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

Daisuke Ishii, Rizwan Jahangir

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.

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

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.

Paper Structure

This paper contains 35 sections, 1 theorem, 1 equation, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Research Questions
  4. Background: The Gap Between Quantum Threats and Trust
  5. Contributions
  6. Novelty
  7. Related Work and Positioning
  8. Formal lattice cryptography.
  9. Complexity theory and cryptography.
  10. Interpretability and assurance communication.
  11. Paper Organization
  12. Three-Part Security Interpretation: Classical, Quantum, and Reduction-Backed
  13. Interpretive Model
  14. Definitions
  15. Worked Example: RSA vs. ML-KEM
  16. ...and 20 more sections

Key Result

Proposition 1

The Classical--Quantum--Reduction Security Interpretation is an interpretive classification tool. It does not by itself imply cryptographic security, average-case hardness, reduction-based guarantees, or concrete parameter security. Its purpose is to provide a structured vocabulary for discussing th

Figures (3)

  • Figure 1: Architecture of the Explainable PQC framework. Three layers connect complexity-theoretic interpretation, mathematical structure, and empirical evidence. The framework is explicitly scoped as an interpretive and communication tool, not a formal proof system.
  • Figure 2: Schematic illustration of lattice basis reduction. A lattice (gray points) with an original basis $(\mathbf{b}_1, \mathbf{b}_2)$ is transformed via reduction algorithms (LLL/BKZ) into a reduced basis $(\mathbf{b}_1', \mathbf{b}_2')$ with shorter, more orthogonal vectors. The dashed vector $\mathbf{v}^*$ represents the shortest lattice vector (SVP solution). In high dimensions, finding $\mathbf{v}^*$ becomes computationally intractable.
  • Figure 3: Computation time versus lattice dimension $n$ for LLL (approximate) and EKZ (exact) solvers. The EKZ solver times out at 40 dimensions, illustrating the rapid growth of exact-solution computation cost even at low dimensions.

Theorems & Definitions (6)

  • Proposition 1: Scope of the Interpretive Framework
  • Definition 1: Explainable PQC Framework
  • Definition 2: P --- Polynomial Time
  • Definition 3: BQP --- Bounded-Error Quantum Polynomial Time
  • Remark 1: On BQP vs. NP
  • Remark 2: On Lattice Problem Hardness