Pseudorandom states, introduced by Ji, Liu, and Song (CRYPTO 2018), are quantum analogues of classical pseudorandom generators. A fundamental property of classical pseudorandom generators is that their output can be stretched to arbitrary polynomial length. Whether an analogous stretching property holds for quantum pseudorandom states has been elusive. Here, we essentially resolve this question for the single-copy variant of pseudorandom states, 1PRS. We show, via a unitary oracle separation, that no fully black-box construction can stretch the output length of a 1PRS by any constant factor greater than one.
Pseudorandom states, introduced by Ji, Liu, and Song (CRYPTO 2018), are quantum analogues of classical pseudorandom generators. A fundamental property of classical pseudorandom generators is that their output can be stretched to arbitrary polynomial length. Whether an analogous stretching property holds for quantum pseudorandom states has been elusive. Here, we essentially resolve this question for the single-copy variant of pseudorandom states, 1PRS. We show, via a unitary oracle separation, that no fully black-box construction can stretch the output length of a 1PRS by any constant factor greater than one.
Pseudorandom Quantum States (PRS) were introduced by Ji, Liu and Song as quantum analogous to Pseudorandom Generators. They are an ensemble of states efficiently computable but computationally indistinguishable from Haar random states. Subsequent works have shown that some cryptographic primitives can be constructed from PRSs. Moreover, recent classical and quantum oracle separations of PRS from One-Way Functions strengthen the interest in a purely quantum alternative building block for quantum cryptography, potentially weaker than OWFs. However, our lack of knowledge of extending or shrinking the number of qubits of the PRS output still makes it difficult to reproduce some of the classical proof techniques and results. Short-PRSs, that is PRSs with logarithmic size output, have been introduced in the literature along with cryptographic applications, but we still do not know how they relate to PRSs. Here we answer half of the question, by showing that it is not possible to shrink the output of a PRS from polynomial to logarithmic qubit length while still preserving the pseudorandomness property, in a relativized way. More precisely, we show that relative to Kretschmer's quantum oracle (TQC 2021) short-PRSs cannot exist (while PRSs exist, as shown by Kretschmer's work).
Pseudorandom functions (PRFs) are one of the most fundamental primitives in classical cryptography. On the other hand, in quantum cryptography, it is possible that PRFs do not exist but their quantum analogues could exist, and still enabling many applications including SKE, MACs, commitments, multiparty computations, and more. Pseudorandom unitaries (PRUs) [Ji, Liu, Song, Crypto 2018], pseudorandom isometries (PRIs) [Ananth, Gulati, Kaleoglu, Lin, Eurocrypt 2024], and pseudorandom function-like state generators (PRFSGs) [Ananth, Qian, Yuen, Crypto 2022] are major quantum analogs of PRFs. PRUs imply PRIs, and PRIs imply PRFSGs, but the converse implications remain unknown. An important open question is whether these natural quantum analogues of PRFs are equivalent. In this paper, we partially resolve this question by ruling out black-box constructions of them: 1. There are no black-box constructions of O(logλ)-ancilla PRUs from PRFSGs. 2. There are no black-box constructions of O(logλ)-ancilla PRIs with O(logλ) stretch from PRFSGs. 3. There are no black-box constructions of O(logλ)-ancilla PRIs with O(logλ) stretch from PRIs with Ω(λ) stretch. Here, O(logλ)-ancilla means that the generation algorithm uses at most O(logλ) ancilla qubits. PRIs with s(λ) stretch is PRIs mapping λ qubits to λ+s(λ) qubits. To rule out the above black-box constructions, we construct a unitary oracle that separates them. For the separations, we construct an adversary based on the quantum singular value transformation, which would be independent of interest and should be useful for other oracle separations in quantum cryptography.
We construct quantum oracles relative to which quantum-secure one-way functions (OWFs) exist but pseudorandom states (PRSs) with superlogarithmic output length do not. At first glance, this appears to contradict the known black-box constructions of PRS generators from quantum-secure OWFs. The distinction lies in the access model to the oracles; our oracle separation uses \emph{classical-accessible} random oracles that can be accessed only classically even by quantum algorithms. In fact, our impossibility of PRSs applies to \emph{any} classically accessible classical oracle in place of the random oracle, while keeping the other oracle component unchanged, showing the need for coherent access in constructing PRSs.
We further show that logarithmic output length pseudorandom function-like states (PRFSs) exist relative to our oracles, giving an oracle separation between classically accessible logarithmic length PRFSs and superlogarithmic length PRSs. This shows that fully black-box PRS length extension from logarithmic to superlogarithmic output length must use coherent access to the underlying short PRS.
Pseudorandom generators (PRGs) are a foundational primitive in classical cryptography, underpinning a wide range of constructions. In the quantum setting, pseudorandom quantum states (PRSs) were proposed as a potentially weaker assumption that might serve as a substitute for PRGs in cryptographic applications. Two primary size regimes of PRSs have been studied: logarithmic-size and linear-size. Interestingly, logarithmic PRSs have led to powerful cryptographic applications, such as digital signatures and quantum public-key encryption with tamper-resilient keys, that have not been realized from their linear counterparts. However, PRGs have only been black-box separated from linear PRSs, leaving open the fundamental question of whether PRGs are also separated from logarithmic PRSs. In this work, we resolve this open problem. We establish a quantum black-box separation between (quantum-evaluable) PRGs and PRSs of either size regime. Specifically, we construct a unitary quantum oracle with inverse access relative to which no black-box construction of PRG from (logarithmic or linear) PRS exists. This does not directly separate PRG from some of the applications of SPRS since these applications involve, as a first step, a non-black-box construction of a notion termed bot-PRGs. To address this, we present another unitary separation showing that PRG are also separated from bot-PRGs. Thus, we obtain separation from digital signatures and quantum public-key encryption.
We show that any quantum pseudorandom state that is secure against single-copy distinguishers, i.e. a 1-PRS, can be amplified to t-copy security, i.e. to a t-PRS, without additional assumptions, for any polynomial t in the security parameter. Prior work (Ananth and Goldin, arXiv 2025) was only able to show this for a restricted class of 1-PRS constructions, namely ones whose generators only use a small number of ancilla qubits. Technically, we show that by carefully accounting for the randomness that is used in the construction, and using quantum extractors, it is possible to eliminate an ancilla register of any length and obtain a meaningful t-PRS outcome.
There are various notions of quantum pseudorandomness, such as pseudorandom unitaries (PRUs), pseudorandom state generators (PRSGs), pseudorandom function-like state generators (PRFSGs) and quantum-computable PRGs. Unlike the different notions of classical pseudorandomness, which are known to be existentially equivalent to each other, the relations among quantum pseudorandomness have yet to be fully established. We present evidence suggesting that some forms of quantum pseudorandomness are unlikely to be constructed from the others. This indicates that quantum pseudorandomness behaves quite differently from classical pseudorandomness. Our main result is a unitary oracle separation where log-length output PRFSGs exist but quantum-computable pseudorandom generators (QPRGs) with negligible correctness error do not. This result suggests that the inverse-polynomial error in the state-of-the-art construction of QPRGs from log-length PRSGs is inherent. To achieve this, we prove a novel differential geometric barrier theorem for the product Haar measure on quantum states which replaces the usual concentration inequalities by certifying a non-negligible ``gap'' between two large trace-separated sets. Our separation is based on an oracle that outputs Haar random quantum states for each bit string, which can be viewed as a quantum version of the random oracle model, where output strings are replaced by quantum states. Variations of this oracle can be used to study other relationships between quantum cryptographic primitives, and we use it to achieve partial separations, that highlight technical difficulties when dealing with ancillary registers, measurements, and adaptivity in the quantum setting.