THE QUANTUM INTELLIGENCER

A Non-Algebraic Foundation for Post-Quantum Cryptography: The Intractability of Chaotic Symbolic Walks via the Symbolic Path Inversion Problem In Plain English: Most current digital security relies on math problems that involve patterns or structures—like multiplying large numbers—that are hard for regular computers to reverse. But quantum computers might break these soon. This paper proposes a new kind of problem based on chaotic, unpredictable paths in a digital system that don’t follow neat mathematical rules. It’s easy to walk forward along such a path, but nearly impossible to figure out where someone started just by looking at the end. The researchers show that even powerful quantum computers wouldn’t be able to speed up guessing the right path, because there are far too many possible routes and no reliable way to check guesses. If developed further, this idea could help build encryption that stays secure even in a future with advanced quantum computers. Summary: The paper introduces the Symbolic Path Inversion Problem (SPIP) as a new computational hardness assumption designed to underpin post-quantum cryptographic systems without relying on algebraic structures. Traditional cryptographic foundations—such as integer factorization, discrete logarithms, and Learning with Errors (LWE)—are vulnerable to quantum algorithms due to their inherent algebraic symmetries. To overcome this, SPIP leverages symbolic trajectories generated by iterated contractive affine maps over the binary field Z2, perturbed by bounded noise and rounded in a way that induces non-injectivity. This results in chaotic symbolic evolution where multiple distinct paths converge to the same endpoint, making inversion computationally infeasible. The authors prove that SPIP is PSPACE-hard and #P-hard, placing it among the most computationally difficult problems in theoretical computer science. Empirical simulations demonstrate that even short symbolic sequences generate hundreds to billions of valid preimage paths, with path counts growing exponentially—reaching 2^256 for moderate parameters—rendering exhaustive search infeasible. A key contribution is the analysis of quantum attack resistance: Grover’s algorithm, while theoretically offering quadratic speedup for unstructured search, is shown to provide no practical advantage against SPIP due to oracle ambiguity (uncertainty in defining a correct check) and verification instability (inability to reliably confirm candidate solutions). These properties suggest SPIP could serve as a robust basis for cryptographic primitives resistant to both classical and quantum adversaries, offering scalability and unpredictability absent in algebraic systems. Key Points: - Most existing cryptographic systems rely on algebraic structures vulnerable to quantum attacks. - The Symbolic Path Inversion Problem (SPIP) is proposed as a non-algebraic alternative based on chaotic symbolic dynamics. - SPIP uses contractive affine maps over Z2 with noise and rounding to generate non-injective symbolic trajectories. - Inversion is computationally hard due to exponential path multiplicity leading to combinatorial explosion. - SPIP is proven to be PSPACE-hard and #P-hard, indicating extreme computational difficulty. - Empirical results show over 500 valid trajectories for small parameters and up to 2^256 paths for moderate ones. - Grover-style quantum search is ineffective due to oracle ambiguity and unstable verification. - The system avoids algebraic symmetries, potentially offering stronger resistance to structural attacks. - SPIP represents a novel intersection of dynamical systems theory and cryptography. - The work aims to establish a new class of post-quantum hardness assumptions grounded in chaos and non-injectivity. Notable Quotes: - "Most classical and post-quantum cryptographic assumptions... rely on algebraic structures such as rings or vector spaces." - "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..." - "Even short symbolic sequences (e.g., n = 3, m = 2) can produce over 500 valid trajectories for a single endpoint, with exponential growth reaching 2^256 paths for moderate parameters." - "Grover-style search offers no practical advantage due to oracle ambiguity and verification instability." Data Points: - Symbolic sequences with parameters n = 3, m = 2 generate over 500 valid trajectories for a single endpoint. - Path counts grow exponentially, reaching 2^256 valid paths for moderate parameter settings. - Hardness classes established: PSPACE-hard and #P-hard. - Operates over the binary field Z2 with bounded noise and contractive affine maps. - Rounding-induced non-injectivity is a core mechanism enabling security. - Quantum analysis focuses on limitations of Grover’s algorithm under oracle ambiguity. Controversial Claims: - The claim that SPIP is PSPACE-hard and #P-hard based solely on the provided abstract lacks detailed proof exposition and may require rigorous peer review to validate, especially given the novelty of the construction. - The assertion that Grover-style quantum search offers "no practical advantage" challenges established expectations in quantum cryptanalysis and would need extensive formal and experimental validation, as quadratic speedups are typically assumed for unstructured search unless verification is prohibitively expensive. - The characterization of SPIP as "inherently non-algebraic" may be debated, as the underlying maps (affine transformations over Z2) still possess algebraic structure, and the claim hinges on the argument that rounding and noise destroy exploitable symmetry. - The scalability and practical implementability of cryptographic schemes based on SPIP remain speculative without concrete constructions (e.g., encryption, signatures) or efficiency analysis. Technical Terms: - Symbolic Path Inversion Problem (SPIP) - Contractive affine maps - Bounded noise - Z2 (binary field) - Symbolic trajectories - Chaotic symbolic evolution - Rounding-induced non-injectivity - PSPACE-hard - #P-hard - Learning with Errors (LWE) - Post-quantum cryptography - Quantum resistance - Grover’s algorithm - Oracle ambiguity - Verification instability - Non-algebraic hardness assumption - Combinatorial explosion - Dynamical systems - Hardness assumption - Inversion infeasibility —Ada H. Pemberley Dispatch from The Prepared E0