Fundamental Limitations of Post-Quantum Cryptographic Architectures

Donghwa Ji; Jiho Jung; Kabgyun Jeong; Mingyu Lee

arxiv: 2605.04582 · v1 · submitted 2026-05-06 · 🪐 quant-ph · cs.CR

Fundamental Limitations of Post-Quantum Cryptographic Architectures

Jiho Jung , Donghwa Ji , Mingyu Lee , Kabgyun Jeong This is my paper

Pith reviewed 2026-05-08 18:21 UTC · model grok-4.3

classification 🪐 quant-ph cs.CR

keywords post-quantum cryptographylattice-based cryptographylearning with errorsquantum error correctiondiscrete Gaussian noiseinformation thermodynamicsquantum learning

0 comments

The pith

Lattice-based cryptography's noise injection does not permanently hide secrets from quantum error correction.

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

The paper argues that modern lattice-based cryptography secures data by adding artificial noise but this approach has fundamental limits that quantum technologies can overcome. Through a thermodynamic lens it shows that the injected discrete Gaussian noise does not erase the secret information permanently because the underlying structure stays intact in the ciphertext. Quantum error correction protocols and quantum learning models can therefore extract the mathematical kernel efficiently. This challenges the claim that such systems are unconditionally post-quantum since their security rests on temporary physical and computational bottlenecks rather than absolute theoretical barriers. A sympathetic reader would care because current standards for quantum-resistant encryption depend on these assumptions and their failure would require entirely new cryptographic foundations.

Core claim

Lattice-based schemes such as learning with errors rely on provisional complexity assumptions and on discrete Gaussian noise whose injection does not equate to permanent information erasure in the thermodynamic sense. Because the structural integrity of the cryptographic secret remains preserved within the ciphertext advanced quantum error correction protocols and quantum learning models can efficiently extract the underlying mathematical kernel. This demonstrates that classifying these frameworks as unconditionally post-quantum is premature since security depends on transient physical limits rather than impenetrable boundaries.

What carries the argument

The discrete Gaussian noise added in the learning with errors paradigm, when mapped to thermodynamic non-erasure, which preserves the secret's structure and thereby enables its extraction by quantum error correction and learning models.

If this is right

Current lattice-based post-quantum standards provide only transitional security that future quantum systems may compromise.
Security classifications as post-quantum rest on transient bottlenecks rather than fundamental impossibilities.
New cryptographic architectures that do not rely on noise injection will be needed for unconditional security.
The boundary between computational hardness and physical extractability must be reevaluated across complexity and quantum theory.

Where Pith is reading between the lines

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

This view could prompt direct tests of quantum learning algorithms on concrete LWE instances to measure extraction efficiency.
It suggests connections to physical limits on information erasure that might apply to other noisy cryptographic systems.
Designers of future standards might need to incorporate assumptions about quantum error correction capabilities explicitly.

Load-bearing premise

Intentionally injected discrete Gaussian noise does not equate to permanent erasure of information and the structural integrity of the cryptographic secret remains preserved within the ciphertext.

What would settle it

An experiment or calculation showing that quantum error correction protocols and quantum learning models cannot recover non-negligible information about the secret key from a standard LWE ciphertext beyond classical computational feasibility.

Figures

Figures reproduced from arXiv: 2605.04582 by Donghwa Ji, Jiho Jung, Kabgyun Jeong, Mingyu Lee.

**Figure 1.** Figure 1: FIG. 1: The algebraic structure of the LWE problem, view at source ↗

**Figure 2.** Figure 2: FIG. 2: Conceptual diagram of computational view at source ↗

**Figure 3.** Figure 3: FIG. 3: Structural correspondence between Shannon’s Communication Model and LWE. While the algorithmic error view at source ↗

read the original abstract

Modern lattice-based cryptography, particularly the learning with errors paradigm, relies on injecting artificial noise to secure data against quantum adversaries. This study systematically examines the theoretical and physical boundaries of this noise-reliant model across four interconnected domains: computational complexity, information-theoretic thermodynamics, quantum error correction, and quantum learning theory. Starting from the algorithmic foundation, our analysis notes that these frameworks rely on provisional complexity-theoretic assumptions that remain vulnerable to future quantum algorithmic advancements. Furthermore, by translating this cryptographic mechanism into physical thermodynamics, we illustrate that intentionally injected discrete Gaussian noise does not equate to the permanent erasure of information. Because the structural integrity of the cryptographic secret remains preserved within the ciphertext, advanced quantum error correction protocols and quantum learning models can efficiently extract the underlying mathematical kernel. Ultimately, we suggest that while lattice-based cryptography provides a robust transitional alternative, definitively classifying these frameworks as unconditionally post-quantum represents a premature classification relying on transient physical bottlenecks rather than impenetrable theoretical boundaries.

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.

Desk Editor's Note private letter to a colleague

The paper correctly notes that LWE noise preserves secrets but provides no mechanism for efficient quantum extraction.

read the letter

This paper points out that the noise added in lattice-based crypto does not erase the secret information, but it stops short of showing how quantum methods would actually extract it efficiently. The authors bring together ideas from thermodynamics, quantum error correction, and learning theory to examine the limits of the LWE paradigm. They do well in clarifying that discrete Gaussian noise leaves the structural integrity of the secret preserved, which follows from basic quantum information but is applied usefully here to challenge the post-quantum label. The cross-domain synthesis is a legitimate extension even if the core observation is not brand new. The central issue is the efficiency claim. The abstract asserts that advanced protocols can efficiently extract the underlying mathematical kernel, yet the stress-test correctly flags the absence of any explicit mechanism or analysis. Standard assumptions about LWE hardness against quantum adversaries are dismissed without new evidence or a counter-reduction. This leaves the load-bearing step unsupported in the provided material. Readers working on post-quantum standards or quantum foundations would get something from the cross-domain view. It highlights that current security relies on transient bottlenecks rather than absolute barriers. The thinking is straightforward and engages the relevant concepts without internal contradiction. I recommend sending it for peer review to let experts verify the missing steps on the extraction side.

Referee Report

2 major / 2 minor

Summary. The manuscript examines fundamental limitations of lattice-based post-quantum cryptography, focusing on the Learning With Errors (LWE) paradigm. It argues across computational complexity, information-theoretic thermodynamics, quantum error correction, and quantum learning theory that injected discrete Gaussian noise does not constitute permanent information erasure, preserving the structural integrity of the secret in the ciphertext. Consequently, advanced quantum error correction protocols and quantum learning models can efficiently extract the underlying mathematical kernel, rendering the classification of these schemes as unconditionally post-quantum premature and reliant on transient physical bottlenecks rather than theoretical boundaries.

Significance. If the claims regarding efficient extraction via QEC and quantum learning hold with concrete support, the work could bridge thermodynamics and cryptography in a novel way, highlighting physical rather than purely computational limits on post-quantum security and potentially affecting standardization efforts. The paper correctly identifies that discrete Gaussian noise preserves secret information (a standard observation in LWE), which is a valid starting point, but the absence of explicit mechanisms limits its current impact.

major comments (2)

[Abstract] Abstract: the assertion that 'advanced quantum error correction protocols and quantum learning models can efficiently extract the underlying mathematical kernel' is load-bearing for the central conclusion yet is stated without any explicit quantum algorithm, circuit construction, reduction from LWE to a QEC decoding problem, runtime analysis, or complexity bound.
[information-theoretic thermodynamics] Section on information-theoretic thermodynamics: the translation of cryptographic noise injection into a thermodynamic argument that 'intentionally injected discrete Gaussian noise does not equate to the permanent erasure of information' is presented interpretively; no first-principles derivation, quantitative bounds on preserved mutual information, or explicit mapping showing how this enables efficient QEC recovery in the LWE setting is supplied.

minor comments (2)

The abstract and domain descriptions could more clearly separate the valid observation that noise preserves information from the unsubstantiated efficiency claim for extraction.
Terminology such as 'mathematical kernel' and 'structural integrity of the cryptographic secret' would benefit from precise definitions tied to the LWE secret vector and error distribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript examining fundamental limitations in post-quantum lattice-based cryptography. We address each major comment point by point below, indicating revisions where appropriate.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that 'advanced quantum error correction protocols and quantum learning models can efficiently extract the underlying mathematical kernel' is load-bearing for the central conclusion yet is stated without any explicit quantum algorithm, circuit construction, reduction from LWE to a QEC decoding problem, runtime analysis, or complexity bound.

Authors: We acknowledge that the abstract makes a strong claim about efficient extraction without providing an explicit quantum algorithm or complexity analysis in the manuscript. Our work is a conceptual study highlighting theoretical vulnerabilities rather than presenting a new attack algorithm. The argument relies on the established preservation of secret information in LWE ciphertexts and the capabilities of existing QEC and quantum learning frameworks to potentially recover it. To strengthen the presentation, we will revise the abstract to clarify that this extraction is a theoretical implication based on information preservation, not a claim of a specific efficient algorithm constructed in this paper. This revision will better reflect the manuscript's focus on limitations. revision: yes
Referee: [information-theoretic thermodynamics] Section on information-theoretic thermodynamics: the translation of cryptographic noise injection into a thermodynamic argument that 'intentionally injected discrete Gaussian noise does not equate to the permanent erasure of information' is presented interpretively; no first-principles derivation, quantitative bounds on preserved mutual information, or explicit mapping showing how this enables efficient QEC recovery in the LWE setting is supplied.

Authors: The thermodynamics section interprets the noise injection through the lens of information theory, noting that discrete Gaussian noise in LWE does not permanently erase the secret due to its structured nature, consistent with standard LWE analyses. While we do not derive this from first principles of thermodynamics in the current draft, we reference the connection via Landauer's principle and information erasure concepts. We will revise this section to include quantitative bounds on preserved mutual information by citing relevant results from LWE literature on noise distributions, and provide a clearer mapping to how this preservation allows QEC protocols to potentially recover the kernel. However, developing a fully explicit reduction or new derivation is outside the scope of this work, which aims to point out the conceptual gap rather than close it with new constructions. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper translates the LWE noise mechanism into a thermodynamic argument that discrete Gaussian noise does not permanently erase information, then asserts that this preservation enables efficient extraction via QEC and quantum learning models. No self-definitional loops appear (e.g., no quantity defined in terms of its own extractability), no fitted parameters are relabeled as predictions, and no load-bearing self-citations or uniqueness theorems from the authors are invoked in the abstract or described chain. The central step is an interpretive claim linking non-erasure to extractability, but it does not reduce by construction to the inputs via equations or prior self-referential results; the derivation remains independent of the target conclusion and can be evaluated against external LWE hardness assumptions and physical thermodynamics without collapsing internally.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard LWE hardness assumptions plus an interpretive mapping of cryptographic noise to thermodynamic non-erasure; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Lattice-based schemes rely on provisional complexity-theoretic assumptions vulnerable to future quantum algorithms
Explicitly stated as the algorithmic foundation in the abstract.
ad hoc to paper Injected discrete Gaussian noise does not equate to permanent erasure of information
This is the load-bearing physical translation used to argue preservation of secret structure.

pith-pipeline@v0.9.0 · 5461 in / 1355 out tokens · 42643 ms · 2026-05-08T18:21:00.641803+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

Cost.FunctionalEquation (J(x)=½(x+x⁻¹)−1) washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the error term e ∈ Z_q is drawn from a specific probability distribution χ over Z_q, typically a discrete Gaussian centered at zero
Foundation.AlexanderDuality / DimensionForcing (D=3 forcing) alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the cryptographic error exhibits a fundamental structural equivalence with continuous-variable displacement errors ... GKP code

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

Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quan- tum computer, SIAM Review41, 303 (1999), https://doi.org/10.1137/S0036144598347011

work page doi:10.1137/s0036144598347011 1999
[2]

O. Regev, On lattices, learning with errors, random lin- ear codes, and cryptography, inProceedings of the Thirty- Seventh Annual ACM Symposium on Theory of Comput- ing, STOC ’05 (Association for Computing Machinery, New York, NY, USA, 2005) pp. 84–93

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Department of Commerce, Washington, D.C., 2024)

National Institute of Standards and Technology,Module- Lattice-Based Key-Encapsulation Mechanism Standard, Federal Information Processing Standards Publication (FIPS) NIST FIPS 203 (U.S. Department of Commerce, Washington, D.C., 2024)

work page 2024
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work page 2025
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Giovannetti, S

V. Giovannetti, S. Lloyd, and L. Maccone, Quantum ran- dom access memory, Phys. Rev. Lett.100, 160501 (2008)

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K. M. R. Audenaert, A sharp continuity estimate for the von neumann entropy, Journal of Physics A: Mathemat- ical and Theoretical40, 8127 (2007)

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[18]

Jeong, Sample-size-reduction of quantum states for the noisy linear problem, Annals of Physics449, 169215 (2023)

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Gottesman, A

D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A64, 012310 (2001)

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P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys- Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, L. Frunzio, M. Mirrahimi, and M. H. Devoret, Quantum error correction of a qubit encoded in grid states of an oscillator, Nature584, 368 (2020)

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L. G. Valiant, A theory of the learnable, Commun. ACM 27, 1134 (1984)

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N. H. Bshouty and J. C. Jackson, Learning dnf over the uniform distribution using a quantum example oracle, in Proceedings of the Eighth Annual Conference on Com- putational Learning Theory, COLT ’95 (Association for Computing Machinery, New York, NY, USA, 1995) pp. 118–127

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W. Song, Y. Lim, K. Jeong, Y.-S. Ji, J. Lee, J. Kim, M. S. Kim, and J. Bang, Quantum solvability of noisy linear problems by divide-and-conquer strategy, Quan- tum Science and Technology7, 025009 (2022)

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W. Song, Y. Lim, K. Jeong, J. Lee, J. J. Park, M. S. Kim, and J. Bang, Polynomial t-depth quantum solv- ability of noisy binary linear problem: from quantum- sample preparation to main computation, New Journal of Physics24, 103014 (2022)

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[26]

Zheng, J

M. Zheng, J. Zeng, W. Yang, P.-J. Chang, Q. Lu, B. Yan, H. Zhang, M. Wang, S. Wei, and G.-L. Long, Quantum- classical hybrid algorithm for solving the learning-with- errors problem on nisq devices, Communications Physics 8, 208 (2025)

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[1] [1]

P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quan- tum computer, SIAM Review41, 303 (1999), https://doi.org/10.1137/S0036144598347011

work page doi:10.1137/s0036144598347011 1999

[2] [2]

O. Regev, On lattices, learning with errors, random lin- ear codes, and cryptography, inProceedings of the Thirty- Seventh Annual ACM Symposium on Theory of Comput- ing, STOC ’05 (Association for Computing Machinery, New York, NY, USA, 2005) pp. 84–93

work page 2005

[3] [3]

Department of Commerce, Washington, D.C., 2024)

National Institute of Standards and Technology,Module- Lattice-Based Key-Encapsulation Mechanism Standard, Federal Information Processing Standards Publication (FIPS) NIST FIPS 203 (U.S. Department of Commerce, Washington, D.C., 2024)

work page 2024

[4] [4]

Peikert, A decade of lattice cryptography, Found

C. Peikert, A decade of lattice cryptography, Found. Trends Theor. Comput. Sci.10, 283 (2016)

work page 2016

[5] [5]

C. E. Shannon, A mathematical theory of communica- tion, The Bell System Technical Journal27, 379 (1948)

work page 1948

[6] [6]

R. M. Fano,The transmission of information, Vol. 65 (Massachusetts Institute of Technology, Research Labo- ratory of Electronics, 1949)

work page 1949

[7] [7]

A. L. Grimsmo and S. Puri, Quantum error correction with the gottesman-kitaev-preskill code, PRX Quantum 2, 020101 (2021)

work page 2021

[8] [8]

K. Noh, C. Chamberland, and F. G. Brand˜ ao, Low- overhead fault-tolerant quantum error correction with the surface-gkp code, PRX Quantum3, 010315 (2022)

work page 2022

[9] [9]

A. B. Grilo, I. Kerenidis, and T. Zijlstra, Learning-with- errors problem is easy with quantum samples, Phys. Rev. A99, 032314 (2019)

work page 2019

[10] [10]

Poremba, Y

A. Poremba, Y. Quek, and P. Shor, The learning stabiliz- ers with noise problem (2025), arXiv:2410.18953 [quant- ph]

work page arXiv 2025

[11] [11]

J. Zeng, M. Zheng, H. Li, S. Wei, and G. Long, Analysis of learning with errors problems with variational quan- tum algorithms, Europhysics Letters150, 58001 (2025). 8

work page 2025

[12] [12]

A. W. Cross, G. Smith, and J. A. Smolin, Quantum learn- ing robust against noise, Phys. Rev. A92, 012327 (2015)

work page 2015

[13] [13]

Bernstein and U

E. Bernstein and U. Vazirani, Quantum complexity the- ory, inProceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, STOC ’93 (Asso- ciation for Computing Machinery, New York, NY, USA,

work page

[14] [14]

Giovannetti, S

V. Giovannetti, S. Lloyd, and L. Maccone, Quantum ran- dom access memory, Phys. Rev. Lett.100, 160501 (2008)

work page 2008

[15] [15]

Landauer, Irreversibility and heat generation in the computing process, IBM Journal of Research and Devel- opment5, 183 (1961)

R. Landauer, Irreversibility and heat generation in the computing process, IBM Journal of Research and Devel- opment5, 183 (1961)

work page 1961

[16] [16]

C. H. Bennett and G. Brassard, Quantum cryptogra- phy: Public key distribution and coin tossing, Theo- retical Computer Science560, 7 (2014), theoretical As- pects of Quantum Cryptography – celebrating 30 years of BB84

work page 2014

[17] [17]

K. M. R. Audenaert, A sharp continuity estimate for the von neumann entropy, Journal of Physics A: Mathemat- ical and Theoretical40, 8127 (2007)

work page 2007

[18] [18]

Jeong, Sample-size-reduction of quantum states for the noisy linear problem, Annals of Physics449, 169215 (2023)

K. Jeong, Sample-size-reduction of quantum states for the noisy linear problem, Annals of Physics449, 169215 (2023)

work page 2023

[19] [19]

Gottesman, A

D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A64, 012310 (2001)

work page 2001

[20] [20]

Campagne-Ibarcq, A

P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys- Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, L. Frunzio, M. Mirrahimi, and M. H. Devoret, Quantum error correction of a qubit encoded in grid states of an oscillator, Nature584, 368 (2020)

work page 2020

[21] [21]

Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics303, 2 (2003)

A. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics303, 2 (2003)

work page 2003

[22] [22]

L. G. Valiant, A theory of the learnable, Commun. ACM 27, 1134 (1984)

work page 1984

[23] [23]

N. H. Bshouty and J. C. Jackson, Learning dnf over the uniform distribution using a quantum example oracle, in Proceedings of the Eighth Annual Conference on Com- putational Learning Theory, COLT ’95 (Association for Computing Machinery, New York, NY, USA, 1995) pp. 118–127

work page 1995

[24] [24]

W. Song, Y. Lim, K. Jeong, Y.-S. Ji, J. Lee, J. Kim, M. S. Kim, and J. Bang, Quantum solvability of noisy linear problems by divide-and-conquer strategy, Quan- tum Science and Technology7, 025009 (2022)

work page 2022

[25] [25]

W. Song, Y. Lim, K. Jeong, J. Lee, J. J. Park, M. S. Kim, and J. Bang, Polynomial t-depth quantum solv- ability of noisy binary linear problem: from quantum- sample preparation to main computation, New Journal of Physics24, 103014 (2022)

work page 2022

[26] [26]

Zheng, J

M. Zheng, J. Zeng, W. Yang, P.-J. Chang, Q. Lu, B. Yan, H. Zhang, M. Wang, S. Wei, and G.-L. Long, Quantum- classical hybrid algorithm for solving the learning-with- errors problem on nisq devices, Communications Physics 8, 208 (2025)

work page 2025