Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Intractability of Chaotic Symbolic Walks: Toward a Non-Algebraic Post-Quantum Hardness Assumption

Mohamed Aly Bouke

TL;DR

The paper introduces SPIP, a structure-free hardness assumption based on chaotic symbolic walks produced by contractive affine maps with bounded noise over $\mathbb{Z}^2$. It argues that inverting symbolic trajectories is PSPACE-hard and #P-hard due to exponential path explosion and many-to-one endpoint mappings, with empirical simulations showing substantial trajectory diversity even for small parameters. Quantum analysis suggests Grover-style search offers limited advantage due to oracle ambiguity and verification instability, positioning SPIP as a robust post-quantum foundation free of algebraic symmetry. The work also connects to prior RNIFS theory, outlines parameter tradeoffs, and discusses potential cryptographic primitives while identifying open questions on encoding, side-channel security, and formal quantum security models.

Abstract

Most classical and post-quantum cryptographic assumptions, including integer factorization, discrete logarithms, and Learning with Errors (LWE), rely on algebraic structures such as rings or vector spaces. While mathematically powerful, these structures can be exploited by quantum algorithms or advanced algebraic attacks, raising a pressing need for structure-free alternatives. To address this gap, we introduce the Symbolic Path Inversion Problem (SPIP), a new computational hardness assumption based on symbolic trajectories generated by contractive affine maps with bounded noise over Z2. Unlike traditional systems, SPIP is inherently non-algebraic and relies on chaotic symbolic evolution and rounding-induced non-injectivity to render inversion computationally infeasible. We prove that SPIP is PSPACE-hard and #P-hard, and demonstrate through empirical simulation that even short symbolic sequences (e.g., n = 3, m = 2) can produce over 500 valid trajectories for a single endpoint, with exponential growth reaching 2256 paths for moderate parameters. A quantum security analysis further shows that Grover-style search offers no practical advantage due to oracle ambiguity and verification instability. These results position SPIP as a viable foundation for post-quantum cryptography that avoids the vulnerabilities of algebraic symmetry while offering scalability, unpredictability, and resistance to both classical and quantum inversion.

On the Intractability of Chaotic Symbolic Walks: Toward a Non-Algebraic Post-Quantum Hardness Assumption

TL;DR

The paper introduces SPIP, a structure-free hardness assumption based on chaotic symbolic walks produced by contractive affine maps with bounded noise over . It argues that inverting symbolic trajectories is PSPACE-hard and #P-hard due to exponential path explosion and many-to-one endpoint mappings, with empirical simulations showing substantial trajectory diversity even for small parameters. Quantum analysis suggests Grover-style search offers limited advantage due to oracle ambiguity and verification instability, positioning SPIP as a robust post-quantum foundation free of algebraic symmetry. The work also connects to prior RNIFS theory, outlines parameter tradeoffs, and discusses potential cryptographic primitives while identifying open questions on encoding, side-channel security, and formal quantum security models.

Abstract

Most classical and post-quantum cryptographic assumptions, including integer factorization, discrete logarithms, and Learning with Errors (LWE), rely on algebraic structures such as rings or vector spaces. While mathematically powerful, these structures can be exploited by quantum algorithms or advanced algebraic attacks, raising a pressing need for structure-free alternatives. To address this gap, we introduce the Symbolic Path Inversion Problem (SPIP), a new computational hardness assumption based on symbolic trajectories generated by contractive affine maps with bounded noise over Z2. Unlike traditional systems, SPIP is inherently non-algebraic and relies on chaotic symbolic evolution and rounding-induced non-injectivity to render inversion computationally infeasible. We prove that SPIP is PSPACE-hard and #P-hard, and demonstrate through empirical simulation that even short symbolic sequences (e.g., n = 3, m = 2) can produce over 500 valid trajectories for a single endpoint, with exponential growth reaching 2256 paths for moderate parameters. A quantum security analysis further shows that Grover-style search offers no practical advantage due to oracle ambiguity and verification instability. These results position SPIP as a viable foundation for post-quantum cryptography that avoids the vulnerabilities of algebraic symmetry while offering scalability, unpredictability, and resistance to both classical and quantum inversion.

Paper Structure

This paper contains 25 sections, 2 theorems, 20 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Positioning SPIP
  3. Mathematical Preliminaries
  4. Contractive Affine Maps over $\mathbb{Z}^2$
  5. Symbolic Trajectories
  6. Noise Distribution and Entropy Source
  7. Symbolic Path Inversion Problem
  8. Problem Definition
  9. Problem Characteristics
  10. Illustrative Numerical Example
  11. Setup:
  12. Complexity Theoretic Analysis
  13. PSPACE-Hardness
  14. #P-Hardness
  15. Growth of the Solution Space
  16. ...and 10 more sections

Key Result

Theorem 1

The SPIP is PSPACE-hard.

Figures (8)

  • Figure 1: Effect of path length $n$ and noise bound $\epsilon$ on symbolic trajectory space size, assuming fixed $m = 10$. Growth is estimated as $\log_2((m \cdot k)^n)$, with $k \sim \lceil \epsilon \cdot 10 \rceil$.
  • Figure 2: Unique endpoint growth vs. step count.
  • Figure 3: Entropy vs. average spatial dispersion.
  • Figure 4: Collision count vs. experiment.
  • Figure 5: Symbolic freedom declines with transformation count.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Definition 1: SPIP
  • Definition 2: Symbolic Reachability Problem
  • Theorem 1
  • proof
  • Definition 3: #PATH COUNT Problem
  • Theorem 2
  • proof