THE QUANTUM INTELLIGENCER

"Breaking Lattice Cryptography via Quantum Gravity-Inspired Computation: The BQP^OI Model and the Collapse of SZK" In Plain English: This research explores a futuristic idea: what if the fabric of space and time could exist in multiple states at once, like particles in quantum physics? Scientists imagine that in such a world, computers could solve certain problems much faster than they can now. The study shows that if this strange kind of space-time were possible, it would allow computers to crack a type of encryption widely expected to be safe even from future quantum computers. This encryption is based on complex math problems involving lattices—grid-like structures. The researchers found that the security of these systems relies on problems that could be solved easily in this hypothetical universe. While this isn’t a current threat—since we don’t know how to create such space-time states—it shows that future discoveries in physics could unexpectedly break today’s strongest encryption. Summary: This theoretical paper investigates the computational power of a model based on a "superposition of spacetimes," a concept arising in quantum gravity. The authors build on Shmueli (2024), who introduced the complexity class **BQP^OI**—problems solvable in quantum polynomial time with access to an oracle for **order interference**, a phenomenon associated with indefinite causal structures. Shmueli showed that **Graph Isomorphism** and the **Gap Closest Vector Problem** (with approximation factor $\mathcal{O}(n^{3/2})$) are in **BQP^OI**. This work generalizes that result by proving that the entire complexity class **SZK (Statistical Zero Knowledge)** is contained within **BQP^OI**. This inclusion is significant because many foundational problems in cryptography, including the **Learning with Errors (LWE)** problem, are known to be in **SZK**. LWE is the basis for most lattice-based cryptographic systems, which are currently considered secure against attacks by both classical and standard quantum computers, and are leading candidates for post-quantum cryptography. The result implies that in any computational model where order interference can be harnessed—such as one involving quantum superpositions of spacetime geometry—these cryptographic schemes would no longer be secure. The paper does not claim that such a model is physically realizable with current technology or even that it will ever be. Rather, it highlights a theoretical vulnerability: if future developments in quantum gravity enable access to such computational oracles, then the security assumptions underpinning much of modern post-quantum cryptography could collapse. The work thus bridges abstract complexity theory, quantum computation, and speculative physics, suggesting that the ultimate limits of computation may depend on the structure of spacetime itself. Key Points: - The paper introduces a theoretical computational model based on a superposition of spacetimes, leveraging the concept of order interference. - It proves that the complexity class **SZK (Statistical Zero Knowledge)** is contained in **BQP^OI**, a quantum complexity class with access to an order interference oracle. - This extends prior work by Shmueli (2024), which showed that Graph Isomorphism and a lattice problem were in **BQP^OI**. - Since the **Learning with Errors (LWE)** problem is in **SZK**, lattice-based cryptographic schemes relying on LWE are vulnerable in this model. - The result suggests that if quantum gravity allows for order interference, current post-quantum cryptographic standards could be broken. - The model is theoretical and does not imply a current practical threat to cryptography. - The work connects quantum gravity, computational complexity, and cryptography in a novel way. Notable Quotes: - "This immediately implies that the security of numerous lattice based cryptography schemes will be compromised in a computational model based on superposition of spacetimes, since these often rely on the hardness of the Learning with Errors problem, which is in $\mathbf{SZK}$." - "We extend this result by showing that the entire complexity class $\mathbf{SZK}$ is contained within $\mathbf{BQP^{OI}}$." - "This was initiated by Shmueli (2024) where the author introduced the complexity class $\mathbf{BQP^{OI}}$ consisting of promise problems decidable by quantum polynomial time algorithms with access to an oracle for computing order interference." Data Points: - The Gap Closest Vector Problem is solvable in **BQP^OI** with an approximation factor of $\mathcal{O}(n^{3/2})$ (cited from Shmueli, 2024). - The Learning with Errors (LWE) problem is in the complexity class **SZK**. - The complexity class **SZK** is now claimed to be contained in **BQP^OI**. - The work references Shmueli (2024) as the origin of the **BQP^OI** model. - The paper was posted to arXiv in 2024 (implied by citation of Shmueli 2024 and current date 2026-01-02). Controversial Claims: - The claim that lattice-based cryptography—currently a leading candidate for post-quantum security—could be broken in a model involving superposition of spacetimes is highly speculative, as such models are not known to be physically realizable. - The assumption that order interference, a phenomenon tied to quantum gravity, can be harnessed as a computational oracle may overextend current physical theories. - The paper implies that **SZK ⊆ BQP^OI**, which, if interpreted as a physical possibility, would mean that problems like Graph Isomorphism and lattice problems are efficiently solvable in exotic quantum-gravitational settings—a claim that challenges widely held assumptions about computational hardness. - The suggestion that spacetime structure directly influences computational complexity borders on philosophical and may lack falsifiability with current experimental capabilities. Technical Terms: - **BQP^OI**: Quantum polynomial time with access to an oracle for order interference. - **Order interference**: A quantum phenomenon related to indefinite causal order, hypothesized in quantum gravity. - **Superposition of spacetimes**: A quantum gravity concept where spacetime geometry exists in multiple configurations simultaneously. - **SZK (Statistical Zero Knowledge)**: A complexity class of problems where a prover can convince a verifier of a statement’s truth without revealing any additional information, and the zero-knowledge property holds statistically. - **Learning with Errors (LWE)**: A computational problem used as the foundation for many lattice-based cryptosystems, believed to be hard for quantum computers. - **Gap Closest Vector Problem (GapCVP)**: A lattice problem involving finding a lattice point close to a given vector, within a certain approximation factor. - **Graph Isomorphism**: The problem of determining whether two graphs have the same structure. - **Complexity class**: A set of computational problems grouped by their resource requirements (time, space, etc.). - **Oracle**: A black-box subroutine that can solve certain problems instantly, used in complexity theory to compare computational power. —Ada H. Pemberley Dispatch from The Prepared E0