Schwartz-Zippel for multilinear polynomials mod N

We derive a tight upper bound on the probability over $\mathbf{x}=(x1,\dots,x\mu) \in \mathbb{Z}^\mu$ uniformly distributed in $ [0,m)^\mu$ that $f(\mathbf{x}) = 0 \bmod N$ for any $\mu$-linear polynomial $f \in \mathbb{Z}[X1,\dots,X\mu]$ co-prime to $N$. We show that for $N=p1^{r1},...,p\ell^{r\ell}$ this probability is bounded by $\frac{\mu}{m} + \prod{i=1}^\ell I{\frac{1}{pi}}(ri,\mu)$ where $I$ is the regularized beta function. Furthermore, we provide an inverse result that for any target parameter $\lambda$ bounds the minimum size of $N$ for which the probability that $f(\mathbf{x}) \equiv 0 \bmod N$ is at most $2^{-\lambda} + \frac{\mu}{m}$. For $\mu =1$ this is simply $N \geq 2^\lambda$. For $\mu \geq 2$, $\log2(N) \geq 8 \mu^{2}+ \log2(2 \mu)\cdot \lambda$ the probability that $f(\mathbf{x}) \equiv 0 \bmod N$ is bounded by $2^{-\lambda} +\frac{\mu}{m}$. We also present a computational method that derives tighter bounds for specific values of $\mu$ and $\lambda$. For example, our analysis shows that for $\mu=20$, $\lambda = 120$ (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 $\mu$ and $\lambda$ values.