal{L}$ , the Hessian evaluated at $v/2$ possesses an eigenvector that aligns closely with $v$ . This relationship allows for the recovery of $v$ through the application of a Bounded Distance Decoding (BDD) algorithm.

🚀 Complexity Analysis

The proposed method represents a substantial leap over the previous state-of-the-art established by Aggarwal, Dadush, Regev, and Stephens-Davidowitz in [STOC'15].

Metric	Previous Best (STOC'15)	New Algorithm (Hhan)
Classical Time	~~$2^{n+o(n)}$~~ $\rightarrow$	$2^{0.6039n+o(n)}$
Quantum Time	N/A	$2^{0.5411n+o(n)}$
Space Complexity	$2^{n+o(n)}$	$2^{0.5n+o(n)}$

🛠️ Technical Methodology

The algorithm operates by searching for the correct parity class of the shortest vector. Because of the periodicity modulo $\mathcal{L}$ , the potential midpoints are categorized by parity classes within the quotient group $\mathcal{L}/2\mathcal{L}$ .

The Process Flow

Implementation Details

To achieve the final complexity, the author implemented several optimizations:

Integration of random sublattice cosets.
Application of advanced sampling techniques.
Utilization of the periodic Gaussian function.

The mathematical core relies on the fact that the Hessian $\mathbf{H}$ at $v/2$ satisfies: $\text{eig}(\mathbf{H}_{v/2}) \approx v$

💻 Algorithmic Logic (Conceptual)

While the full paper provides the rigorous proof, the logic can be summarized in the following pseudo-structure:

def solve_svp(lattice_L):
    # Iterate through parity classes in L/2L
    for parity_class in L_over_2L:
        # Estimate the Hessian using discrete Gaussian samples
        hessian_est = estimate_hessian(parity_class, samples=GAUSSIAN_SAMPLES)
        
        # Find the principal eigenvector
        v_approx = compute_eigenvector(hessian_est)
        
        # Use BDD to snap the approximation to the actual lattice vector
        v_final = BDD_algorithm(lattice_L, v_approx)
        
        if is_shortest(v_final):
            return v_final

📅 Submission History & Metadata

First Version (v1): Submitted Monday, 3 Aug 2026.
Revised Version (v2): Updated Tuesday, 4 Aug 2026 (Corrected title).
DOI: https://doi.org/10.48550/arXiv.2608.02478

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This work is hosted on arXiv and is part of the broader effort to optimize lattice-based cryptographic analysis.

Solving the Shortest Vector Problem in 20.6039n2^{0.6039n}20.6039n Time via Mid-point Hessian