Logarithmic-Depth Pseudorandom Functions from Well-Founded Code-Based Assumptions

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We give the first $\mathsf{NC}^1$-computable Pseudorandom Function (PRF) constructions from well-founded (non-ad-hoc) code-based assumptions. Specifically, we give two constructions based on two different classical variants of the learning parity with noise (LPN) assumption: (1) An $\mathsf{NC}^1$-computable PRF from hardness of Sparse-LPN [Alekhnovich, FOCS~'03] (with respect to a sublinear-depth expander graph family). This PRF is also key-homomorphic. (2) An $\mathsf{NC}^1$-computab