We construct explicit pseudorandom generators that fool $n$-variate polynomials of degree at most $d$ over a finite field $\mathbb{F}_q$. The seed length of our generators is $O(d \log n + \log q)$, over fields of size exponential in $d$ and characteristic at least $d(d-1)+1$. Previous constructions such as Bogdanov's (STOC 2005) and Derksen and Viola's (FOCS 2022) had either suboptimal seed length or required the field size to depend on $n$. Our approach follows Bogdanov's paradigm while incorporating techniques from Lecerf's factorization algorithm (J. Symb. Comput. 2007) and insights from the construction of Derksen and Viola regarding the role of indecomposability of polynomials.

翻译：我们构造了显式伪随机生成器，能够欺骗有限域 $\mathbb{F}_q$ 上次数至多为 $d$ 的 $n$ 元多项式。在域大小关于 $d$ 呈指数级且特征至少为 $d(d-1)+1$ 的条件下，我们的生成器种子长度为 $O(d \log n + \log q)$。先前的工作，如 Bogdanov (STOC 2005) 以及 Derksen 与 Viola (FOCS 2022) 的构造，要么种子长度非最优，要么要求域大小依赖于 $n$。我们的方法遵循 Bogdanov 的范式，同时融合了 Lecerf 因式分解算法 (J. Symb. Comput. 2007) 的技术，以及 Derksen 与 Viola 关于多项式不可分解性作用的见解。