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
Full article114 words · extracted from arxiv.org · click to collapse
A 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\!\left(1 - \frac{1}{2r}\right) \, . \end{align} Combining this with standard sphere packing bounds, for large $k$ we obtain an almost-tight asymptotic comparison with the known lower bounds: \begin{align} \left(\tfrac{1}{2}-o(1)\right) \, \frac{\log_2 k}{k} \le \mathcal{R}_k \le (1 + o(1)) \, \frac{\log_2 k}{k} \, . \end{align}
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.08190