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REVIEW 5 major objections 5 minor 19 references

Malleability of transformations on the ciphertext in noisy Quantum public key encryption

T0 review · 5 major / 5 minor · reviewed 2026-07-31 · deepseek-v4-flash

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Pith's one-line read Everlasting security survives noise in quantum encryption

desk verdict The central everlasting-security claim is the definition itself, proved by assuming the conclusion and manipulating undefined ratios of kets; not ready for review. read the letter →

arxiv 2607.26086 v1 pith:RAURWFB4 submitted 2026-07-27 quant-ph math.PR

Pete Rigas This is my paper · ORCID
classification quant-phmath.PR MSC 81P0281Q02
keywords quantumpublickeyencryptioneverlastingsecuritydistributionnoiseGentleMeasurementLemmatracedistancemalleabilitynegligibilityfunction
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 paper aims to show that everlasting security—the guarantee that an adversary with a quantum computer cannot later break a key exchanged in two rounds—survives the injection of noise into a quantum public-key encryption protocol for quantum key distribution. It constructs noisy counterparts of the protocol's key generation, signature, verification, and encryption steps, and proves that the trace distance between the real and ideal noisy experiments is bounded by a negligibility function NEGL(λ′) at a security threshold λ′ > λ, extending a noiseless result. The proof works by adapting the Gentle Measurement Lemma to convert a lower bound on the trace of a ratio of noisy and noiseless state differences into an upper bound on the trace distance. A sympathetic reader would care because noise is unavoidable in real quantum hardware, and this is a first step toward showing that the strong 'everlasting' post-quantum guarantee degrades gracefully rather than failing outright. The main caveat, acknowledged implicitly by the presentation, is that the noisy projection operator at the heart of the construction is never given a concrete, normalized definition.

What carries the argument

The argument is carried by two objects. First, the noisy projection operator eΠ, which replaces each pure signature state |σ⟩ in the noiseless projection Π by |σ+noise⟩ (with 'noise' left unspecified), is used to inject noise into the protocol's acceptance check. Second, an adaptation of the Gentle Measurement Lemma from quantum information theory is the mechanism that converts a lower bound on the trace of a ratio of noisy/noiseless state differences—derived through a chain of up-to-constant inequalities with constants C, C′, C′′—into an upper bound of 2√ϵ on the trace distance between hybrid experiments. The negligibility function NEGL(λ′) then absorbs the security threshold gap λ′−λ. Thes

What would settle it

Take any specific, normalized noise model (for instance, a depolarizing channel applied to the signature register) and compute the trace distance Td(gExp_{A_{λ′}}(1^{λ′},1), gExp_{A_{λ′}}(1^{λ′},0)) for a finite security parameter λ′. If the value exceeds the claimed upper bound NEGL(λ′) (or the intermediate lower-bound step Tr ≥ 1−ϵ fails), the theorem is false.

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Extended reading notes

Core claim

The paper's central claim is Theorem: if the noisy quantum public-key encryption protocol is correct (decoding never aborts erroneously and always succeeds), then the trace distance between the two noisy experiments, Td(gExp_{A_{λ′}}(1^{λ′},1), gExp_{A_{λ′}}(1^{λ′},0)), is bounded, up to constants, by NEGL(λ′), where λ′ > λ is a higher security threshold. In other words, noisy everlasting security holds in the sense that the real and ideal worlds are indistinguishable even to a computationally unbounded adversary who later receives the quantum secret key. The derivation rests on an adaptation of the Gentle Measurement Lemma: the paper first proves a lower bound on a trace involving ratios of

Load-bearing premise

The load-bearing premise is that the noisy projection operator eΠ—which adds an undefined 'noise' to each signature state without normalization or proof that it is a valid quantum operation—is a legitimate mathematical object; if that operator is not well-defined, the entire noisy protocol and the derived trace-distance bound collapse.

Editorial extensions

If this is right

  • If correct, noisy everlasting security holds for a two-message quantum key distribution protocol, meaning the post-quantum guarantee degrades to a higher security threshold rather than disappearing under noise.
  • The trace-distance bound Td ≲ NEGL(λ′) with NEGL(λ′) related to exp(λ−λ′) gives an explicit quantitative tradeoff: more noise (larger λ′ gap) costs more resources to maintain the same level of security.
  • The three noisy advantage functions Adv(0), Adv(1), Adv(2) are all bounded by NEGL(λ′), so hybrid proofs over noisy experiments continue to work.
  • The proof technique—using the Gentle Measurement Lemma to turn a trace lower bound into a trace-distance upper bound—is portable to other noisy cryptographic settings.
  • The alternative definition of NEGL(λ′) via strong unforgeability of the noisy one-time signature scheme means that unforgeability and indistinguishability are tied in the noisy setting, as in the noiseless case.

Reading between the lines

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

  • If the noisy projection operator were made concrete—say, with additive noise on the signature register followed by renormalization—the same proof structure would likely yield an explicit NEGL(λ′) in terms of the noise amplitude; this would turn the current up-to-constants claim into a testable quantitative bound.
  • The higher security threshold λ′ > λ suggests a resource overhead: to achieve the same level of everlasting security as the noiseless protocol, the noisy protocol must run with a larger security parameter, implying a concrete cost in qubits or rounds that could be optimized.
  • The malleability framing (forwarding/deleting/reconstruction error probabilities) borrowed from symmetric encryption suggests a route to proving that noisy everlasting security composes under protocol composition, since those error probabilities are designed to track adversarial transformations of ciphertexts.
  • One could test the central claim numerically: instantiate the protocol with a specific noise channel and check whether the trace distance between the noisy experiments stays within the claimed bound; a counterexample for a finite λ′ would falsify the theorem.
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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

5 major / 5 minor

Summary. The paper claims to characterize a noisy variant of the Quantum Public Key Encryption (QPKE) protocol introduced by Malavolta and Walter, with the goal of extending everlasting security to noisy channels. To this end it introduces a noisy projection operator eΠ (Section 1.1), noisy versions of QPTs, OTSs, and QPKE protocols (Definitions 32–40), and a Lemma asserting that upper bounds on the trace distance follow from lower bounds on traces via the Gentle Measurement Lemma. The main Theorem concludes Td(gExp^{A_{λ′}}(1^{λ′},1), gExp^{A_{λ′}}(1^{λ′},0)) ≲ NEGL(λ′), i.e. noisy everlasting security with a higher security threshold λ′ > λ. The proof is in Sections 4.1–4.2; a Corollary compares three noisy advantage functions. The paper does not provide machine-checked proofs or reproducible code.

Significance. If the main theorem were established, it would be a meaningful extension of the Malavolta–Walter noiseless everlasting-security result to a noisy setting, and it would connect trace-distance bounds to the malleability framework of Maurer–Tackmann. The paper also identifies a plausible technical tool (Gentle Measurement Lemma) for such an extension. However, as written, no such result is actually derived: the theorem's conclusion is identical to Definition37 and is assumed as a premise in the proof; the central Lemma manipulates undefined ratios of kets and operators; and the Gentle Measurement Lemma is applied with the negligibility direction inverted. The claimed theorem is therefore either circular or rests on ill-posed mathematical objects. The potential significance is real, but the manuscript as it stands does not achieve it.

major comments (5)
  1. [§4.2, Theorem proof] The proof states: 'it suffices to argue that the desired statement holds from the conditions Td(gExp^{A_{λ′}}(1^{λ′},1), gExp^{A_{λ′}}(1^{λ′},0)) ≲ NEGL(λ′) ... which encapsulate everlasting security in the noisy and noiseless settings.' The displayed inequality is exactly the theorem's conclusion and Definition37. Taking it as a 'condition' makes the proof circular. The subsequent chain of implications rewrites the target, and the only non-tautological input, the Lemma, does not independently establish the inequality.
  2. [§1.1 and Lemma (B.1)–(B.4)] The noisy projection eΠ = Σ_{σ0∈Σ0} |0⟩⊗|σ0+noise⟩⟨σ0+noise|⊗⟨0| + Σ_{σ1∈Σ1} |1⟩⊗|σ1+noise⟩⟨σ1+noise|⊗⟨1| is not a well-defined projector: 'noise' is never specified, |σ+noise⟩ is not normalized, and the summands are not guaranteed orthogonal. The Lemma proof then manipulates ratios such as |c,σc⟩/|ec,fσc⟩, Πρ*Π/(eΠ fρ* eΠ), Tr(Πρ*)/Tr(eΠ fρ*), and |σ0/(σ0+noise)⟩ as if they were scalars or operator quotients. These objects are undefined in Hilbert-space operator formalism, so the estimates (B.1)–(B.4) and the subsequent GML step are not valid mathematical statements.
  3. [§4.1 Lemma, item (C)] The proof assumes a lower bound of the form 1/2 Tr(...) ≳ 1 − NEGL(λ′−λ)^{-1}. Since NEGL is supposed to be negligible, NEGL(λ′−λ)^{-1} is super-polynomially large, so '1 − NEGL^{-1}' is not close to 1 and may even be negative. The later step 'Td ≤ 2√(NEGL(λ′−λ)^{-1}) ≲ exp(λ−λ′)' is directionally wrong: 2√(NEGL^{-1}) is exponentially large whenever NEGL is exponentially small. The parameter ϵ is effectively chosen after the constants to make the final bound fit, rather than being derived from the GML hypotheses.
  4. [§3.1, Definitions 37 and 40] Definition37 defines noisy everlasting security as exactly Td(gExp^{A_{λ′}}(1^{λ′},1), gExp^{A_{λ′}}(1^{λ′},0)) ≲ NEGL(λ′). Definition40 defines NEGL(λ′) as the probability of strong unforgeability of the NOTS scheme. The Theorem then uses Definition40 as an 'alternative formulation' of the desired bound, and the Lemma's assumption (C) already contains NEGL(λ′−λ)^{-1}. Thus the main result is a restatement of definitions plus an assumption involving the same negligible function, rather than a derivation of trace-distance security from correctness and unforgeability.
  5. [§4.1, trace lower bound] The proof of the trace lower bound asserts 1 ≲ Tr(A)/Tr(B) and then '1 ≤ ...', applies Jensen's inequality four times to the square root, introduces constants C, C′, C″ (and repeated C, C′, C″) 'taken sufficiently small', and multiplies by inf_{ρ,τ,eρ,eτ} of a list of traces. No argument shows this infimum is positive, nor that the square-root inequality is valid for operators rather than scalars. These steps are load-bearing because they produce the 1−ϵ lower bound on which the GML invocation depends. Without a valid operator inequality, the constants and the parameter ϵ are free parameters, not derived quantities.
minor comments (5)
  1. [Definition37] The second random experiment is written gExp^{A_λ}(1^λ,0) instead of gExp^{A_{λ′}}(1^{λ′},0), making the security parameter inconsistent.
  2. [Lemma, (B.1)–(B.4)] Expressions such as |c_j⟩/|f c_{j′}⟩ appear in sums and tensor products; these are not defined quantum states, and the '≲' relation is never given a precise norm or ordering. This makes the inequalities unverifiable in their current form.
  3. [Section 3.2] The lengthy 'Run Time' comparisons and subsequentiality assumptions are heuristic and are not formally connected to the Theorem or Corollary. They should either be made precise or removed from the main development.
  4. [Definitions 34–36 and Theorem proof] The noisy objects (gVer, fvk, esk, ^SKGen, eρ, ect) are introduced without specifying a concrete noise model or CPTP map; the assertion '^SKGen ↔ SKGen⊗noise' in the Theorem proof is a definitional stipulation, not a construction.
  5. [Definition32 and References [14]–[17]] The cited Lemma3.2.3 from [17] is not stated precisely enough for the reader to verify the CPTP contractivity claim; several self-citations are to unpublished or not-yet-available preprints and should be replaced by citable versions or detailed statements.

Circularity Check

2 steps flagged · score 9.0 of 10

Theorem's proof assumes its own conclusion: the target inequality Td(gExp_Aλ'(1^λ',1), gExp_Aλ'(1^λ',0)) ≲ NEGL(λ') is Definition37 and is listed verbatim among the 'conditions' from which the proof says it follows.

  1. self definitional [Definition37; Theorem statement and Proof of Theorem, §4.2]
    "namely that it suffices to argue that the desired statement holds from the conditions, Td(gExpAλ′(1λ′,1), gExpAλ′(1λ′,0)) ≲ NEGL(λ′), Td(ExpAλ(1λ,1), ExpAλ(1λ,0)) = negl(λ), which encapsulate everlasting security in the noisy and noiseless settings, respectively."

    The first 'condition' is exactly the inequality the Theorem must prove and is exactly Definition37's definition of everlasting security. The proof therefore assumes its own conclusion; the remaining text is a sequence of restatements of the same trace-distance inequality. The correctness and unforgeability assumptions are not used to derive the inequality from more basic data.

  2. other [Proof of Theorem, §4.2 implication chain after Lemma application]
    "=⇒ {Td(gExpAλ′(1λ′,1) ⊗ ExpAλ(1λ,1), gExpAλ′(1λ′,0) ⊗ ExpAλ(1λ,0)) ≤ NEGL(λ′−λ)} =⇒ {Td(...) ≤ NEGL(λ′)} =⇒ {Td(...) ≲ NEGL(λ′−λ)} =⇒ {Td(...) ≲ NEGL(λ′)}"

    This '=⇒' chain moves the same trace-distance inequality between experimentation notations and between the parameters λ′−λ and λ′ without proving any new bound. It terminates at the theorem's conclusion, which was already assumed in the preceding 'conditions' sentence. Thus the central everlasting-security claim reduces to its own statement plus the paper's definitions of NEGL.

full rationale

The main result is circular in the strict sense: Definition37 defines everlasting security as Td(gExp_Aλ′(1^λ′,1), gExp_Aλ′(1^λ′,0)) ≲ NEGL(λ′), and the Theorem claims exactly that inequality, while its proof says the 'desired statement holds from the conditions' that include the same inequality. The surrounding Lemma does not break the circle: its lower-bound assumption already contains (NEGL(λ′−λ))^{-1}, and the GML application gives at best a sqrt-type bound, not the claimed NEGL decay; the proof then overwrites this with the desired inequality via the displayed =⇒ chain. I also note that the noisy projection eΠ is introduced as |σ0+noise⟩ without a definition of 'noise' or normalization; this makes the formal objects ill-posed, but the decisive circularity is the theorem's self-assumption rather than the noise model. There is no load-bearing self-citation here: the author's prior work [14,15,16] is background context, and Malavolta–Walter [3] is an external source. Score 9 because the central theorem is forced by the paper's own definition and proof structure.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The central claim rests almost entirely on definitions and adjustable constants: NEGL is defined in Definitions 33 and 40, the theorem is Definition37, and the proof's bounds are made to work by choosing C,C′,C′′ and ϵ. No external or independent evidence is introduced.

free parameters (4)
  • Constants C, C′, C′′ (and repeated C, C′, C′′) = "taken sufficiently small"
    In §4.1 the constants are introduced and later chosen so that the big constant term dominates the inf over states; this is fitting the bound after the fact.
  • Noisy security parameter λ′ = λ′ > λ, unspecified
    All noisy definitions and NEGL(λ′) depend on λ′, but no relation to a concrete noise model or channel is given.
  • Malleability error probabilities δ̃1(q), δ̃2(q), δ̃3(q) = unspecified positive functions
    The noisy forwarding, deleting, and reconstruction error probabilities are assumed to exist; the final bounds depend on their ratios without any quantitative value.
  • GML parameter ϵ = chosen after constants
    In §4.1, ϵ is set equal to the expression derived after choosing C,C′,C′′, making the 2√ϵ upper bound dependent on the same constants.
assumptions (6)
  • standard math Gentle Measurement Lemma: detection probability ≥1−ϵ implies trace distance ≤2√ϵ
    Invoked in §4.1; the lemma is standard, but the application here uses an invalid measurement operator and reverses the role of ϵ.
  • ad hoc to paper Jensen's inequality can be applied four times to the square root inside traces
    Used in §4.1 without proving that the relevant operators are positive semidefinite or that Jensen applies to operator products under the trace.
  • ad hoc to paper There exist C,C′,C′′ sufficiently small such that the constant term is ≥ the inf expression >0
    Stated just before 'the above constant can be taken sufficiently small'; this is an existence assumption chosen to make the bound hold.
  • ad hoc to paper By construction ^SKGen ↔ SKGen⊗noise
    The Theorem's protocol steps assume the noisy secret-key generation is the noiseless one tensored with an unspecified 'noise' object.
  • ad hoc to paper NEGL(λ′−λ)^{-1} can appear inside a 1−δ lower bound that is close to 1
    Lemma (C) assumes 1/2 Tr sqrt(...) ≳ 1 − NEGL(λ′−λ)^{-1}; since NEGL is negligible, its inverse is large, so the right-hand side is not close to 1 and the direction is inconsistent.
  • domain assumption Two-universal hashing and collision bounds from [10] carry over unchanged to the noisy setting
    Background from Ostrev is imported without proving that the noisy projection or noisy CPTP maps preserve the collision-bound and security arguments.
invented entities (1)
  • Ad hoc 'noise' added to signature states in eΠ
    purpose: Models the noisy QKD channel/protocol inside the projection operator
    |σ+noise⟩ is never defined; no channel, error model, or rate is specified, and the security threshold depends on this term.

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Cite this review

Pith. "Pith review of Malleability of transformations on the ciphertext in noisy Quantum public key encryption." pith.science (2026). https://pith.science/paper/RAURWFB4

@misc{pith2026260726086,
  author       = {Pith},
  title        = {Pith review of: Malleability of transformations on the ciphertext in noisy Quantum public key encryption},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RAURWFB4}},
  note         = {Machine review of arXiv:2607.26086}
}
read the original abstract

We characterize a noisy variant of a Quantum public encryption protocol recently introduced by Malavolta and Walter which demonstrated that the notion of everlasting security can be rigorously formulated for Quantum key distribution after two rounds of interaction between Alice and Bob. To address one possible direction of research that is related to injecting noise in the cryptographic protocol related to Quantum key distribution we formulate arguments for further examining the notion of everlasting security through malleability assumptions on transformations of the ciphertext. Assumptions surrounding malleability were introduced by Maurer and Tackmann for the purposes of comparing how authenticate then encrypt, and encrypt then authenticate, protocols behave through a variety of expressions for the forwarding error, deleting error, and reconstruction probabilities. Such probabilities are put to further use for obtaining connections between the indistinguishability and security threshold for a cryptographic protocol of interests. To further build upon such associations we demonstrate, through an adaptation of the Gentle Measurement Lemma from Quantum information theory, how upper bounds on the trace distance can be used to generalize the negligibility function obtained by Malavolta and Walter in the noiseless setting. Besides the fact that the negligibility function in the noisy setting is related to a higher security threshold it continues to remain of interest to determine whether computations provided in this work for upper bounding the trace distance can be related to other settings that are centered more on game-theoretic approaches.

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