Lattice Based Crypto breaks in a Superposition of Spacetimes
Divesh Aggarwal, Shashwat Agrawal, Rajendra Kumar
TL;DR
This work formalizes a quantum-computational model inspired by superpositions of spacetime geometries, introducing the complexity class $ extbf{BQP^{OI}}$ with access to an order-interference oracle. It shows that $ extbf{GI}$ and $ extbf{GapCVP}$ (with a $O(n^{3/2})$-approximation) reside in $ extbf{BQP^{OI}}$, and then proves the entire class $ extbf{SZK}$ is contained in $ extbf{BQP^{OI}}$ via reductions from Statistical Difference to SISD and a sequence of invertible-circuit constructions. Since $ extbf{LWE}$ lies in $ extbf{SZK}$ for relevant lattice-cryptography parameter regimes, this implies that lattice-based cryptography could be efficiently compromised in the $ extbf{BQP^{OI}}$ model. The results thereby bridge a physics-inspired computational paradigm with cryptographic security, underscoring parameter-sensitive vulnerabilities and motivating new hardness assumptions beyond $ extbf{SZK}$-based settings.
Abstract
We explore the computational implications of a superposition of spacetimes, a phenomenon hypothesized in quantum gravity theories. 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. In this work, it was shown that the Graph Isomorphism problem and the Gap Closest Vector Problem (with approximation factor $\mathcal{O}(n^{3/2})$) are in $\mathbf{BQP^{OI}}$. We extend this result by showing that the entire complexity class $\mathbf{SZK}$ (Statistical Zero Knowledge) is contained within $\mathbf{BQP^{OI}}$. 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 $\mathbf{SZK}$.