{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,21]],"date-time":"2025-06-21T04:07:18Z","timestamp":1750478838618,"version":"3.41.0"},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2003,5,20]],"date-time":"2003-05-20T00:00:00Z","timestamp":1053388800000},"content-version":"unspecified","delay-in-days":19,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2003,5]]},"abstract":"<jats:p>This paper shows that the largest possible contrast <jats:inline-formula>$C_{k,n}$<\/jats:inline-formula> in a <jats:inline-formula>$k$<\/jats:inline-formula>-out-of-<jats:inline-formula>$n$<\/jats:inline-formula> secret sharing scheme is approximately <jats:inline-formula>$4^{-(k-1)}$<\/jats:inline-formula>. More precisely, we show that <jats:inline-formula>$4^{-(k-1)} \\leq C_{k,n} \\leq 4^{-(k-1)}n^k\/(n(n-1)\\cdots(n-(k-1)))$<\/jats:inline-formula>. This implies that the largest possible contrast equals <jats:inline-formula>$4^{-(k-1)}$<\/jats:inline-formula> in the limit when <jats:inline-formula>$n$<\/jats:inline-formula> approaches infinity. For large <jats:inline-formula>$n$<\/jats:inline-formula>, the above bounds leave almost no gap. For values of <jats:inline-formula>$n$<\/jats:inline-formula> that come close to <jats:inline-formula>$k$<\/jats:inline-formula>, we will present alternative bounds (being tight for <jats:inline-formula>$n=k$<\/jats:inline-formula>). The proofs of our results proceed by finding a relationship between the largest possible contrast in a secret sharing scheme and the smallest possible approximation error in problems occurring in approximation theory.<\/jats:p>","DOI":"10.1017\/s096354830200559x","type":"journal-article","created":{"date-parts":[[2003,10,15]],"date-time":"2003-10-15T15:59:58Z","timestamp":1066233598000},"page":"285-299","source":"Crossref","is-referenced-by-count":37,"title":["Determining the Optimal Contrast for Secret Sharing Schemes in Visual Cryptography"],"prefix":"10.1017","volume":"12","author":[{"given":"MATTHIAS","family":"KRAUSE","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"HANS ULRICH","family":"SIMON","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2003,5,20]]},"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S096354830200559X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,20]],"date-time":"2025-06-20T22:46:47Z","timestamp":1750459607000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S096354830200559X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,5]]},"references-count":0,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2003,11]]}},"alternative-id":["S096354830200559X"],"URL":"https:\/\/doi.org\/10.1017\/s096354830200559x","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"type":"print","value":"0963-5483"},{"type":"electronic","value":"1469-2163"}],"subject":[],"published":{"date-parts":[[2003,5]]}}}