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REVIEW 2 major objections 3 minor 89 references

Quantum One-Way Functions and Related Cryptographic Primitives

T0 review · 2 major / 3 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Quantum one-wayness is not one primitive but a spectrum of state-based constructions whose security depends on how many copies an adversary receives.

desk verdict A useful and mostly sound review that deserves refereeing; the coherent-state section has a genuine scaling typo in Eq. (15), but the intended result survives. read the letter →

arxiv 2608.05754 v1 pith:IW5IYINE submitted 2026-08-06 quant-ph cs.CRmath-phmath.MP

Georgios M. Nikolopoulos This is my paper · ORCID
classification quant-phcs.CRmath-phmath.MP MSC 81P6881P4594A60 PACS 03.67.Dd03.67.-a
keywords quantumone-wayfunctionsstategeneratorspseudorandomstatesEFIpairsweakcoherentbounded-copysecuritycryptographyinformation-theoreticone-wayness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This review tries to establish that quantum one-wayness is not a single primitive but a spectrum of state-based constructions—quantum one-way functions, one-way state generators, pseudorandom quantum states, and efficiently indistinguishable state pairs—that differ in the adversarial model, the number of accessible copies, and the physical basis of their security. It argues that information-theoretic one-wayness can hold only in bounded-copy regimes, whereas security against polynomially many copies requires computational assumptions, and it organizes the primitives into the conceptual hierarchy pseudorandom-state generator $\Rightarrow$ one-way state generator $\Rightarrow$ computational quantum one-way function. The review further argues that the path toward practical quantum cryptography should run through simple, physically realizable state families, in particular phase-encoded weak coherent states, which in the effective-qubit regime recover the logarithmic Holevo bound $\log_2(c+1)$ on accessible information. A sympathetic reader would take away that security definitions should be matched to what near-term optical hardware can actually prepare and verify.

What carries the argument

The central mechanism is the symmetric subspace of $c$ identical copies of a finite-dimensional state: for qubits it has dimension $c+1$, which yields the Holevo bound $I_{\mathrm{acc}} \le \log_2(c+1)$ and the exact-label recovery bound $\Pr[\mathrm{succ}] \le (c+1)/2^n$. The review transfers this mechanism to continuous variables by truncating each coherent state to its vacuum and single-photon sector, so that under $\sqrt{c}\,\mu \ll 1$ the state behaves as an effective qubit; the paper bounds the trace-distance error of this truncation and then uses an entropy-continuity inequality to convert that closeness into a bound on the entropy of the average state, giving the $\log_2(c+1)$ scaling.

What would settle it

Compute the exact trace distance between $c$ copies of the true coherent state and $c$ copies of its truncated effective qubit at parameters near $\sqrt{c}\,\mu \approx 1$ (for example $\mu=0.01$, $c=10^4$) and compare with the paper's estimate $\mathcal{O}(\sqrt{c}\,\mu)$. If the distance is not small in that regime, or if the entropy of the true average state exceeds $\log_2(c+1)$ by a non-negligible amount, the claim that weak coherent states recover the logarithmic Holevo bound fails.

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Extended reading notes

Core claim

The central claim is that the relations among QOWFs, OWSGs, PRSGs, and EFI pairs are complementary security regimes rather than interchangeable definitions. In the bounded-copy information-theoretic model, one-wayness is enforced by physical limits: non-orthogonal states cannot be perfectly discriminated, measurement disturbs the state, and $c$ copies of a qubit state live in a symmetric subspace of dimension $c+1$, so the Holevo-accessible information is at most $\log_2(c+1)$ and exact-label recovery succeeds with probability at most $(c+1)/2^n$. Against adversaries with polynomially many copies, these limits no longer suffice—shadow tomography yields a generic attack on information-theoretic OWSG security—so OWSGs, PRSGs, and EFI pairs must rely on computational hardness, and the review states the hierarchy PRSG $\Rightarrow$ OWSG $\Rightarrow$ computational QOWF. For coherent states, the paper's own analysis claims that in the weak-photon regime $\sqrt{c}\,\mu \ll 1$ the ensemble behaves as an effective qubit ensemble, recovering the same logarithmic copy scaling and making coherent states the most experimentally accessible candidate family for quantum one-wayness.

Load-bearing premise

That the multiphoton sector of weak coherent states can be neglected when $\sqrt{c}\,\mu \ll 1$, so that the state behaves as an effective qubit and the accessible information is governed by the $\log_2(c+1)$ symmetric-subspace bound; the paper's trace-distance estimate for this approximation is the fragile step.

Editorial extensions

If this is right

  • Information-theoretically secure quantum one-way functions exist in the bounded-copy regime from quantum limits alone (conjugate coding, quantum fingerprints, single-qubit rotations), and their security degrades as the number of copies grows.
  • Against polynomially many copies, quantum one-wayness requires computational assumptions: shadow tomography provides a generic attack on information-theoretic OWSG security, and unconditionally secure OWSGs would imply a major complexity separation.
  • The conceptual hierarchy PRSG $\Rightarrow$ OWSG $\Rightarrow$ computational QOWF holds, with the converse implications generally false.
  • Weak coherent states, already used in QKD-style hardware, are the most promising near-term platform for QOWFs, provided the adversary's copy count stays in the regime $\sqrt{c}\,\mu \ll 1$.
  • Verification by fidelity threshold requires ensembles whose maximum pairwise fidelity is bounded away from 1; constructions with arbitrarily close neighboring states need protocol-level verification instead.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the paper's trace-distance scaling is corrected, the effective-qubit regime for coherent-state QOWFs may be narrower than stated, so proposed protocol parameters should be re-derived from the exact distance rather than from the approximate bound.
  • The same symmetric-subspace logic should extend to other phase-invariant continuous-variable ensembles (for instance squeezed or thermal states), suggesting a general principle: copy-limited one-wayness arises whenever the accessible states effectively live in a finite-dimensional subspace.
  • Bounded-copy security connects naturally to noisy- and bounded-storage models, where hardware constraints on stored quantum states turn the copy restriction into a physical resource.
  • The noise sensitivity of pseudorandom quantum states suggests that indistinguishability from Haar-random states is a poor target for photonic implementations; bounded-copy hiding based on simple states may be the more realistic near-term goal.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. This paper is a topical review of quantum one-way functions and related quantum-state primitives. It develops a common framework for classical-to-quantum QOWFs with two inversion objectives (exact-label recovery and fidelity-threshold approximate inversion) and two adversarial models (computational QPT adversaries and bounded-copy information-theoretic adversaries). It then reviews representative constructions, including conjugate coding, quantum fingerprints, single-qubit rotations, and phase-encoded coherent states, and connects them to OWSGs, EFI pairs, PRSGs, quantum t-designs, quantum money, PUFs/QPUFs, and post-quantum cryptography. The central thesis is that these primitives form a complementary spectrum whose security depends on the number of available copies and on whether the hardness is information-theoretic or computational, and that practical progress requires simple, robust, physically realizable state families with context-dependent security guarantees.

Significance. The review's contribution is synthetic and pedagogical rather than a new set of theorems. Its strengths are the explicit separation of exact-label recovery from preimage resistance, the systematic comparison of QOWFs/OWSGs/EFI/PRSGs with a summary table, the candid treatment of the gap between complexity-theoretic primitives and experimental feasibility, and the useful appendix material on symmetric subspaces and trace-distance operational interpretations. If the technical illustrations are corrected, this will be a valuable reference for physics-oriented readers. However, the main technical derivation in Section 3.2.4 contains a scaling error and an unjustified truncation step, so the review needs revision before publication.

major comments (2)
  1. [§3.2.4, Eq. (15)] The small-µ expansion in Eq. (15) is incorrect. Since |⟨ϕ_x|ψ_x⟩|² = (1+µ)e^{−µ}, the exact c-copy trace norm is 2√(1−(1+µ)^c e^{−cµ}) ≈ √(2c) µ, not √(2cµ). The same correction propagates to Eq. (16), which should read D ≲ √(c/2) µ, and to the error term D log₂ c in Eq. (18). This is not cosmetic: under the stated condition √c µ ≪ 1, the printed formula with c=100 and µ=0.01 gives D≈0.7, while the exact value is ≈0.07, so the claimed small trace-distance approximation would fail. The exact expression supports the intended effective-qubit conclusion, so I view this as a correctable error, but the formula, its validity condition, and the constants in Eq. (18) must be fixed.
  2. [§3.2.4, Eqs. (17)–(18) and footnote 7] The Audenaert–Fannes step is not justified as written. The inequality in Eq. (17) requires both states to live in the same d-dimensional Hilbert space, but the actual state ρ̄^{(c)} is supported on the infinite-dimensional symmetric subspace of (H∞)^{⊗c}, while σ̄^{(c)} is supported on the c+1-dimensional qubit symmetric subspace. Footnote 7 says the multiphoton components 'can be ignored,' but ignoring them changes the state, and the trace-distance bound of Eqs. (15)–(16) does not by itself control the entropy of the discarded component. Even after correcting the scaling in Eq. (15), Eq. (18)'s bound S(ρ̄^{(c)}) ≲ log₂(c+1) + D log₂ c + h₂(D) therefore needs an explicit projection/truncation argument, with finite-c constants stated. The block structure of the phase-averaged coherent state should make such an argument straightforward, but the manuscript should provide it rather than relying on the current footnote.
minor comments (3)
  1. [Appendix D, Eq. (30)] The displayed bound Σ_n √(P(µ,n)P(µ,n+qN)) ≤ e^{−µ}(eµ/(qN))^{qN} is too strong; the n=0 term alone is approximately e^{−µ}√(P(µ,qN)), which for µ=0.01 and N=100 exceeds the claimed right-hand side. The qualitative conclusion that the discrete phase average is close to the fully dephased state is plausible, but this inequality should be rederived or replaced by a correct tail bound.
  2. [§3.2.1] The sentence claiming that 'the optimal single-qubit measurements yield the optimal probability of recovering the entire string' appears before the commuting structure of ρ₀ and ρ₁ is explained; since the product structure alone does not rule out collective measurements, the argument would be clearer if the simultaneous diagonalizability in the Breidbart basis were invoked immediately.
  3. [§4, Eq. (24)] Equation (24) is presented as the c-copy average state for the coherent-state construction, but it is derived from the truncated effective-qubit approximation of Eq. (14), not from the exact coherent-state ensemble; an explicit reminder in the text near Eq. (24) would prevent readers from mistaking it for the exact average state.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the review's central comparisons are supported by in-line derivations and external published results; self-citations are exemplary, not load-bearing.

full rationale

I walked the paper's derivation chain. Definitions 1-2 set up an inversion game, and the one-wayness claims for Constructions 1-4 are obtained by explicit calculations (Helstrom discrimination, the symmetric-subspace dimension D_sym = c+1, Holevo and accessible-information bounds, fidelity and trace-distance estimates), not by assuming the target conclusion. In Construction 4 (Sec. 3.2.4), the effective-qubit approximation is derived rather than presupposed: Eq. (14) defines the truncated state, Eq. (15) gives the exact trace-distance expression 2 sqrt(1 - ((1+mu)e^{-mu})^c), and Eqs. (16)-(18) use it to bound S(rho_bar^(c)). The printed asymptotic approximation 'approximately sqrt(2 c mu)' in Eq. (15) is a genuine scaling typo: the small-mu expansion is sqrt(2c) mu. This is a correctness defect, not a circular step, and the exact expression supports the intended log(c+1) leading scaling. Self-citations to [38], [39], and [41] are used to point to applications (single-qubit QOWFs, quantum public-key encryption, weak-coherent-state key exchange) and are not inputs to the review's central comparisons or taxonomy. Claims about the impossibility of information-theoretic OWSGs with many copies and the PRSG => OWSG hierarchy are attributed to external works [42], [44], [65]; no fitted parameter is renamed as a prediction, and no uniqueness theorem or ansatz is imported from the author's prior work. The review is therefore self-contained against external benchmarks and contains no circular derivation chain.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The review introduces no new fitted parameters or invented entities. Its technical derivations rely on standard quantum information theorems (Holevo bound, no-cloning, trace-distance operational interpretation, symmetric subspace dimension) and on external hardness assumptions from the surveyed literature, such as the conjectured difficulty of graph automorphism and of approximating boson-sampling amplitudes. The original coherent-state analysis in Section 3.2.4 assumes a truncated two-level approximation in the weak-photon regime, a technical assumption stated explicitly.

assumptions (6)
  • standard math Holevo bound: at most log2(d) bits of classical information can be extracted per copy of a d-dimensional quantum state.
    Used throughout Section 3.2 to argue information-theoretic one-wayness from limited accessible information.
  • standard math No-cloning theorem and quantum measurement disturbance.
    Foundational physical principles invoked in Section 3.1 to motivate QOWF security.
  • standard math Uniform ensemble of 2^n states in a D-dimensional subspace has average exact-label recovery probability at most D/2^n.
    Used in Section 3.2.3 for the single-qubit construction; stated without proof in the paper.
  • domain assumption Graph Automorphism is not solvable in quantum polynomial time (conjecture).
    Basis for the hardness of the Kawachi et al. EFI-like pair in Section 5.
  • domain assumption Existence of post-quantum secure pseudorandom functions.
    Used in Section 6.3 as the basis for PRSG constructions, e.g., by Ji-Liu-Song and follow-ups.
  • domain assumption Average-case hardness of computing or approximating permanents underlying boson sampling.
    Security foundation for boson-sampling-based OWFs in Section 7.3.

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Cite this review

Pith. "Pith review of Quantum One-Way Functions and Related Cryptographic Primitives." pith.science (2026). https://pith.science/paper/IW5IYINE

@misc{pith2026260805754,
  author       = {Pith},
  title        = {Pith review of: Quantum One-Way Functions and Related Cryptographic Primitives},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IW5IYINE}},
  note         = {Machine review of arXiv:2608.05754}
}
read the original abstract

Quantum cryptographic primitives beyond key distribution remain a less well understood area of research. In classical cryptography, one-way functions underpin nearly all standard cryptographic protocols, motivating the search for meaningful quantum analogues and for a clear understanding of the physical and computational mechanisms that could enforce one-wayness. In this article, we review quantum one-way functions and a range of closely related quantum-state primitives, including one-way state generators, pseudorandom quantum states, and efficiently indistinguishable pairs of states. We discuss both computational and information-theoretic notions of quantum one-wayness, emphasizing the different adversarial models and security assumptions that underlie these constructions. We compare and contrast the various proposed primitives, and clarify their conceptual relationships. Particular emphasis is placed on questions of physical realizability, experimental feasibility, and robustness to noise. Finally, we outline open problems and future directions toward the development of practical quantum cryptographic primitives beyond key distribution, and the emergence of a broader quantum-cryptographic ecosystem.

Figures

Figures reproduced from arXiv: 2608.05754 by the authors.

Figure 1
Figure 1. Schematic comparison of injective and non-injective mappings in the classical and quantum settings. In the quantum case, the shaded regions represent states with finite operational overlap; distinct labels may therefore correspond to distinct, but nonorthogonal states. assumptions. Since the present review discusses constructions originating from both the cryptographic and quantum-information communities, it is usef… view at source ↗

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.