A Note on Output Length of One-Way State Generators and EFIs

We study the output length of one-way state generators (OWSGs), their weaker variants, and EFIs. - Standard OWSGs. Recently, Cavalar et al. (arXiv:2312.08363) give OWSGs with $m$-qubit outputs for any $m=\omega(\log \lambda)$, where $\lambda$ is the security parameter, and conjecture that there do not exist OWSGs with $O(\log \log \lambda)$-qubit outputs. We prove their conjecture in a stronger manner by showing that there do not exist OWSGs with $O(\log \lambda)$-qubit outputs. This means that their construction is optimal in terms of output length. - Inverse-polynomial-advantage OWSGs. Let $\epsilon$-OWSGs be a parameterized variant of OWSGs where a quantum polynomial-time adversary's advantage is at most $\epsilon$. For any constant $c\in \mathbb{N}$, we construct $\lambda^{-c}$-OWSGs with $((c+1)\log \lambda+O(1))$-qubit outputs assuming the existence of OWFs. We show that this is almost tight by proving that there do not exist $\lambda^{-c}$-OWSGs with at most $(c\log \lambda-2)$-qubit outputs. - Constant-advantage OWSGs. For any constant $\epsilon>0$, we construct $\epsilon$-OWSGs with $O(\log \log \lambda)$-qubit outputs assuming the existence of subexponentially secure OWFs. We show that this is almost tight by proving that there do not exist $O(1)$-OWSGs with $((\log \log \lambda)/2+O(1))$-qubit outputs. - Weak OWSGs. We refer to $(1-1/\mathsf{poly}(\lambda))$-OWSGs as weak OWSGs. We construct weak OWSGs with $m$-qubit outputs for any $m=\omega(1)$ assuming the existence of exponentially secure OWFs with linear expansion. We show that this is tight by proving that there do not exist weak OWSGs with $O(1)$-qubit outputs. - EFIs. We show that there do not exist $O(\log \lambda)$-qubit EFIs. We show that this is tight by proving that there exist $\omega(\log \lambda)$-qubit EFIs assuming the existence of exponentially secure PRGs.