排列之和的不可区分性：通往经典与量子安全的傅里叶分析路径

中文摘要 AI 辅助

我们研究独立随机排列之和以及从排列到函数的相关变换的经典与量子不可区分性。设 $G$ 为阶为 $N$ 的有限阿贝尔群，并设 $\npi^k_+(x)=\npi_1(x)+\ncdots+\npi_k(x)$，其中 $k\ngeq2$ 个独立的均匀随机排列作用于 $G$。我们给出统一的傅里叶分析处理，其中构造由其概率密度表示，区分器由其接受函数表示，经典与量子查询模型对后者的傅里叶支撑施加不同限制。在经典情形下，对于每个 $q<N$，我们得到界 $O_k(q/N^{k-1/2})$，并在生日阈值以下将其改进为 $O_k(q^2/N^k)$。在量子模型中，模拟论证给出 $q\nleq(N-1)/2$ 时的界 $O_k(N^{-(k-3/2)})$，而傅里叶插值给出 $q\nleq4N/15$ 的具体有限界，以及在整个 $1\nleq q\nleq(N-1)/2$ 范围内，对于 $k=2$ 和 $k\ngeq3$ 分别给出依赖于查询数的界 $O\nleft(\nmin\nleft\{N^{-1/2},q^3/N^2 + 1/N\nright\}\nright)$ 和 $O_k\nleft(\nmin\nleft\{q^3/N^k,N^{-(k-3/2)}\nright\}\nright)$。对于 $q=1$，第一个界改进为 $O(N^{-2})$。在 $G=\nmathbb F_2^n$ 上，单查询傅里叶攻击达到我们单查询界的阶，而优势为 $1/2$ 的 $N/2$ 查询奇偶攻击表明我们的界达到常数优势查询阈值。我们进一步研究二元向量空间上排列之和的两种变体。首先，我们允许任意满射线性后处理（包括截断），并获得保留输出大小依赖性的经典与量子界。其次，我们分析 Dinur 的可变输出单排列构造 $\mathsf{LXoP}$，针对每个固定输出宽度，推导其经典与量子安全界；对于单块和双块输出，我们给出具体的量子安全界。

英文摘要

We study classical and quantum indistinguishability of sums of independent random permutations and related transformations from permutations to functions. Let $G$ be a finite abelian group of order $N$, and let $π^k_+(x)=π_1(x)+\cdots+π_k(x)$ for $k\geq2$ independent uniform random permutations of $G$. We give a unified Fourier analytic treatment in which the construction is represented by its probability density and a distinguisher by its acceptance function, with the classical and quantum query models imposing different restrictions on the Fourier support of the latter. Classically, we obtain the bound $O_k(q/N^{k-1/2})$ for every $q<N$, and refine it below the birthday threshold to $O_k(q^2/N^k)$. In the quantum model, a simulation argument gives $O_k(N^{-(k-3/2)})$ for $q\leq(N-1)/2$, while Fourier interpolation gives concrete finite bounds up to $q\leq4N/15$ and the query-dependent bounds $O\left(\min\left\{N^{-1/2},q^3/N^2 + 1/N\right\}\right)$ and $O_k\left(\min\left\{q^3/N^k,N^{-(k-3/2)}\right\}\right)$, for $k=2$ and $k \geq 3$, respectively, throughout $1\leq q\leq(N-1)/2$. For $q = 1$, the first bound sharpens to $O(N^{-2})$. Over $G=\mathbb F_2^n$, a one-query Fourier attack matches the order of our one-query bound, while an $N/2$-query parity attack with advantage $1/2$ shows that our bounds reach the constant-advantage query threshold. We further study two variants of sum of permutations over binary vector spaces. First, we allow arbitrary surjective linear postprocessing, which includes truncation, and obtain classical and quantum bounds that retain the output-size dependence. Second, we analyse Dinur's variable-output single-permutation construction, $\mathsf{LXoP}$, for every fixed output width, and derive its classical and quantum security bounds; for one- and two-block outputs, we give concrete quantum security bounds.

发表机构

• CRC, TII（阿布扎比计算研究中心与技术创新研究所）
• Shandong University（山东大学）
• University of Wuppertal（伍珀塔尔大学）

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