aiqbench analysis
This research fundamentally challenges the security assumptions of post-quantum cryptographic schemes relying on non-abelian group rings, demonstrating that unit inversion can be solved efficiently via block decomposition rather than requiring hard Hidden Subgroup Problem solvers. By providing an explicit reversible quantum circuit for this process, the study delineates the precise structural boundaries where current defenses fail and proposes new constructions based on revised security assumptions. The work is critical for cryptographers and quantum algorithm designers seeking to understand the true hardness limits of lattice-based and group-ring-based encryption in a post-quantum era.
Quantum Inversion of Units in Group Rings: Block Dimension, Not Commutativity, Governs Hardness
arXiv:2609.10596v1 Announce Type: new Abstract: Several public-key schemes base their security on the belief that inverting a unit of a group ring is hard. A recent result showed that this belief is false on a quantum computer when the group is abelian. To restore security, designers moved to non-abelian groups, especially dihedral groups, believing that the hardness of the dihedral hidden subgroup problem (HSP) would protect the scheme. This paper shows that unit inversion is a different problem and does not require an HSP solver. Instead, it can be solved by a change of basis that splits the group ring into small matrix blocks. We prove that unit inversion is polynomial-time, classically and quantumly, when an efficient generalized Fourier transform exists, the group ring is semisimple, and the largest matrix block has polynomial size. Dihedral group rings satisfy these conditions because their irreducible representations have dimension at most 2 and an efficient Fourier transform exists. We give an explicit reversible quantum circuit for the block-inversion step and validate it in a register-level simulator. We also identify the exact structural boundary where the method stops and propose a candidate construction in the surviving regime under a new, clearly stated security assumption. The constructive results are supported by reproducible software artifacts and experiments.
Abstract:Several public-key schemes base their security on the belief that inverting a unit of a group ring is hard. A recent result showed that this belief is false on a quantum computer when the group is abelian. To restore security, designers moved to non-abelian groups, especially dihedral groups, believing that the hardness of the dihedral hidden subgroup problem (HSP) would protect the scheme. This paper shows that unit inversion is a different problem and does not require an HSP solver. Instead, it can be solved by a change of basis that splits the group ring into small matrix blocks. We prove that unit inversion is polynomial-time, classically and quantumly, when an efficient generalized Fourier transform exists, the group ring is semisimple, and the largest matrix block has polynomial size. Dihedral group rings satisfy these conditions because their irreducible representations have dimension at most 2 and an efficient Fourier transform exists. We give an explicit reversible quantum circuit for the block-inversion step and validate it in a register-level simulator. We also identify the exact structural boundary where the method stops and propose a candidate construction in the surviving regime under a new, clearly stated security assumption. The constructive results are supported by reproducible software artifacts and experiments.
Quantum Physics (quant-ph); Cryptography and Security (cs.CR)
arXiv:2609.10596 [quant-ph]
(or arXiv:2609.10596v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2609.10596 Focus to learn more arXiv-issued DOI via DataCite
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