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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.

Black-box separations have been successfully used to identify the limits of a powerful set of tools in cryptography, namely those of black-box reductions. They allow proving that a large set of techniques are not capable of basing one primitive 𝒫 on another 𝒬. Such separations, however, do not say anything about the power of the combination of primitives 𝒬₁,𝒬₂ for constructing 𝒫, even if 𝒫 cannot be based on 𝒬₁ or 𝒬₂ alone. By introducing and formalizing the notion of black-box uselessness, we develop a framework that allows us to make such conclusions. At an informal level, we call primitive 𝒬 black-box useless (BBU) for 𝒫 if 𝒬 cannot help constructing 𝒫 in a black-box way, even in the presence of another primitive 𝒵. This is formalized by saying that 𝒬 is BBU for 𝒫 if for any auxiliary primitive 𝒵, whenever there exists a black-box construction of 𝒫 from (𝒬,𝒵), then there must already also exist a black-box construction of 𝒫 from 𝒵 alone. We also formalize various other notions of black-box uselessness, and consider in particular the setting of efficient black-box constructions when the number of queries to 𝒬 is below a threshold. Impagliazzo and Rudich (STOC'89) initiated the study of black-box separations by separating key agreement from one-way functions. We prove a number of initial results in this direction, which indicate that one-way functions are perhaps also black-box useless for key agreement. In particular, we show that OWFs are black-box useless in any construction of key agreement in either of the following settings: (1) the key agreement has perfect correctness and one of the parties calls the OWF a constant number of times; (2) the key agreement consists of a single round of interaction (as in Merkle-type protocols). We conjecture that OWFs are indeed black-box useless for general key agreement. We also show that certain techniques for proving black-box separations can be lifted to the uselessness regime. In particular, we show that the lower bounds of Canetti, Kalai, and Paneth (TCC'15) as well as Garg, Mahmoody, and Mohammed (Crypto'17 & TCC'17) for assumptions behind indistinguishability obfuscation (IO) can be extended to derive black-box uselessness of a variety of primitives for obtaining (approximately correct) IO. These results follow the so-called "compiling out" technique, which we prove to imply black-box uselessness. Eventually, we study the complementary landscape of black-box uselessness, namely black-box helpfulness. We put forth the conjecture that one-way functions are black-box helpful for building collision-resistant hash functions. We define two natural relaxations of this conjecture, and prove that both of these conjectures are implied by a natural conjecture regarding random permutations equipped with a collision finder oracle, as defined by Simon (Eurocrypt'98). This conjecture may also be of interest in other contexts, such as amplification of hardness.