Pseudorandom Functions in $\mathsf{NC}^1$ from LWE/LPN/CDH (Or: How to Build Low-Depth PRFs, Generically)
Youlong Ding, Aayush Jain, Ilan Komargodski
We obtain new $\mathsf{NC}^1$-computable pseudorandom functions (PRFs) from various standard 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 \textbf{Learning Parity with Noise (LPN)} assumption, removing the need for structured LPN variants [Boyle et al., FOCS 2020], [Ding-Jain-Komargodski, STOC 2025], [Ding-Jain-Komargodski, TCC 2026]; (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]. Technically, our results follow from a new generic transformation that converts weak PRFs computable in depth $d(n) = Ω(\log n)$ to (strong) PRFs computable in depth $O(d(n))$. Roughly speaking, our approach combines \emph{tapering} the internal state of the classical GGM tree (so that the per-leveldepth decreases geometrically) with a two-stage security proof that transitions from a computational to an information-theoretic argument.
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