Abstract
We introduce structured‑seed local pseudorandom generators (SSL-PRGs), pseudorandom generators whose seed is drawn from an efficiently sampleable, structured distribution rather than uniformly. This seemingly modest relaxation turns out to capture many known applications of local PRGs, yet it can be realized from a broader family of hardness assumptions. Our main technical contribution is a generic template for constructing SSL-PRGs that combines the following two ingredients: [i.] 1) noisy‑NC⁰ PRGs, computable by constant‑depth circuits fed with sparse noise, with 2) new local compression schemes for sparse vectors derived from combinatorial batch codes. Instantiating the template under the sparse Learning‑Parity‑with‑Noise (LPN) assumption yields the first SSL-PRGs with polynomial stretch and constant locality from a subquadratic‑sample search hardness assumption; a mild strengthening of sparse‑LPN gives strong SSL-PRGs of arbitrary polynomial stretch. We further show that for all standard noise distributions, noisy‑local PRGs cannot be emulated by ordinary local PRGs, thereby separating the two notions. Plugging SSL-PRGs into existing frameworks, we revisit the canonical applications of local PRGs and demonstrate that SSL-PRGs suffice for: (i) indistinguishability obfuscation, (ii) constant-overhead secure computation, (iii) compact homomorphic secret sharing, and (iv) deriving hardness results for PAC‑learning DNFs from sparse‑LPN. Our work thus broadens the landscape of low‑depth pseudorandomness and anchors several primitives to a common, well‑motivated assumption.
We introduce structured‑seed local pseudorandom generators (SSL-PRGs), pseudorandom generators whose seed is drawn from an efficiently sampleable, structured distribution rather than uniformly. This seemingly modest relaxation turns out to capture many known applications of local PRGs, yet it can be realized from a broader family of hardness assumptions. Our main technical contribution is a generic template for constructing SSL-PRGs that combines the following two ingredients: [i.] 1) noisy‑NC⁰ PRGs, computable by constant‑depth circuits fed with sparse noise, with 2) new local compression schemes for sparse vectors derived from combinatorial batch codes. Instantiating the template under the sparse Learning‑Parity‑with‑Noise (LPN) assumption yields the first SSL-PRGs with polynomial stretch and constant locality from a subquadratic‑sample search hardness assumption; a mild strengthening of sparse‑LPN gives strong SSL-PRGs of arbitrary polynomial stretch. We further show that for all standard noise distributions, noisy‑local PRGs cannot be emulated by ordinary local PRGs, thereby separating the two notions. Plugging SSL-PRGs into existing frameworks, we revisit the canonical applications of local PRGs and demonstrate that SSL-PRGs suffice for: (i) indistinguishability obfuscation, (ii) constant-overhead secure computation, (iii) compact homomorphic secret sharing, and (iv) deriving hardness results for PAC‑learning DNFs from sparse‑LPN. Our work thus broadens the landscape of low‑depth pseudorandomness and anchors several primitives to a common, well‑motivated assumption.
