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Kazuki Ogitsuka

Publications and source records attributed to Kazuki Ogitsuka.

2 recordsLinked to original sources
arXiv2026

On the Complexity of Locally Dense Lattices

\emph{Locally dense lattices} are central gadgets used to prove the hardness of the Shortest Vector Problem and related lattice problems. Informally, a locally dense lattice is a lattice $\mathcal{L}$ that contains exponentially many lattice vectors inside some $\ell_p$ ball centered at $\vec{s}$ with radius at most an $α< 1$ fraction of the length of its shortest nonzero lattice vector. In this paper, taking a ``meta'' viewpoint on locally dense lattices, we introduce the \emph{Locally Dense Lattice Problem} (LDLP), the decision problem of determining whether a given input specifies a locally dense lattice. Our main result is that LDLP in $\ell_p$ norms for all finite $p \geq \log_2 3$ and for the infinity norm is complete for the second level of the polynomial hierarchy. We also compare two standard definitions of local density that appear in prior work. Micciancio's original definition (FOCS 1998 and SICOMP 2001) uses integer coefficient vectors, while later work by Micciancio (ToC 2012) and by Bennett and Peikert (RANDOM 2023) uses short vectors in a shifted coset. We show that the corresponding promise problems are mutually reducible in deterministic polynomial time, which shows that the two formulations are robust.

cs.CC↗
arXiv2026

One-Sided-Error Parameterized Reductions for the Minimum Distance and Shortest Vector Problems

It is notoriously difficult to obtain deterministic reductions for the Minimum Distance Problem (MDP) and the Shortest Vector Problem (SVP). Under two-sided-error randomized reductions, Bennett, Cheraghchi, Guruswami, and Ribeiro (STOC 2023) proved parameterized hardness of approximation for these problems. We partially derandomize their reductions and present one-sided-error randomized reductions: MDP is W[1]-hard to approximate within an arbitrary constant factor under FPT many-one one-sided-error randomized reductions; For every $p \ge 1$, SVP in the $\ell_p$ norm is W[1]-hard to approximate within an arbitrary constant factor below $2^{1/p}$. We demonstrate the usefulness of one-sided-error randomized reductions by showing that they can be conditionally derandomized when the target problem has an OR function. Under a standard hardness-vs-randomness assumption, namely a plausible lower-bound assumption against nondeterministic circuits, we prove a general theorem formalizing this derandomization. Here, an OR function combines several instances into one instance that preserves their disjunction. We construct such OR functions for the relevant MDP and SVP gap problems, and thereby obtain deterministic W[1]-hardness for approximating MDP over every fixed finite field within every constant factor, and for approximating SVP in $\ell_p$ norms for every fixed integer $p$ within every factor below $2^{1/p}$. Applying the same framework to Micciancio's one-sided-error randomized reduction (ToC 2012) yields, under the same circuit lower-bound assumption, deterministic polynomial-time NP-hardness of approximating Euclidean SVP within every constant factor.

cs.CC↗