Quantum Inversion of Units in Group Rings: Block Dimension, Not Commutativity, Governs Hardness

Quantum Physics arXiv:2609.10596 (quant-ph) [Submitted on 7 Sep 2026] Title:Quantum Inversion of Units in Group Rings: Block Dimension, Not Commutativity, Governs Hardness Authors:Bhanwar Gupta View a PDF of the paper titled Quantum Inversion of Units in Group Rings: Block Dimension, Not Commutativity, Governs Hardness, by Bhanwar Gupta View PDF HTML (experimental) 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. Comments: Subjects: Quantum Physics (quant-ph); Cryptography and Security (cs.CR) Cite as: 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 Submission history From: Bhanwar Gupta [view email] [v1] Mon, 7 Sep 2026 09:50:40 UTC (153 KB) Full-text links: Access Paper: View a PDF of the paper titled Quantum Inversion of Units in Group Rings: Block Dimension, Not Commutativity, Governs Hardness, by Bhanwar GuptaView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-09 Change to browse by: cs cs.CR References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)