{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:44:56Z","timestamp":1787323496936,"version":"build-2736575974"},"reference-count":17,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2003,1]]},"abstract":"<jats:p>A (k,n)-threshold visual cryptography scheme (VCS) is a method to encode a secret image SI into n shadow images called shares such that any k or more shares enable the \"visual\" recovery of the secret image. However, by inspecting less than k shares one cannot gain any information on the secret image. The \"visual\" recovery consists of copying the shares onto transparencies and then stacking them. Any k shares will reveal the secret image without any cryptographic computation. In this paper we analyze the contrast of the reconstructed image for a (k,n)-threshold VCS. We define a canonical form for a (k,n)-threshold VCS and provide a characterization of a (k,,n)-threshold VCS. We completely characterize a contrast optimal (n-1,n)-threshold VCS in canonical form. Moreover, for $n\\geq 4$, we provide a contrast optimal (3,n)-threshold VCS in canonical form. We first describe a family of (3,n)-threshold VCS achieving various values of contrast and pixel expansion. Then we prove an upper bound on the contrast of any (3,n)-threshold VCS and show that a scheme in the described family has optimal contrast. Finally, for k=4,5 we present two schemes with contrast asymptotically equal to 1\/64 and 1\/256, respectively.<\/jats:p>","DOI":"10.1137\/s0895480198336683","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"224-261","source":"Crossref","is-referenced-by-count":126,"title":["Contrast Optimal Threshold Visual Cryptography Schemes"],"prefix":"10.1137","volume":"16","author":[{"given":"C.","family":"Blundo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"P.","family":"D'Arco","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A.","family":"De Santis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D. R.","family":"Stinson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,8,1]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1006\/inco.1996.0076"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(99)00127-9"},{"key":"R3","unstructured":"G. Ateniese, C. Blundo, A. De Santis, and D. R. Stinson,\n                      Constructions and bounds for visual cryptography\n                      , in Proc. of the 23rd International Colloquium on Automata, Languages and Programming (ICALP \u201996), F. M. auf der Heide and B. Monien, eds., Lecture Notes in Comput. Sci. 1099, Springer\u2010Verlag, Berlin, 1996, pp. 416\u2013428."},{"key":"R4","unstructured":"E. Biham,\n                      Visual cryptography with polarization\n                      , talk presented at the Weizmann Workshop on Cryptography, Weizmann Institute, Rehovot, Israel, June 8\u20139, 1997."},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/s001459900057"},{"key":"R6","unstructured":"AnnalisaDe Bonis, AlfredoDe Santis, Randomness in visual cryptography, Lecture Notes in Comput. Sci., Vol. 1770, Springer, Berlin, 2000, 626\u20136382001e:94012"},{"key":"R7","unstructured":"S. Droste,\n                      New results on visual cryptography\n                      , in Advances in Cryptology \u2014 CRYPTO \u201996, N. Koblitz, ed., Lecture Notes in Comput. Sci. 1109, Springer\u2010Verlag, Berlin, 1996, pp. 401\u2013415."},{"key":"R8","volume-title":"Concrete mathematics","author":"Graham Ronald","year":"1989"},{"key":"R9","unstructured":"T. Hofmeister, M. Krause, and H. U. Simon,\n                      Contrast\u2010optimalkout ofnsecret sharing schemes in visual cryptography\n                      , in COCOON \u201997, T. Jiang and D. T. Lee, eds., Lecture Notes in Comput. Sci. 1276, Springer\u2010Verlag, Berlin, 1997, pp. 176\u2013185."},{"key":"R10","unstructured":"D. Naccache,\n                      Colorful cryptography\u2014a purely physical secret\u2010sharing scheme based on chromatic filters\n                      , talk presented at Coding and Information Integrity, French\u2010Israeli Workshop, December 1994."},{"key":"R11","unstructured":"M. Naor and B. Pinkas,\n                      Visual authentication and identification\n                      , in Advances in Cryptology\u2014CRYPTO \u201997, B. S. Kaliski Jr., ed., Lecture Notes in Comput. Sci. 1294, Springer\u2010Verlag, Berlin, 1997, pp. 322\u2013336."},{"key":"R12","unstructured":"M. Naor and A. Shamir,\n                      Visual cryptography\n                      , in Advances in Cryptology\u2014Eurocrypt \u201994, A. De Santis, ed., Lecture Notes in Comput. Sci. 950, Springer\u2010Verlag, Berlin, 1995, pp. 1\u201312."},{"key":"R13","unstructured":"MoniNaor, AdiShamir, Visual cryptography. II. Improving the contrast via the cover base, Lecture Notes in Comput. Sci., Vol. 1189, Springer, Berlin, 1997, 197\u20132021477715"},{"key":"R14","unstructured":"V. Rijmen,\n                      Efficient colour visual encryption or \u201c\u201dhared colors of Benetton,\"\n                      talk presented at EUROCRYPT \u201996 Rump Session. Text available online at http:\/\/www.iacr.org\/conferences\/ec96\/rump\/index.html."},{"key":"R15","unstructured":"D. R. Stinson,\n                      An introduction to visual cryptography\n                      , talk presented at Public Key Solutions \u201997, Toronto, April 28\u201330, 1997."},{"key":"R15","unstructured":"Text available online at http:\/\/www.cacr.math.uwaterloo.ca\/\u223cdstinson\/."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008280705142"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895480198336683","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:05:42Z","timestamp":1787321142000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895480198336683"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1]]},"references-count":17,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["10.1137\/S0895480198336683"],"URL":"https:\/\/doi.org\/10.1137\/s0895480198336683","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,1]]}}}