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.

翻译：我们研究单向状态生成器（OWSGs）及其弱变体以及EFI的输出长度。 - **标准OWSGs**：近期，Cavalar等人（arXiv:2312.08363）给出了任意$m=\omega(\log \lambda)$量子比特输出的OWSGs（$\lambda$为安全参数），并猜想不存在$O(\log \log \lambda)$量子比特输出的OWSGs。我们通过证明不存在$O(\log \lambda)$量子比特输出的OWSGs，以更强形式证实该猜想。这意味着他们的构造在输出长度方面是最优的。 - **逆多项式优势OWSGs**：令$\epsilon$-OWSGs为OWSGs的参数化变体，其中量子多项式时间敌手的优势不超过$\epsilon$。对于任意常数$c\in \mathbb{N}$，我们基于OWFs的存在性，构造了输出长度为$((c+1)\log \lambda+O(1))$量子比特的$\lambda^{-c}$-OWSGs。通过证明不存在输出长度不超过$(c\log \lambda-2)$量子比特的$\lambda^{-c}$-OWSGs，我们表明该结果近乎紧致。 - **常数优势OWSGs**：对于任意常数$\epsilon>0$，我们基于亚指数安全OWFs的存在性，构造了输出长度为$O(\log \log \lambda)$量子比特的$\epsilon$-OWSGs。通过证明不存在输出长度为$((\log \log \lambda)/2+O(1))$量子比特的$O(1)$-OWSGs，我们表明该结果近乎紧致。 - **弱OWSGs**：我们将$(1-1/\mathsf{poly}(\lambda))$-OWSGs称为弱OWSGs。基于具有线性扩展的指数安全OWFs的存在性，我们构造了任意$m=\omega(1)$量子比特输出的弱OWSGs。通过证明不存在$O(1)$量子比特输出的弱OWSGs，我们表明该结果是紧致的。 - **EFIs**：我们证明不存在$O(\log \lambda)$量子比特的EFIs。通过证明在指数安全PRGs存在下存在$\omega(\log \lambda)$量子比特的EFIs，我们表明该结果是紧致的。