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=ω(logλ), where λ is the security parameter, and conjecture that there do not exist OWSGs with O(loglogλ)-qubit outputs. We prove their conjecture in a stronger manner by showing that there do not exist OWSGs with O(logλ)-qubit outputs. This means that their construction is optimal in terms of output length. - Inverse-polynomial-advantage OWSGs. Let ϵ-OWSGs be a parameterized variant of OWSGs where a quantum polynomial-time adversary's advantage is at most ϵ. For any constant c∈ℕ, we construct λ^-c-OWSGs with ((c+1)logλ+O(1))-qubit outputs assuming the existence of OWFs. We show that this is almost tight by proving that there do not exist λ^-c-OWSGs with at most (clogλ-2)-qubit outputs. - Constant-advantage OWSGs. For any constant ϵ>0, we construct ϵ-OWSGs with O(loglogλ)-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 ((loglogλ)/2+O(1))-qubit outputs. - Weak OWSGs. We refer to (1-1/𝗉𝗈𝗅𝗒(λ))-OWSGs as weak OWSGs. We construct weak OWSGs with m-qubit outputs for any m=ω(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(logλ)-qubit EFIs. We show that this is tight by proving that there exist ω(logλ)-qubit EFIs assuming the existence of exponentially secure PRGs.