{
  "abstract": "In this article, we investigate whether combining Regev's reduction with the Cheng-Wan reduction can yield an alternative quantum algorithm for the discrete logarithm problem (DLOG) in finite abelian groups. Cheng and Wan reduce DLOG over finite fields to bounded-distance decoding of low-rate Reed-Solomon codes at a large decoding radius. Regev's reduction, in turn, exploits a decoder for the high-rate dual code with a lower noise level to obtain a quantum algorithm for decoding the original low-rate code at a larger radius. The Cheng-Wan instances therefore provide a natural setting in which to explore the power of Regev's framework. We first extend the Cheng-Wan reduction to the discrete logarithm problem in finite abelian groups, assuming access to embeddings of their cyclic factors. We also establish NP-hardness of Reed-Solomon bounded-distance decoding at asymptotically vanishing rate, reaching the rate regime relevant to Cheng-Wan, while the radius at which we establish hardness remains beyond the Cheng-Wan radius. We then instantiate Regev's reduction at the Cheng-Wan parameters and, under a conjecture on the performance of the induced BDD algorithm on the structured Cheng-Wan instances, characterize the noise threshold for the quantum decoding problem(QDP) sufficient to solve DLOG. We evaluate known decoders in this regime and show that a quantum decoder based on unambiguous state discrimination(USD) enables Regev's reduction to reach a strictly stronger decoding regime than the standard classical Reed-Solomon decoders we consider. Nevertheless, the QDP noise threshold required by the Cheng-Wan reduction is asymptotically 4/3 times that achieved by USD. Going beyond the USD framework, we show that every Cheng-Wan instance can be solved through Regev's reduction using a suitable Pretty Good Measurement and its inverse, although no efficient implementation of it is known.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.03972",
  "author_total": 2,
  "authors": [
    {
      "name": "M. Isabel Franco Garrido"
    },
    {
      "name": "André Chailloux"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3nkys57s5z9h9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.03972",
  "fulltext_url": null,
  "id": "3nkys57s5z9h9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.03972",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7160373198",
  "pdf_url": "https://arxiv.org/pdf/2605.03972",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3nkys57s5z9h9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2605.03972",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.03972"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Regev's reduction as a candidate quantum algorithm for the discrete logarithm problem in finite abelian groups",
  "title_text": "Regev's reduction as a candidate quantum algorithm for the discrete logarithm problem in finite abelian groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:03Z",
  "url": "https://research.draconitic.com/paper/3nkys57s5z9h9",
  "venue": "arXiv",
  "versions": []
}