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The library offers a range of two-source, seeded and deterministic randomness extractors, together with parameter calculation modules, making it easy to use and suitable for a variety of applications. We also present theoretical results, including new extractor constructions and improvements to existing extractor parameters. The extractor implementations are efficient in practice and tolerate input sizes of up to <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mn>2<\/mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>40<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>&amp;#x003E;<\/mml:mo><mml:msup><mml:mn>10<\/mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>12<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math> bits. Contrary to alternatives using the fast Fourier transform, we implement convolutions efficiently using the number-theoretic transform to avoid rounding errors, making them well suited to cryptography. The algorithms and parameter calculation are described in detail, including illustrative code examples and performance benchmarking.<\/jats:p>","DOI":"10.22331\/q-2025-01-08-1584","type":"journal-article","created":{"date-parts":[[2025,1,8]],"date-time":"2025-01-08T17:33:47Z","timestamp":1736357627000},"page":"1584","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":["Cryptomite: A versatile and user-friendly library of randomness extractors"],"prefix":"10.22331","volume":"9","author":[{"given":"Cameron","family":"Foreman","sequence":"first","affiliation":[{"name":"Quantinuum, Partnership House, Carlisle Place, London SW1P 1BX, United Kingdom"},{"name":"Department of Computer Science, University College London, London, United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Richie","family":"Yeung","sequence":"additional","affiliation":[{"name":"Quantinuum, 17 Beaumont Street, Oxford OX1 2NA, United Kingdom"},{"name":"Department of Computer Science, University of Oxford, Oxford, United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alec","family":"Edgington","sequence":"additional","affiliation":[{"name":"Quantinuum, Terrington House, 13\u201315 Hills Road, Cambridge CB2 1NL, United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Florian J.","family":"Curchod","sequence":"additional","affiliation":[{"name":"Quantinuum, Terrington House, 13\u201315 Hills Road, Cambridge CB2 1NL, United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2025,1,8]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Ronen Shaltiel. 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On circulant matrices. Notices of the AMS, 59 (3): 368\u2013377, 2012. http:\/\/dx.doi.org\/10.1090\/noti804.","DOI":"10.1090\/noti804"},{"key":"75","doi-asserted-by":"crossref","unstructured":"Umesh Vazirani. Efficiency considerations in using semi-random sources. 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