基于LWE/LPN/CDH的NC¹中的伪随机函数（或：如何通用地构造NC¹中的PRF）
Pseudorandom Functions in $\mathsf{NC}^1$ from LWE/LPN/CDH (Or: How to Build PRFs in $\mathsf{NC}^1$, Generically)
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中文总结 AI 辅助
该研究提出通用变换将弱PRF转为低深度强PRF，基于LWE、LPN、CDH假设首次构造出NC¹可计算的PRF，解决了多个长期开放问题。
中文摘要 AI 辅助
我们提出一种新的通用变换，将深度d(n)=Ω(log n)下可计算的弱伪随机函数（weak PRF）转换为深度O(d(n))下可计算的强伪随机函数（strong PRF）。该构造通过对内部状态进行“锥形化”处理，使得每一层的深度呈几何级数递减，从而改进了经典的GGM树基范式。我们还补充了基于各类标准假设的新的低深度弱PRF构造。作为推论，我们从各类经典假设中获得了新的NC¹可计算PRF，解决了若干长期存在的开放问题。具体而言，我们首次获得了以下NC¹可计算PRF：（1）基于带误差学习（Learning With Errors, LWE）假设，且具有多项式模数噪声比，改进了此前需要带超多项式比率的环-LWE的低深度构造[Banerjee-Peikert-Rosen, EUROCRYPT 2012]；（2）基于标准的带噪声奇偶学习（Learning Parity with Noise, LPN）假设，消除了对结构化LPN变体的需求[Boyle等人，FOCS 2020]、[Ding-Jain-Komargodski，STOC 2025]；（3）基于计算性Diffie-Hellman（Computational Diffie-Hellman, CDH）假设，而此前的工作依赖于更强的判定性Diffie-Hellman（Decisional Diffie-Hellman, DDH）或广义Diffie-Hellman（generalized Diffie-Hellman, GDH）假设[Naor-Reingold, FOCS '97, J. ACM '04]。
英文摘要
We present a new generic transformation from weak PRFs computable in depth $d(n) = Ω(\log n)$ to strong PRFs computable in depth $O(d(n))$. This construction refines the classical tree-based paradigm of GGM by {tapering} the internal state so the per-level depth decreases geometrically. We complement the above with new depth-efficient weak PRF constructions based on various standard assumptions. As a corollary, we obtain new $\mathsf{NC}^1$-computable PRFs from various classical assumptions, resolving several long-standing open problems. Concretely, for the first time, we obtain $\mathsf{NC}^1$-computable PRFs: (1) from the \textbf{Learning With Errors (LWE)} assumption with a polynomial modulus-to-noise ratio, improving upon prior low-depth constructions that required Ring-LWE with super-polynomial ratios [Banerjee-Peikert-Rosen, EUROCRYPT 2012]; (2)from the standard \textbf{Learning Parity with Noise (LPN)} assumption, removing the need for structured LPN variants [Boyle et al., FOCS 2020], [Ding-Jain-Komargodski, STOC 2025]; (3) from the \textbf{Computational Diffie-Hellman (CDH)} assumption; prior works relied on the stronger Decisional Diffie-Hellman (DDH) or generalized Diffie-Hellman (GDH) assumptions [Naor-Reingold, FOCS '97, J. ACM '04].
