Explainable PQC: A Layered Interpretive Framework for Post-Quantum Cryptographic Security Assumptions

Daisuke Ishii; Rizwan Jahangir

arxiv: 2604.03665 · v1 · submitted 2026-04-04 · 💻 cs.CR · math.AG

Explainable PQC: A Layered Interpretive Framework for Post-Quantum Cryptographic Security Assumptions

Daisuke Ishii , Rizwan Jahangir This is my paper

Pith reviewed 2026-05-13 17:35 UTC · model grok-4.3

classification 💻 cs.CR math.AG

keywords post-quantum cryptographylattice-based cryptographyHodge theorypolyhedral geometrylattice basis reductionML-KEMsecurity assumptionsinterpretive framework

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The pith

A three-layer framework called Explainable PQC can interpret post-quantum cryptographic security assumptions through complexity classes, geometric structures, and low-dimensional experiments.

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

The paper proposes an interdisciplinary framework to organize and communicate the meaning of post-quantum cryptographic security assumptions, especially those underlying lattice-based schemes such as ML-KEM and ML-DSA. It connects three layers: a complexity-based model that separates classical security, quantum security, and reduction-backed hardness; an exploratory mathematical layer that applies combinatorial Hodge theory and polyhedral geometry to lattice problems; and an empirical layer that runs Julia experiments on algorithms like LLL and BKZ in small dimensions. A sympathetic reader would care because formal reduction proofs are technically dense, and an interpretive structure might make it easier to understand what the assumptions actually protect against. The contribution stays conceptual and organizational rather than producing new hardness proofs or attacks.

Core claim

The central claim is that PQC security assumptions can be represented and communicated through the Explainable PQC framework, whose three layers link a complexity-theoretic interpretive model, combinatorial Hodge theory and polyhedral geometry for structural analysis of lattice hardness, and Julia-based empirical measurement of basis reduction behavior, thereby clarifying the scope of existing security arguments without replacing formal proofs.

What carries the argument

The Explainable PQC three-layer framework, which uses a complexity-based interpretive model to distinguish security notions, combinatorial Hodge theory and polyhedral geometry to examine lattice structure, and a Julia platform to observe LLL and BKZ behavior in low dimensions.

If this is right

Security assumptions can be described by separating classical, quantum, and reduction-backed hardness using complexity classes as language.
Combinatorial Hodge theory and polyhedral geometry can reveal structural features of lattice hardness problems.
Low-dimensional Julia experiments on LLL and BKZ can supply empirical observations about basis reduction that complement theoretical arguments.
The framework remains strictly interpretive and does not generate new cryptographic hardness results or concrete parameter recommendations.

Where Pith is reading between the lines

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

The same layered approach might later be tested on non-lattice PQC families to see whether geometric or empirical layers transfer.
If the Hodge-theoretic layer identifies new invariants, it could suggest alternative ways to bound the cost of lattice attacks.
Low-dimensional experimental outputs could be scaled or extrapolated to check consistency with known high-dimensional security estimates.
The framework's emphasis on communication could support clearer explanations of PQC migration timelines for non-specialist audiences.

Load-bearing premise

That linking a complexity-language layer, a Hodge-theory layer, and a low-dimensional Julia experiment layer will produce useful technical interpretation of existing PQC security assumptions beyond what reduction proofs already supply.

What would settle it

Running the framework on the module-learning-with-errors assumption in ML-KEM and finding that none of the three layers produces any concrete interpretive statement or visualization that is not already present in standard reduction proofs would falsify the claimed added value.

Figures

Figures reproduced from arXiv: 2604.03665 by Daisuke Ishii, Rizwan Jahangir.

**Figure 2.** Figure 2: Schematic illustration of lattice basis reduction. A lattice (gray points) with an original basis [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Computation time versus lattice dimension [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

read the original abstract

This paper studies how post-quantum cryptographic (PQC) security assumptions can be represented and communicated through a structured, layered framework that is useful for technical interpretation but does not replace formal cryptographic proofs. We propose ``Explainable PQC,'' an interdisciplinary framework connecting three layers: (1) a complexity-based interpretive model that distinguishes classical security, quantum security, and reduction-backed hardness, drawing on computational complexity classes as supporting language; (2) an exploratory mathematical investigation applying combinatorial Hodge theory and polyhedral geometry to study structural aspects of lattice hardness; and (3)~an empirical experimentation platform, implemented in Julia, for measuring the behavior of lattice basis reduction algorithms (LLL, BKZ) in low-dimensional settings. The motivating case study throughout the paper is lattice-based PQC, including ML-KEM (FIPS 203) and ML-DSA (FIPS 204). The contribution of this paper is conceptual and organizational: it defines a layered interpretive framework, clarifies its scope relative to formal cryptographic proofs and reduction-based security arguments, and identifies mathematical and implementation-level directions through which PQC security claims may be more transparently communicated. This paper does not claim new cryptographic hardness results, new attacks, or concrete security parameter estimates.

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

A clear but modest proposal for a three-layer organizational framework on PQC assumptions, with no new results or demonstrated interpretive gains.

read the letter

The main point is that this paper defines an 'Explainable PQC' framework with three layers for discussing lattice-based security assumptions, but it produces no new hardness results, attacks, or parameter changes. It stays strictly conceptual and says so directly. The first layer uses complexity classes to separate classical security, quantum security, and reduction-backed hardness. The second applies combinatorial Hodge theory and polyhedral geometry to lattice problems in an exploratory manner. The third runs low-dimensional Julia experiments on LLL and BKZ for basis reduction, using ML-KEM and ML-DSA as examples. The authors keep the scope narrow and consistent, which avoids the usual overreach in this area. That clarity is the main strength. The paper draws on existing literature for each piece without claiming to derive anything fresh from the combination. The soft spots are straightforward. The layers are assembled but not shown to generate interpretations that improve on standard reduction arguments. The Hodge and polyhedral section is labeled exploratory and does not deliver concrete new links to security levels. The Julia experiments stay in low dimensions, so they function as illustrations rather than tests of the actual high-dimensional hardness that matters for PQC. No equations or predictions are introduced that could be checked independently. This is the kind of work that might interest readers who want structured ways to talk about crypto assumptions across fields, such as complexity theorists or people teaching lattice crypto. It will not change how practitioners choose parameters or prove security. The structure is internally consistent and the disclaimers are explicit, so it deserves a serious referee in a venue open to conceptual pieces rather than a desk reject.

Referee Report

0 major / 2 minor

Summary. The paper proposes 'Explainable PQC,' a three-layer interdisciplinary framework for representing and communicating post-quantum cryptographic security assumptions without replacing formal proofs. Layer (1) is a complexity-based interpretive model distinguishing classical security, quantum security, and reduction-backed hardness using computational complexity classes. Layer (2) applies combinatorial Hodge theory and polyhedral geometry to explore structural aspects of lattice hardness. Layer (3) is an empirical platform in Julia for low-dimensional experiments with lattice basis reduction algorithms (LLL, BKZ). The motivating case study is lattice-based PQC including ML-KEM (FIPS 203) and ML-DSA (FIPS 204). The contribution is explicitly conceptual and organizational: defining the framework, clarifying its scope relative to reduction proofs, and identifying future directions, with no new hardness results, attacks, or quantitative security estimates claimed.

Significance. If the framework is coherently adopted, it offers a structured way to organize existing ideas from complexity theory, algebraic topology, and empirical cryptanalysis for discussing PQC assumptions, potentially aiding communication across subfields. The paper appropriately scopes its contribution as definitional rather than predictive and draws on external literature for each layer. No machine-checked proofs, parameter-free derivations, or falsifiable quantitative predictions are present, consistent with the stated scope; the value is therefore primarily organizational rather than advancing new technical insight beyond what reduction arguments already supply.

minor comments (2)

Abstract, layer (2) description: the application of combinatorial Hodge theory to lattice hardness is stated at a high level; adding one concrete illustrative example (e.g., how a specific polyhedral property relates to a known lattice problem) would improve clarity without altering the conceptual scope.
Throughout: ensure that the interactions or hand-off points between the three layers are described with the same level of explicitness used for the individual layer definitions, to avoid any impression that the layers remain disconnected.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript. The recommendation for minor revision is noted; we will incorporate improvements to presentation and clarity in the revised version while preserving the explicitly conceptual scope of the work.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper explicitly frames its contribution as conceptual and organizational: it defines a three-layer interpretive framework (complexity model, Hodge/polyhedral investigation, and Julia experimentation platform) without introducing any equations, predictions, fitted parameters, or quantitative claims that could reduce to self-defined inputs by construction. No load-bearing steps rely on self-citation chains, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation; each layer draws on external literature, and the text distinguishes its scope from formal reduction proofs. The central claim is self-contained through coherent definition and scoping rather than any derivation that collapses to its own premises.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper rests on standard domain assumptions from lattice-based cryptography and introduces one new organizational construct; no free parameters are fitted and no new physical or mathematical entities are postulated.

axioms (1)

domain assumption Lattice problems remain hard on average for appropriately chosen parameters in both classical and quantum settings
Invoked as the foundation for the complexity-based interpretive layer and the choice of ML-KEM/ML-DSA as motivating case studies.

invented entities (1)

Explainable PQC framework no independent evidence
purpose: To connect complexity language, geometric analysis, and empirical observation for clearer communication of PQC assumptions
Newly defined three-layer structure whose value is asserted rather than demonstrated through external validation or new results.

pith-pipeline@v0.9.0 · 5524 in / 1417 out tokens · 49283 ms · 2026-05-13T17:35:06.057750+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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SIAM Journal on Computing 26(5), 1484– 1509 (Oct 1997)

P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,”SIAM Journal on Computing, vol. 26, no. 5, pp. 1484–1509, Oct. 1997.doi: 10.1137/S0097539795293172 11

work page doi:10.1137/s0097539795293172 1997

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Module-Lattice-Based Key-Encapsulation Mechanism Standard,

National Institute of Standards and Technology (NIST), “Module-Lattice-Based Key-Encapsulation Mechanism Standard,” FIPS 203, Aug. 2024.https://csrc.nist.gov/pubs/fips/203/final

work page 2024

[3] [3]

Module-Lattice-Based Digital Signature Standard,

National Institute of Standards and Technology (NIST), “Module-Lattice-Based Digital Signature Standard,” FIPS 204, Aug. 2024.https://csrc.nist.gov/pubs/fips/204/final

work page 2024

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On lattices, learning with errors, random linear codes, and cryptography,

O. Regev, “On lattices, learning with errors, random linear codes, and cryptography,”Journal of the ACM, vol. 56, no. 6, Art. 34, 2009. (Preliminary version inProc. 37th ACM STOC, pp. 84–93, 2005.) doi: 10.1145/1568318.1568324

work page doi:10.1145/1568318.1568324 2009

[5] [5]

In: Computer Vision – ECCV 2018

D. Micciancio and O. Regev, “Lattice-based cryptography,” inPost-Quantum Cryptography(D. J. Bernstein, J. Buchmann, and E. Dahmen, eds.), pp. 147–191, Springer, Berlin, 2009.doi: 10.1007/978- 3-540-88702-7_5

work page doi:10.1007/978- 2009

[6] [6]

A decade of lattice cryptography,

C. Peikert, “A decade of lattice cryptography,”Foundations and Trends in Theoretical Computer Science, vol. 10, no. 4, pp. 283–424, 2016.doi: 10.1561/0400000074

work page doi:10.1561/0400000074 2016

[7] [7]

BQP and the polynomial hierarchy,

S. Aaronson, “BQP and the polynomial hierarchy,” inProc. 42nd ACM STOC, pp. 141–150, 2010. doi: 10.1145/1806689.1806711

work page doi:10.1145/1806689.1806711 2010

[8] [8]

Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems,

C.-P. Schnorr and M. Euchner, “Lattice basis reduction: Improved practical algorithms and solving sub- set sum problems,”Mathematical Programming, vol. 66, pp. 181–199, 1994.doi: 10.1007/BF01581144

work page doi:10.1007/bf01581144 1994

[9] [9]

Predicting lattice reduction,

N. Gama and P. Q. Nguyen, “Predicting lattice reduction,” inAdvances in Cryptology — EUROCRYPT 2008, LNCS vol. 4965, pp. 31–51, Springer, 2008.doi: 10.1007/978-3-540-78967-3_3

work page doi:10.1007/978-3-540-78967-3_3 2008

[10] [10]

BKZ 2.0: Better lattice security estimates,

Y. Chen and P. Q. Nguyen, “BKZ 2.0: Better lattice security estimates,” inAdvances in Cryptology — ASIACRYPT 2011, LNCS vol. 7073, pp. 1–20, Springer, 2011.doi: 10.1007/978-3-642-25385-0_1

work page doi:10.1007/978-3-642-25385-0_1 2011

[11] [11]

The intersection cohomology of a fan and the Hodge conjecture for toric varieties,

R. Jahangir, “The intersection cohomology of a fan and the Hodge conjecture for toric varieties,” arXiv preprint arXiv:2512.06755 [math.AG], 2025.https://arxiv.org/abs/2512.06755

work page arXiv 2025

[12] [12]

Canonical lattices and integer relations associated to rational fans,

R. Jahangir, “Canonical lattices and integer relations associated to rational fans,” arXiv preprint arXiv:2601.05678 [math.CO], 2026.https://arxiv.org/abs/2601.05678

work page arXiv 2026

[13] [13]

doi: 10.1007/BF01457454

A. K. Lenstra, H. W. Lenstra, and L. Lovász, “Factoring polynomials with rational coefficients,” Mathematische Annalen, vol. 261, pp. 515–534, 1982.doi: 10.1007/BF01457454 12

work page doi:10.1007/bf01457454 1982