{
  "abstract": "The Elliptic Curve Discrete Logarithm Problem (ECDLP) is a fundamental problem in cryptography, and reducing the resource requirements of quantum algorithms for solving ECDLP is an important goal. In this work, we present a space-efficient quantum algorithm for solving the ECDLP over prime fields, achieving an implementation with only $3n+6\\lfloor \\log_2 n \\rfloor+O(1)$ logical qubits and $1056n^3/\\log_2 n+O(n^2)$ Toffoli gates, where $n$ is the bit-length of the prime. For a 256-bit prime-field curve, our construction requires only 835 logical qubits, reducing the previous best estimates of 1098 and 1175 logical qubits by Chevignard et al. [EUROCRYPT 2026] and Babbush et al. [ArXiv Preprint 2026], respectively. The key to our improvement is a new space-efficient reversible modular inversion circuit, which addresses the dominant space bottleneck in affine-coordinate point addition. Starting from the extended Euclidean algorithm (EEA), we refine the register-sharing technique of Proos and Zalka by introducing length registers and location-controlled arithmetic to compactly store and update intermediate variables. We further optimize the reversible update procedures and construct the corresponding controlled arithmetic circuits, resulting in a modular inversion circuit implemented by only $2n+6\\lfloor \\log_2 n \\rfloor+O(1)$ logical qubits and $229n^2+O(n\\log_2 n)$ Toffoli gates. This modular inversion circuit together with mid-circuit measurements and classical feed-forward operations provides a space-efficient controlled affine point-addition circuit and a complete implementation of Shor's algorithm for ECDLP.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.13816",
  "author_total": 8,
  "authors": [
    {
      "name": "Han Luo"
    },
    {
      "name": "Ziyi Yang"
    },
    {
      "name": "Jingquan Luo"
    },
    {
      "name": "Ziruo Wang"
    },
    {
      "name": "Yuexin Su"
    },
    {
      "name": "Xiaoming Sun"
    },
    {
      "name": "Lvzhou Li"
    },
    {
      "name": "Tongyang Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2c807zje05p1t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.13816",
  "fulltext_url": null,
  "id": "2c807zje05p1t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.13816",
  "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",
      "cs.CR"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7168456217",
  "pdf_url": "https://arxiv.org/pdf/2607.13816",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2c807zje05p1t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:48Z",
      "native_id": "2607.13816",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.13816"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Quantum Algorithm for Elliptic Curve Discrete Logarithms with Space-Efficient Point Addition",
  "title_text": "Quantum Algorithm for Elliptic Curve Discrete Logarithms with Space-Efficient Point Addition",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:48Z",
  "url": "https://research.draconitic.com/paper/2c807zje05p1t",
  "venue": "arXiv",
  "versions": []
}