Unconditional correctness of recent quantum algorithms for factoring and computing discrete logarithms

[2404.16450] Unconditional correctness of recent quantum algorithms for factoring and computing discrete logarithms Skip to main content Search Submit Donate Log in Mathematics > Number Theory arXiv:2404.16450 (math) [Submitted on 25 Apr 2024 ( v1 ), last revised 24 Aug 2026 (this version, v3)] Title: Unconditional correctness of recent quantum algorithms for factoring and computing discrete logarithms Authors: Cédric Pilatte View PDF HTML (experimental) Abstract: In 1994, Shor introduced his famous quantum algorithm to factor integers and compute discrete logarithms in polynomial time. In 2023, Regev proposed a multi-dimensional version of Shor's algorithm that requires far fewer quantum gates. His algorithm relies on a number-theoretic conjecture on the elements in $(\mathbb{Z}/N\mathbb{Z})^{\times}$ that can be written as short products of very small prime numbers. We prove a version of this conjecture using tools from analytic number theory such as zero-density estimates. As a result, we obtain an unconditional proof of correctness of this improved quantum algorithm and of subsequent variants. Comments: Minor typos corrected. 25 pages Subjects: Number Theory (math.NT) ; Computational Complexity (cs.CC); Quantum Physics (quant-ph) MSC classes: 68Q12, 11H06, 11M26, 11L40, 11Y05 Cite as: arXiv:2404.16450 [math.NT] (or arXiv:2404.16450v3 [math.NT] for this version) https://doi.org/10.48550/arXiv.2404.16450 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Forum of Mathematics, Pi 14 (2026) e5 Related DOI : https://doi.org/10.1017/fmp.2025.10023 Focus to learn more DOI(s) linking to related resources Submission history From: Cédric Pilatte [ view email ] [v1] Thu, 25 Apr 2024 09:30:19 UTC (23 KB) [v2] Thu, 11 Dec 2025 15:59:26 UTC (27 KB) [v3] Mon, 24 Aug 2026 18:17:43 UTC (27 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2024-04 Change to browse by: cs cs.CC math quant-ph References & Citations INSPIRE HEP SciX Google Scholar Semantic Scholar export BibTeX citation 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? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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