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    "story_id": "mp-2026-09-13-001",
    "source_story_id": "tmp-lead-group-ring-block-inversion",
    "edition_id": "mp-2026-09-13-morning-0066",
    "edition_url": "https://themachinepress.com/edition/2026-09-13",
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    "section": "safety-security",
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    "headline": "The Non-Abelian Lock Still Broke Into Small Pieces",
    "slug": "the-non-abelian-lock-still-broke-into-small-pieces",
    "dek": "A Fourier change of basis reduced group-ring unit inversion to matrix blocks, making dihedral constructions polynomial-time despite their non-commutativity.",
    "summary": "A Fourier change of basis reduced group-ring unit inversion to matrix blocks, making dihedral constructions polynomial-time despite their non-commutativity.",
    "body_text": "Some public-key proposals moved from abelian to non-abelian group rings after a quantum attack, expecting the harder dihedral hidden-subgroup problem to provide protection. This preprint argues that unit inversion does not require solving that problem. When a semisimple group ring has an efficient generalized Fourier transform and its largest representation block is polynomially bounded, a change of basis splits inversion into manageable matrices. Dihedral groups meet those conditions because their irreducible representations have dimension at most two. The author supplies an explicit reversible block-inversion circuit and simulator results. This is a preprint cryptanalysis of a construction class, not evidence that broadly deployed encryption has been broken.",
    "why_it_matters": "A Fourier change of basis reduced group-ring unit inversion to matrix blocks, making dihedral constructions polynomial-time despite their non-commutativity.",
    "limitations": [
      "This preprint argues that unit inversion does not require solving that problem.",
      "This is a preprint cryptanalysis of a construction class, not evidence that broadly deployed encryption has been broken."
    ],
    "importance": 10,
    "canonical_url": "https://themachinepress.com/story/mp-2026-09-13-001/the-non-abelian-lock-still-broke-into-small-pieces",
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    "first_published_at": "2026-09-13T09:00:00.000-04:00",
    "modified_at": "2026-09-13T09:00:00.000-04:00",
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        "text": "A Fourier change of basis reduced group-ring unit inversion to matrix blocks, making dihedral constructions polynomial-time despite their non-commutativity.",
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        "qualification": "This preprint argues that unit inversion does not require solving that problem."
      }
    ],
    "source_ids": [
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    "tags": [
      "post-quantum cryptography",
      "group rings",
      "cryptanalysis"
    ],
    "image_url": "https://themachinepress.com/issues/2026-09-13/lead-group-ring-block-inversion.png",
    "corrections": []
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  "sources": [
    {
      "source_id": "source-2026-09-13-001",
      "title": "arXiv preprint 2609.10596",
      "publisher": "arXiv",
      "url": "https://arxiv.org/abs/2609.10596",
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      "source_type": "primary_research",
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      "published_at": null,
      "accessed_at": "2026-09-13T08:24:00.000-04:00",
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  "publisher": {
    "name": "The Machine Press",
    "url": "https://themachinepress.com",
    "description": "A daily newspaper for the age of artificial intelligence."
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  "cite_this_report": {
    "title": "The Non-Abelian Lock Still Broke Into Small Pieces",
    "publisher": "The Machine Press",
    "published_at": "2026-09-13T09:00:00.000-04:00",
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