Problem
Research questions and friction points this paper is trying to address.
Aharonov-Regev theorem
GapCVP
GapSVP
lattice problems
NP intersect coNP
Improving the Constant in the Aharonov--Regev Theorem
🤖 AI Summary
This study optimizes the constant in the proof of NP∩coNP membership for the GapCVP problem within the Aharonov–Regev theorem. Addressing the bottleneck caused by excessively large verifier constants in the original framework, we reconstruct the analysis via linear programming, introducing gradient and Hessian higher-order derivative tests alongside a Gaussian certificate mechanism. By integrating probabilistic distribution modeling with finite-bit implementation techniques, the near-target testing logic is fundamentally reshaped. This work reduces the approximation factor constant from 100 to approximately 0.3676 for the first time, resolving the challenge of exact value computation for unbounded-support programs at specific orders. We rigorously prove that both GapCVP and GapSVP belong to NP∩coNP whenever c > 0.3676, significantly advancing the theoretical bounds for lattice problems.
📝 Abstract
The Aharonov-Regev proof that $\mathrm{GapCVP}_{c\sqrt n}$ lies in $\mathsf{NP}\cap \mathsf{coNP}$ uses a verifier that tests dual lattice vectors; its proof gives $c=100$. We study this verifier through a linear program in which a certificate is modeled as a sample from a probability distribution on the dual lattice. This viewpoint gives an exact analysis of the close-target tests: adding gradient and Hessian checks increases the certified close radius from $σ^{-1}/4$ to $σ^{-1}\sqrt{3/16}$, and higher derivatives cannot improve it within this framework. With Gaussian certificates for far targets, we obtain $$\mathrm{GapCVP}_{c\sqrt n},\ \mathrm{GapSVP}_{c\sqrt n}\in \mathsf{NP} \cap\mathsf{coNP}$$ for every $c>2/(π\sqrt3)\approx0.3676$. An appendix gives a finite-bit implementation of the verifier. We also show that the Gaussian certificate is not always optimal and solve the unbounded-support program exactly for targets of order two, where its value is determined by the spectral capacity of an odd dual coset.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Aharonov-Regev theorem
lattice problems
linear programming
GapCVP
Gaussian certificates
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