低次多项式在多项式大小域上的最优伪随机生成器
Optimal PRGs for Low-Degree Polynomials over Polynomial-Size Fields
AI总结：
本文提出了一种在多项式大小域上构造低次多项式最优种子长度PRG的方法，展示了域大小与特征之间的阈值现象。
摘要：
伪随机生成器（PRGs）对于低次多项式是伪随机性中的核心对象，应用于电路下界和去随机化。Viola的著名构造给出了二元域上的PRG，但种子长度与次数d呈指数关系。这种指数依赖性可以在足够大的域上避免。特别是，Dwivedi、Guo和Volk构造了在域大小为指数级的d上的最优种子长度PRG。后者基于Derksen和Viola的框架，后者在域大小为多项式级的d上获得了最优种子构造，尽管随着变量数n的增长而增长。在本工作中，我们构造了第一个在多项式大小域上具有最优种子长度的度d多项式PRG，具体为q≈d⁴，假设足够大的特征。我们的构造遵循先前工作的框架，并通过用新的伪随机对象替代先前构造中的击中集生成器来减少所需的域大小。我们还观察到域大小依赖性中的阈值现象。具体来说，我们证明了在域大小为亚线性规模（例如q=d^0.99，其中q是2的幂）的情况下，通过我们的减少，已经可以为二元域构造具有可比种子长度的PRG，只要构造不施加对特征的限制。尽管已知现有技术之前曾出现过崩溃，但我们证明这种现象本身是问题固有的，与所用技术无关。
英文摘要：
We construct the first pseudorandom generator (PRG) with optimal seed length for degree-$d$ polynomials over fields of size $q \approx d^4$, with no restriction on the characteristic. This improves on the construction of Dwivedi, Guo, and Volk (RANDOM 2024), which requires fields of size exponential in $d$ and sufficiently large characteristic, and on that of Derksen and Viola (FOCS 2022), which requires a field size that grows with the number of variables $n$. A recent breakthrough by Chattopadhyay, Hatami, Lee, Lovett, Tal, and Viola (ECCC 2026) gives a PRG over the binary field with seed length $\widetilde{O}(d^2\log^2 n)$, with subsequent work by Cohen (ECCC 2026) achieving seed length $O(d^4\log n)$. Both constructions use polarized random walks, and the current analyses incur at least a quadratic dependence on $d$. Our second result is a reduction that provides an alternative route to nearly optimal seed length over the binary field through improved constructions over larger fields. Specifically, we prove that, for any $δ\in (0,1)$, an optimal-seed PRG for degree-$d$ polynomials over fields of characteristic two and size $q \approx d^δ$ would yield a PRG over the binary field with seed length $O(d^{1/(1-δ)}\log n)$.
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