We introduce a novel construction of quantum-resistant one-way puzzles based on the uncomputability of Kolmogorov complexity and the obfuscation of pseudorandom generators. The security of our scheme relies on the minimal assumption that quantum polynomial-time (QPT) adversaries cannot efficiently approximate the Kolmogorov complexity 𝑲(𝒔) of strings sampled from specific distributions. We define a new cryptographic primitive-Kolmogorov puzzles-and formally prove that inversion implies a nontrivial compression of random strings, contradicting the Incompressibility Lemma. To support this, we provide a theoretical security reduction and simulation-based evidence. Compression algorithms (e.g., gzip) serve as heuristic estimators of 𝑲(𝒔), and Grover-based quantum circuits implemented in Qiskit validate the infeasibility of brute-force inversion. While our obfuscation uses lightweight heuristics, the reduction remains valid under any black-box obfuscator. This work offers a complexity-theoretic approach to post-quantum cryptography without algebraic assumptions, aligning with NIST's call for diverse post-quantum primitives.
We introduce a novel construction of quantum-resistant one-way puzzles based on the uncomputability of Kolmogorov complexity and the obfuscation of pseudorandom generators. The security of our scheme relies on the minimal assumption that quantum polynomial-time (QPT) adversaries cannot efficiently approximate the Kolmogorov complexity 𝑲(𝒔) of strings sampled from specific distributions. We define a new cryptographic primitive-Kolmogorov puzzles-and formally prove that inversion implies a nontrivial compression of random strings, contradicting the Incompressibility Lemma. To support this, we provide a theoretical security reduction and simulation-based evidence. Compression algorithms (e.g., gzip) serve as heuristic estimators of 𝑲(𝒔), and Grover-based quantum circuits implemented in Qiskit validate the infeasibility of brute-force inversion. While our obfuscation uses lightweight heuristics, the reduction remains valid under any black-box obfuscator. This work offers a complexity-theoretic approach to post-quantum cryptography without algebraic assumptions, aligning with NIST's call for diverse post-quantum primitives.
We initiate the time-bounded meta-complexity program for quantum cryptography. Recent work characterizes one-way puzzles, the minimal search primitive of quantum cryptography without one-way functions, by the average-case hardness of approximating the plain, uncomputable Kolmogorov complexity over quantumly samplable distributions; the classical program of Liu and Pass, by contrast, lives at polynomial time bounds. We define a probabilistic time-bounded quantum program complexity pKq^t for classical strings and prove two unconditional theorems. First, a quantum coding theorem: any string output by a quantum polynomial-time sampler with probability delta admits a description of the information-theoretically optimal length log(1/delta) plus logarithmic terms, decodable by a quantum machine in time O(sqrt(1/delta)) times a polynomial, via amplitude amplification over the coherently executed sampler. Second, an exact characterization at subexponential time: one-way puzzles exist if and only if the gap problem for pKq at time bound 2^(n/2) poly(n) is weakly quantum-average-hard, refining the plain-complexity characterizations. We then isolate the polynomial-time coding theorem as the single load-bearing open conjecture of the program, prove that it implies the full polynomial-time characterization, analyze why the classical derandomization proof resists quantization, and formulate a relativized barrier conjecture delimiting string-valued meta-complexity at one-way puzzles. Conjectures are labeled as such throughout.
From Worst-Case Hardness of NP to Quantum Cryptography via Quantum Indistinguishability Obfuscation
Indistinguishability obfuscation (iO) has emerged as a powerful cryptographic primitive with many implications. While classical iO, combined with the infinitely-often worst-case hardness of NP, is known to imply one-way functions (OWFs) and a range of advanced cryptographic primitives, the cryptographic implications of quantum iO remain poorly understood. In this work, we initiate a study of the power of quantum iO. We define several natural variants of quantum iO, distinguished by whether the obfuscation algorithm, evaluation algorithm, and description of obfuscated program are classical or quantum. For each variant, we identify quantum cryptographic primitives that can be constructed under the assumption of quantum iO and the infinitely-often quantum worst-case hardness of NP (i.e., NP⊆i.o.BQP). In particular, we construct pseudorandom unitaries, QCCC quantum public-key encryption and (QCCC) quantum symmetric-key encryption, and several primitives implied by them such as one-way state generators, (efficiently-verifiable) one-way puzzles, and EFI pairs, etc. While our main focus is on quantum iO, even in the classical setting, our techniques yield a new and arguably simpler construction of OWFs from classical (imperfect) iO and the infinitely-often worst-case hardness of NP.
Founding Quantum Cryptography on Quantum Advantage, or, Towards Cryptography from #P-Hardness
Recent oracle separations [Kretschmer, TQC'21, Kretschmer et. al., STOC'23] have raised the tantalizing possibility of building quantum cryptography from sources of hardness that persist even if the polynomial hierarchy collapses. We realize this possibility by building quantum bit commitments and secure computation from unrelativized, well-studied mathematical problems that are conjectured to be hard for P#P -- such as approximating the permanents of complex Gaussian matrices, or approximating the output probabilities of random quantum circuits. Indeed, we show that as long as any one of the conjectures underlying sampling-based quantum advantage (e.g., BosonSampling, Random Circuit Sampling, IQP, etc.) is true, quantum cryptography can be based on the extremely mild assumption that P#P⊆(io)BQP/qpoly. We prove that the following hardness assumptions are equivalent. (1) The hardness of approximating the probability assigned to a randomly chosen string in the support of certain efficiently sampleable distributions (upto inverse polynomial multiplicative error).(2) The existence of one-way puzzles, where a quantum sampler outputs a pair of classical strings -- a puzzle and its key -- and where the hardness lies in finding the key corresponding to a random puzzle. These are known to imply quantum bit commitments [Khurana and Tomer, STOC'24]. (3) The existence of state puzzles, or one-way state synthesis, where it is hard to synthesize a secret quantum state given a public classical identifier. These capture the hardness of search problems with quantum inputs (secrets) and classical outputs (challenges). These are the first constructions of quantum cryptographic primitives (one-way puzzles, quantum bit commitments, state puzzles) from concrete, well-founded mathematical assumptions that do not imply the existence of classical cryptography.
One-way puzzles (OWPuzzs) introduced by Khurana and Tomer [STOC 2024] are a natural quantum analogue of one-way functions (OWFs), and one of the most fundamental primitives in ''Microcrypt'' where OWFs do not exist but quantum cryptography is possible. OWPuzzs are implied by almost all quantum cryptographic primitives, and imply several important applications such as non-interactive commitments and multi-party computations. A significant goal in the field of quantum cryptography is to base OWPuzzs on plausible assumptions that will not imply OWFs. In this paper, we base OWPuzzs on hardness of non-collapsing measurements. To that end, we introduce a new complexity class, SampPDQP, which is a sampling version of the decision class PDQP introduced in [Aaronson, Bouland, Fitzsimons, and Lee, ITCS 2016]. We show that if SampPDQP is hard on average for quantum polynomial time, then OWPuzzs exist. SampPDQP is the class of sampling problems that can be solved by a classical polynomial-time algorithm that can make a single query to a non-collapsing measurement oracle, which is a ''magical'' oracle that can sample measurement results on quantum states without collapsing the states. Such non-collapsing measurements are highly unphysical operations that should be hard to realize in quantum polynomial-time. We also study upperbounds of the hardness of SampPDQP. We introduce a new primitive, distributional collision-resistant puzzles (dCRPuzzs), which are a natural quantum analogue of distributional collision-resistant hashing [Dubrov and Ishai, STOC 2006]. We show that dCRPuzzs imply average-case hardness of SampPDQP (and therefore OWPuzzs as well). We also show that two-message honest-statistically-hiding commitments with classical communication and one-shot signatures [Amos, Georgiou, Kiayias, Zhandry, STOC 2020] imply dCRPuzzs.
Post-quantum cryptography currently rests on a small number of hardness assumptions, posing significant risks should any one of them be compromised. This vulnerability motivates the search for new and cryptographically versatile assumptions that make a convincing case for quantum hardness. In this work, we argue that decoding random quantum stabilizer codes---a quantum analog of the well-studied LPN problem---is an excellent candidate. This task occupies a unique middle ground: it is inherently native to quantum computation, yet admits an equivalent formulation with purely classical input and output, as recently shown by Khesin et al. (STOC '26). We prove that the average-case hardness of quantum stabilizer decoding implies the core primitives of classical Cryptomania, including public-key encryption (PKE) and oblivious transfer (OT), as well as one-way functions. Our constructions are moreover practically efficient: our PKE scheme achieves essentially the same efficiency as state-of-the-art LPN-based PKE. We also provide substantial evidence that stabilizer decoding does not reduce to LPN, suggesting that the former problem constitutes a genuinely new post-quantum assumption. Our primary technical contributions are twofold. First, we give a reduction from random quantum stabilizer decoding to an average-case problem closely resembling LPN, but which is equipped with additional symplectic algebraic structure. While this structure is essential to the quantum nature of the problem, it raises significant barriers to cryptographic security reductions. Second, we develop a new suite of scrambling techniques for such structured linear spaces, and use them to produce rigorous security proofs for all of our constructions.
In classical cryptography, one-way functions are necessary for most of cryptography. In the quantum world, however, Kretschmer at {TQC'21} and subsequent works have demonstrated that quantum versions of various primitives in minicrypt, such as signatures and private-key encryption, do not require quantum-computable one-way functions (\textsf{qOWF}s). However, the question of the necessity of \textsf{qOWF}s for primitives in Cryptomania like public-key encryption with classical public keys and ciphertexts (\textsf{qPKE}) has remained open.
We provide a comprehensive picture of the relationship between \textsf{qPKE} and \textsf{qOWF}. First, our main result is a black-box separation between \textsf{qOWF}s and \textsf{qPKE}. In particular, we show that relative to a unitary, real, and self-inverse oracle, there exists a \textsf{qPKE} scheme (with classical keys and ciphertexts), but \textsf{qOWF}s do not exist. Further, our oracles admit a point-function obfuscation construction, thus ruling out black-box construction of \textsf{qOWF} from point function obfuscations.
On the positive side, we construct \textsf{qPKE} from quantum indistinguishability obfuscation for quantum circuits with classical obfuscated descriptions and efficiently verifiable one-way puzzles, which is known to be a potentially weaker assumption than \textsf{qOWF}s, due to an existing black-box separation. Our construction is the quantum analog of the classical construction of \textsf{PKE} from one-way functions and indistinguishability obfuscation in the quantum computation with classical communication (QCCC) setting.