← 研踪 PaperTrailarXiv:2610.12357 · v2
Improved Local Leakage Resilience of Shamir Secret Sharing and Worst-Case Optimal Polynomial Intersection
首次提交 2026-10-09 01:23
版本更新 2026-11-03 08:43
雷达观测 2026-11-04 23:40
摘要
We study two problems: Local Leakage Resilience (LLR) for Shamir secret sharing, and worst-case Optimal Polynomial Intersection (OPI), over a prime field $\mathbb{F}_p$. For a polynomial $Q(X)$ of degree less than $k$, LLR for Shamir secret sharing asks what a few leaked bits from each of the $n$ shares $Q(α_1), \ldots, Q(α_n)$ reveal about $Q(0)$. Given (worst-case) input lists $S_1, \ldots, S_n \subset \mathbb{F}_p$, OPI seeks a polynomial $Q(X)$ so that $Q(α_i) \in S_i$ for as many $i$ as possible. Leveraging recent connections between these two problems due to (Sun, Wootters 2026), we improve the state-of-the-art for both problems. For LLR, we show that there is some constant $δ> 0$ so that Shamir secret-sharing is one-bit LLR (only a negligible amount about $Q(0)$ is learned) whenever $R := k/n \geq 1/2 - δ$. This is the first result to break the so-called ``one-half barrier'' for LLR, and improves over the previous best known result, requiring $R \geq 0.668$ (Kasser, 2025). For OPI, we give a quantum algorithm that finds a polynomial $Q(X)$ that agrees with at least a $\mathsf{SCL}_ρ(R)-\varepsilon$ fraction of the lists, for every fixed $\varepsilon>0$, where $\mathsf{SCL}_ρ$ is the \emph{semicircle law} of (Jordan et al., 2025). This improves previous algorithmic (and existential) results of (Jo, 2026) and (Horinaga, Yamakawa, 2026). We also give further improved existential results for all OPI instances; and when the evaluation points $α_i$ are chosen at random, for every rate $R > 0$ we show that, with high probability, for worst-case $S_i$ there exists \emph{perfect} solution. Finally, we adapt the hardness result of (Yamakawa, Zhandry, 2024) to apply to OPI (rather than a folded version); this gives an unconditional separation between the quantum and classical hardness of OPI relative to a membership oracle.
作者备注：82 pages, 3 figures; v2 adds near-optimal OPI discrepancy for random Reed-Solomon codes
观测到的版本
相邻记录可能跳过中间版本；首次发现 v2 及以上也会标为版本更新。
v2 · 2026-11-03 08:43 · 最新观测版本
Improved Local Leakage Resilience of Shamir Secret Sharing and Worst-Case Optimal Polynomial Intersection
We study two problems: Local Leakage Resilience (LLR) for Shamir secret sharing, and worst-case Optimal Polynomial Intersection (OPI), over a prime field $\mathbb{F}_p$. For a polynomial $Q(X)$ of degree less than $k$, LLR for Shamir secret sharing asks what a few leaked bits from each of the $n$ shares $Q(α_1), \ldots, Q(α_n)$ reveal about $Q(0)$. Given (worst-case) input lists $S_1, \ldots, S_n \subset \mathbb{F}_p$, OPI seeks a polynomial $Q(X)$ so that $Q(α_i) \in S_i$ for as many $i$ as possible. Leveraging recent connections between these two problems due to (Sun, Wootters 2026), we improve the state-of-the-art for both problems. For LLR, we show that there is some constant $δ> 0$ so that Shamir secret-sharing is one-bit LLR (only a negligible amount about $Q(0)$ is learned) whenever $R := k/n \geq 1/2 - δ$. This is the first result to break the so-called ``one-half barrier'' for LLR, and improves over the previous best known result, requiring $R \geq 0.668$ (Kasser, 2025). For OPI, we give a quantum algorithm that finds a polynomial $Q(X)$ that agrees with at least a $\mathsf{SCL}_ρ(R)-\varepsilon$ fraction of the lists, for every fixed $\varepsilon>0$, where $\mathsf{SCL}_ρ$ is the \emph{semicircle law} of (Jordan et al., 2025). This improves previous algorithmic (and existential) results of (Jo, 2026) and (Horinaga, Yamakawa, 2026). We also give further improved existential results for all OPI instances; and when the evaluation points $α_i$ are chosen at random, for every rate $R > 0$ we show that, with high probability, for worst-case $S_i$ there exists \emph{perfect} solution. Finally, we adapt the hardness result of (Yamakawa, Zhandry, 2024) to apply to OPI (rather than a folded version); this gives an unconditional separation between the quantum and classical hardness of OPI relative to a membership oracle.
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