{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T07:49:03Z","timestamp":1781077743857,"version":"3.54.1"},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2012,12,7]],"date-time":"2012-12-07T00:00:00Z","timestamp":1354838400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2013,1]]},"abstract":"<jats:p>E. Thorp introduced the following card shuffling model. Suppose the number of cards is even. Cut the deck into two equal piles, then interleave them as follows. Choose the first card from the left pile or from the right pile according to the outcome of a fair coin flip. Then choose from the other pile. Continue this way, flipping an independent coin for each pair, until both piles are empty.<\/jats:p><jats:p>We prove an upper bound of <jats:italic>O<\/jats:italic>(<jats:italic>d<\/jats:italic><jats:sup>3<\/jats:sup>) for the mixing time of the Thorp shuffle with 2<jats:sup><jats:italic>d<\/jats:italic><\/jats:sup> cards, improving on the best known bound of <jats:italic>O<\/jats:italic>(<jats:italic>d<\/jats:italic><jats:sup>4<\/jats:sup>). As a consequence, we obtain an improved bound on the time required to encrypt a binary message of length <jats:italic>d<\/jats:italic> using the Thorp shuffle.<\/jats:p>","DOI":"10.1017\/s0963548312000478","type":"journal-article","created":{"date-parts":[[2012,12,7]],"date-time":"2012-12-07T15:43:24Z","timestamp":1354895004000},"page":"118-132","source":"Crossref","is-referenced-by-count":13,"title":["Improved Mixing Time Bounds for the Thorp Shuffle"],"prefix":"10.1017","volume":"22","author":[{"given":"BEN","family":"MORRIS","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2012,12,7]]},"reference":[{"key":"S0963548312000478_ref17","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1973.10481434"},{"key":"S0963548312000478_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-09444-0_5"},{"key":"S0963548312000478_ref14","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-28628-8_7"},{"key":"S0963548312000478_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-69710-1_8"},{"key":"S0963548312000478_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/PL00003817"},{"key":"S0963548312000478_ref9","doi-asserted-by":"publisher","DOI":"10.1214\/08-AOP409"},{"key":"S0963548312000478_ref8","first-page":"484","article-title":"The mixing time of the Thorp shuffle","volume":"38","author":"Morris","year":"2008","journal-title":"STOC 2005, SIAM J. Comput."},{"key":"S0963548312000478_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-39200-9_34"},{"key":"S0963548312000478_ref7","volume-title":"Mathematical Aspects of Mixing Times in Markov Chains","author":"Montenegro","year":"2006"},{"key":"S0963548312000478_ref4","doi-asserted-by":"publisher","DOI":"10.1137\/0217022"},{"key":"S0963548312000478_ref1","unstructured":"Bellare M. , Rogaway P. and Spies T. (2010) The FFX mode of operation and format preserving encryption (draft 1.1). NIST submission."},{"key":"S0963548312000478_ref16","unstructured":"Rudich S. (1989) Limits on the provable consequences of one-way functions. PhD Thesis, UC Berkeley."},{"key":"S0963548312000478_ref12","first-page":"301","volume-title":"Advances in Cryptology: CRYPTO 1991","author":"Patarin","year":"1991"},{"key":"S0963548312000478_ref3","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v16-912"},{"key":"S0963548312000478_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-03356-8_17"},{"key":"S0963548312000478_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-47555-9_21"},{"key":"S0963548312000478_ref2","doi-asserted-by":"publisher","DOI":"10.1002\/0471200611"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548312000478","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,24]],"date-time":"2019-04-24T20:05:59Z","timestamp":1556136359000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548312000478\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,12,7]]},"references-count":17,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2013,1]]}},"alternative-id":["S0963548312000478"],"URL":"https:\/\/doi.org\/10.1017\/s0963548312000478","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,12,7]]}}}