达到香农上限的秘密共享
Secret Sharing at the Shannon Ceiling
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中文总结 AI 辅助
本文针对n≥9且为3的倍数的参与者构造显式访问结构，通过改进Csirmaz的下界，利用香农不等式和平均论证使完美秘密共享方案达到总份额与最大份额的香农上限。
中文摘要 AI 辅助
对于每个满足n≥9且为3的倍数的n，我们构造了一个针对n个参与者的显式访问结构。在实现该访问结构的每个完美秘密共享方案中，若S表示随机秘密，则份额熵的总和至少为（n²/9 + 2n/3）H(S)，且存在某个参与者的份额熵至少为（n/6 + 1/2）H(S)。经H(S)归一化后，这些分别是Ω(n²)和Ω(n)级别的下界，同时也为份额的总预期二进制长度和最大预期二进制长度提供了相同的渐近下界。这将Csirmaz提出的长期通用下界（总份额大小为Ω(n²/log n)，最大份额大小为Ω(n/log n)）改进了一个对数因子。证明仅使用基本的香农不等式及一些平均论证。香农信息方法可证明的总归一化熵下界和最大归一化熵下界分别具有通用的O(n²)和O(n)上限，因此我们的构造在常数因子范围内达到了这两个上限。
英文摘要
For every $n\geq 9$ that is a multiple of 3, we construct an explicit access structure on $n$ participants. In every perfect secret-sharing scheme realising this access structure, if $S$ denotes the random secret, then the sum of the share entropies is at least $\left(\frac{n^2}{9}+\frac{2n}{3}\right)H(S)$, and some participant has share entropy at least $\left(\frac{n}{6}+\frac12\right)H(S)$. After normalisation by $H(S)$, these are respectively $Ω(n^2)$ and $Ω(n)$ lower bounds and also give the same asymptotic lower bounds on the total and largest expected binary lengths of the shares. This improves by a logarithmic factor the longstanding general lower bounds of $Ω(n^2/\log n)$ for total share size and $Ω(n/\log n)$ for maximum share size due to Csirmaz. The proof uses only elementary Shannon inequalities, together with some averaging arguments. The Shannon-information method has universal $O(n^2)$ and $O(n)$ ceilings for the total and maximum normalised entropy lower bounds it can certify, so our construction reaches both ceilings up to constant factors.
