Schwartz-Zippel for multilinear polynomials mod N

Benedikt Bünz and Ben Fisch

Abstract

We derive a tight upper bound on the probability over $\mathbf{x}=(x_1,\dots,x_\mu) \in \mathbb{Z}^\mu$ uniformly distributed in $[0,m)^\mu$ that $f(\mathbf{x}) = 0 \bmod N$ for any $\mu$-linear polynomial $f \in \mathbb{Z}[X_1,\dots,X_\mu]$ co-prime to $N$. We show that for $N=\prod_{i=1}^\ell p_i^{r_i}$ this probability is bounded by $\frac{\mu}{m} + \prod_{i=1}^\ell I_{\frac{1}{p_i}}(r_i,\mu)$ 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 $\mu \geq 2$, $\log_2(N) \geq 8 \mu^{2}+ \log_2(2 \mu)\cdot \lambda$ 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.

Available format(s)
Category
Foundations
Publication info
Preprint. Minor revision.
Keywords
Schwartz-Zippel
Contact author(s)
benedikt @ cs stanford edu
benafisch @ gmail com
History
2022-05-04: revised
See all versions
Short URL
https://ia.cr/2022/458

CC BY

BibTeX

@misc{cryptoeprint:2022/458,
author = {Benedikt Bünz and Ben Fisch},
title = {Schwartz-Zippel for multilinear polynomials mod N},
howpublished = {Cryptology ePrint Archive, Paper 2022/458},
year = {2022},
note = {\url{https://eprint.iacr.org/2022/458}},
url = {https://eprint.iacr.org/2022/458}
}

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