A pr 2 02 2 Schwartz-Zippel for multilinear polynomials mod N

We derive a tight upper bound on the probability over x = (x1, . . . , xμ) ∈ Z uniformly distributed in [0, m) that f(x) = 0 mod N for any μ-linear polynomial f ∈ Z[X1, . . . , Xμ] coprime toN . We show that forN = p1 1 , ..., p rl l this probability is bounded by μ m + ∏l i=1 I 1 pi (ri, 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(x) ≡ 0 mod N is at most 2 + μ m . For μ = 1 this is simply N ≥ 2. For μ ≥ 2, log2(N) ≥ 8μ + log2(2μ) ·λ the probability that f(x) ≡ 0 mod N is bounded by 2 + μ 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 log2(N) ≥ 416 the probability is bounded by 2 + 20 m . We provide a table of computational bounds for a large set of μ and λ values.