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arxiv: 2605.22569 · v1 · pith:3DNEHQWSnew · submitted 2026-05-21 · 💻 cs.CR · quant-ph

A Formal Basis for Quantum Cryptographic Exposure Measurement under HNDL Threat

Matheus Rufino , Rafael Duarte Marcelino , Julio Smanioto Garcia This is my paper

Pith reviewed 2026-05-22 04:54 UTC · model grok-4.3

classification 💻 cs.CR quant-ph
keywords HNDL threatquantum cryptographic exposurecompromise probabilityvulnerability-exposure planedefense-attack ratioadversarial productionvalue-decay dynamicsexposure measurement
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The pith

The probability of future quantum decryption of stored encrypted traffic factorizes into temporal hazard, vulnerability-exposure product and saturation term under three assumptions on adversary behavior.

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

The paper establishes that the risk an adversary will store encrypted traffic today and decrypt it later with a quantum computer takes a specific mathematical form. This form is required by three assumptions about how adversaries generate attacks and how the value of decrypted information decays over time. A sympathetic reader would care because the resulting expression makes the sensitivity to changes in cryptography or operations depend on the organization's current location in a vulnerability-exposure plane rather than treating every improvement as equally valuable. Additive scoring methods cannot produce this structure because they omit the interaction between vulnerability and exposure by design. The framework therefore supplies a structurally justified way to rank exposure reduction actions when only partial information is available.

Core claim

Under three assumptions about adversarial production and value-decay dynamics, the HNDL compromise probability factorises into a temporal hazard, a multiplicative cryptographic-vulnerability and operational-exposure term, and a saturation denominator governed by the defense-attack intensity ratio; the marginal sensitivity to each dimension is endogenous to the organisation's position in the vulnerability-exposure plane, not a fixed global constant. Additive scoring frameworks cannot reproduce this structure because the interaction between cryptographic vulnerability and operational exposure is absent by construction, regardless of calibration.

What carries the argument

The factorization of the HNDL compromise probability into a temporal hazard multiplied by a combined vulnerability-exposure term and divided by a saturation term set by the defense-attack intensity ratio.

If this is right

  • The functional form of the compromise probability is determined by the assumptions rather than by free parameter calibration.
  • Marginal sensitivity to improvements in cryptography or operations changes with the organization's current position in the vulnerability-exposure plane.
  • Additive scoring models are structurally unable to capture the required interaction between cryptographic vulnerability and operational exposure.
  • The framework supports prioritisation of exposure-reduction actions even when only partial observability of the adversary and the data is available.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Organisations could simulate shifts in their position on the vulnerability-exposure plane to predict how future quantum timeline updates would change their ranking of defensive actions.
  • The same structural approach might apply to other delayed-threat settings where data collected today is processed by a more powerful adversary later.
  • Empirical tracking of how measured exposure changes after new post-quantum migration steps could provide a practical check on whether the position-dependent sensitivities appear in real systems.

Load-bearing premise

Three assumptions about how adversaries produce attacks and how the value of decrypted data decays are sufficient to fix the exact functional form of the compromise probability.

What would settle it

A direct derivation showing that the three assumptions on adversarial production and value decay do not produce the claimed factored form, or observed exposure data in which marginal sensitivities fail to vary with position in the vulnerability-exposure plane.

Figures

Figures reproduced from arXiv: 2605.22569 by Julio Smanioto Garcia, Matheus Rufino, Rafael Duarte Marcelino.

Figure 1
Figure 1. Figure 1: Structural organisation of the HNDL exposure assessment problem. Observable and declarative signals are mapped to the structural variables 𝑉 (vulnerability fraction) and 𝐸 (operational exposure); the proportional-hazards contest (Hypotheses 1–3) and temporal hazard 𝐻 (Definition 1) produce the exposure score of Eq. (8). The contribution of this paper is the structural form of the lower two stages; the sign… view at source ↗
Figure 2
Figure 2. Figure 2: Structural properties of the IEQ under Hypotheses 1–3 with 𝑎 = 1.0, 𝑏 = 0.5, 𝜃 = 1.0, 𝐻 = 0.6, 𝑀 = 1.15. (a) IEQ surface over the (𝑉 , 𝐸) unit square (Eq. (8)). Contour density reflects score sensitivity: high in the defense-dominant regime (𝑉 𝑎𝐸𝑏 ≪ 𝜃, lower left) and low near saturation (𝑉 𝑎𝐸𝑏 ≫ 𝜃, upper right). The observed range reflects the representative parameter 𝜃 = 1.0; the full [0, 100] scale is r… view at source ↗
read the original abstract

An adversary copies your encrypted traffic today and waits for a quantum computer to decrypt it later. How exposed are you? We show that the functional form of the answer is not merely a calibration choice -- it is structurally justified by three assumptions about adversarial production and value-decay dynamics. Under those assumptions, the HNDL compromise probability factorises into a temporal hazard, a multiplicative cryptographic-vulnerability and operational-exposure term, and a saturation denominator governed by the defense-attack intensity ratio; the marginal sensitivity to each dimension is endogenous to the organisation's position in the vulnerability-exposure plane, not a fixed global constant. Additive scoring frameworks cannot reproduce this structure because the interaction between cryptographic vulnerability and operational exposure is absent by construction, regardless of calibration. The resulting framework provides a structurally grounded basis for operational HNDL exposure prioritisation under partial observability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a formal framework for quantifying organizational exposure to Harvest-Now-Decrypt-Later (HNDL) attacks. It asserts that three assumptions on adversarial production and value-decay dynamics structurally determine the functional form of the compromise probability, which factorizes into a temporal hazard, a multiplicative cryptographic-vulnerability and operational-exposure term, and a saturation denominator set by the defense-attack intensity ratio. Marginal sensitivities are claimed to be endogenous to the organization's location in the vulnerability-exposure plane, and additive scoring frameworks are argued to be incapable of reproducing this interaction structure.

Significance. If the derivation from the three assumptions to the exact factorization is rigorous and unique, the work would supply a theoretically grounded alternative to heuristic exposure metrics for post-quantum risk prioritization. The position-dependent sensitivities constitute a distinctive prediction that could inform adaptive defense allocation under partial observability.

major comments (2)
  1. [Model derivation section (near Eq. for compromise probability)] The central claim that the three assumptions 'structurally justify' the specific factorization (temporal hazard × multiplicative term / saturation denominator) rather than permitting arbitrary functional forms requires an explicit step-by-step derivation. The abstract invokes the assumptions but does not display how each one excludes alternatives such as additive interactions or non-saturating forms; this derivation must be added and shown to be unique.
  2. [Comparison with additive frameworks] The statement that additive scoring frameworks 'cannot reproduce this structure because the interaction ... is absent by construction' needs a formal demonstration. Provide a short proof or explicit counter-example showing that no choice of weights or calibration in an additive model can recover the endogenous marginal sensitivities described in the vulnerability-exposure plane.
minor comments (2)
  1. [Assumptions subsection] Clarify the precise mathematical definitions of the three assumptions at the point they are introduced, including any implicit regularity conditions needed for the factorization to hold.
  2. [Results or illustrative examples] Add a short table or diagram illustrating how the compromise probability surface changes with position in the vulnerability-exposure plane to make the endogenous sensitivity claim concrete.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight opportunities to strengthen the presentation of the core derivations. We address each major comment below and will incorporate the requested additions in the revised manuscript.

read point-by-point responses

  1. Referee: [Model derivation section (near Eq. for compromise probability)] The central claim that the three assumptions 'structurally justify' the specific factorization (temporal hazard × multiplicative term / saturation denominator) rather than permitting arbitrary functional forms requires an explicit step-by-step derivation. The abstract invokes the assumptions but does not display how each one excludes alternatives such as additive interactions or non-saturating forms; this derivation must be added and shown to be unique.

    Authors: We agree that an explicit step-by-step derivation is required to establish uniqueness. In the revised manuscript we will add a dedicated subsection that derives the factorization directly from the three assumptions. The assumption on adversarial production will be shown to enforce the multiplicative interaction between cryptographic vulnerability and operational exposure (compromise occurs only when both are jointly satisfied). The value-decay dynamics will be shown to produce the saturation denominator governed by the defense-attack intensity ratio. The temporal hazard separates as an independent time-dependent factor. We will then demonstrate uniqueness by exhibiting the functional forms that become admissible once any assumption is relaxed (e.g., additive interactions appear when the joint-production requirement is dropped). revision: yes

  2. Referee: [Comparison with additive frameworks] The statement that additive scoring frameworks 'cannot reproduce this structure because the interaction ... is absent by construction' needs a formal demonstration. Provide a short proof or explicit counter-example showing that no choice of weights or calibration in an additive model can recover the endogenous marginal sensitivities described in the vulnerability-exposure plane.

    Authors: We accept that a formal demonstration is necessary. In the revision we will insert a short appendix containing a proof that any additive model of the form S = f(vuln) + g(exposure) + … yields marginal sensitivities that are independent of the orthogonal coordinate. Consequently, no choice of weights or monotonic transformations can reproduce the position-dependent sensitivities that arise from the saturation term in our factorization. We will also supply a brief counter-example in which an additive model fitted to the same vulnerability-exposure plane fails to match the endogenous marginals at interior points. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation presented as consequence of stated assumptions

full rationale

The paper asserts that three assumptions on adversarial production and value-decay dynamics structurally entail the specific factorization of HNDL compromise probability (temporal hazard × multiplicative term / saturation denominator). No equations or steps are exhibited that reduce the claimed form to a fitted parameter, self-citation, or definitional tautology by construction. The contrast with additive frameworks is derived from the presence of the interaction term under the assumptions rather than from renaming or smuggling prior results. The derivation remains self-contained against external benchmarks as the functional form is tied directly to the listed assumptions without load-bearing self-citation chains or uniqueness theorems imported from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on three unspecified assumptions about adversarial production and value-decay dynamics that are invoked to justify the functional form; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • ad hoc to paper Three assumptions about adversarial production and value-decay dynamics
    These are stated to structurally justify the factorization of HNDL compromise probability.

pith-pipeline@v0.9.0 · 5671 in / 1325 out tokens · 46011 ms · 2026-05-22T04:54:12.213973+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Under Hypotheses 1–3, the HNDL compromise probability factors as P_HNDL = H ⋅ V^a E^b / (V^a E^b + θ), θ = μ/λ0. ... The multiplicative structure follows directly from the intersection principle... Axiomatically, this family is further supported by three structural axioms established by Skaperdas (1996) for contest success functions: (A1) Anonymity... (A2) Independence... (A3) Homogeneity...

  • IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    P_D = θ / (u + θ) = e^{-u/θ}_{q=2} ... The value q=2 is not a fitted parameter; it is determined by the binary contest structure of Hypothesis 3

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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