A Note on Sphere Packing Bounds for Tuple Lattice Sieving
Proves upper bounds on k-irreducible unit vector set rates, yielding nearly tight asymptotics relevant to tuple lattice sieving in cryptanalysis.
The paper bounds the maximal asymptotic rate of k-irreducible sets of unit vectors via spherical code packing bounds. It shows R_k is sandwiched between (1/2 - o(1)) log2(k)/k and (1 + o(1)) log2(k)/k for large k. These almost-tight bounds inform subexponential complexity analyses of tuple lattice sieving, which underpins security estimates for lattice-based cryptography.
- Defines k-irreducible vector sets where signed sums of two to k elements exceed unit norm
- Relates maximal rate R_k to spherical codes with bounded pairwise inner products
- Derives upper bounds via sphere packing for all k at least 2
- For large k, upper and lower bounds are tight up to roughly a factor of two
From the original article
1 min read at arxiv.orgA finite set of unit vectors is $k$-irreducible if every signed sum of between two and $k$ distinct elements has norm greater than one. Let $\mathcal{R}_k$ be the maximal asymptotic rate of such sets, and let $κ(α)$ be the maximal asymptotic rate of spherical codes with pairwise inner products at most $α$. For $k \ge 2$ we show: \begin{align} \mathcal{R}_k \le \min_{1 \le r \le \lfloor k/2 \rfloor} \frac{1}{r} \, κ\!
By Thijs Laarhoven · arxiv.org
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