论文标题
Schwartz-Zippel用于多项式多项式mod n
Schwartz-Zippel for multilinear polynomials mod N
论文作者
论文摘要
我们在$ \ mathbf {x} =（x_1，\ dots，x_μ）\ in \ mathbb {z}^μ$中均匀分布在$ [0，m）^μ$ th $ f（\ mathbf {x}）= 0 \ bmod n $ - in \ n $ - \ Mathbb {Z} [x_1，\ dots，x_μ] $ co-Prime至$ n $。我们证明，对于$ n = p_1^{r_1}，...，p_ \ ell^{r_ \ ell} $，此概率由$ \fracμ{m} + \ prod_ {i = 1}^\ ell I _ {i = 1}^\ ell I _ {此外，我们提供了一个反向结果：对于任何目标参数，$λ$限制了$ f（\ mathbf {x}}）\ equiv 0 \ bmod n $的最小尺寸的最小尺寸，最多最多是$ 2^{ - λ} + \fracμ{m} $。对于$μ= 1 $，这只是$ n \ geq 2^λ$。对于$μ\ geq 2 $，$ \ log_2（n）\ geq8μ^{2} + \ log_2（2μ）\cdotλ$ $ f（\ mathbf {x}）\ equiv 0 \ bmod n $的可能性由$ 2^{ - λ} + \ frac}限制。我们还提出了一种计算方法，该方法针对$μ$和$λ$的特定值更加范围。例如，我们的分析表明，$μ= 20 $，$λ= 120 $（典型的加密应用程序中值），$ \ log_2（n）\ geq 416 $该概率受$ 2^{ - 120}+\ frac {20}} {M} $的限制。我们为大型$μ$和$λ$值提供了一张计算界表。
We derive a tight upper bound on the probability over $\mathbf{x}=(x_1,\dots,x_μ) \in \mathbb{Z}^μ$ uniformly distributed in $ [0,m)^μ$ that $f(\mathbf{x}) = 0 \bmod N$ for any $μ$-linear polynomial $f \in \mathbb{Z}[X_1,\dots,X_μ]$ co-prime to $N$. We show that for $N=p_1^{r_1},...,p_\ell^{r_\ell}$ this probability is bounded by $\fracμ{m} + \prod_{i=1}^\ell I_{\frac{1}{p_i}}(r_i,μ)$ where $I$ is the regularized beta function. Furthermore, we provide an inverse result that for any target parameter $λ$ bounds the minimum size of $N$ for which the probability that $f(\mathbf{x}) \equiv 0 \bmod N$ is at most $2^{-λ} + \fracμ{m}$. For $μ=1$ this is simply $N \geq 2^λ$. For $μ\geq 2$, $\log_2(N) \geq 8 μ^{2}+ \log_2(2 μ)\cdot λ$ the probability that $f(\mathbf{x}) \equiv 0 \bmod N$ is bounded by $2^{-λ} +\fracμ{m}$. We also present a computational method that derives tighter bounds for specific values of $μ$ and $λ$. For example, our analysis shows that for $μ=20$, $λ= 120$ (values typical in cryptography applications), and $\log_2(N)\geq 416$ the probability is bounded by $ 2^{-120}+\frac{20}{m}$. We provide a table of computational bounds for a large set of $μ$ and $λ$ values.
