{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,16]],"date-time":"2025-09-16T21:05:05Z","timestamp":1758056705632,"version":"3.44.0"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"9","license":[{"start":{"date-parts":[[2025,5,24]],"date-time":"2025-05-24T00:00:00Z","timestamp":1748044800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,5,24]],"date-time":"2025-05-24T00:00:00Z","timestamp":1748044800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100014374","name":"Universitat Polit\u00e8cnica de Catalunya","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100014374","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2025,9]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We consider access structures over a set of <jats:italic>n<\/jats:italic> participants, defined by a parameter <jats:italic>k<\/jats:italic> with <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$1 \\le k \\le n$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> in the following way: a subset is authorized if it contains at least <jats:italic>k<\/jats:italic> consecutive participants. Depending on whether we consider the participants placed in a line (that is, participant 1 is not next to participant <jats:italic>n<\/jats:italic>) or in a circle, we obtain two different families, that we call (<jats:italic>k<\/jats:italic>,\u00a0<jats:italic>n<\/jats:italic>)-line-consecutive and (<jats:italic>k<\/jats:italic>,\u00a0<jats:italic>n<\/jats:italic>)-circle-consecutive access structures, respectively. Such access structures can appear in real-life situations involving distributed cryptography, which makes it more interesting to look for the best secret sharing schemes that can realize them. For both families, we characterize which are the configurations (<jats:italic>k<\/jats:italic>,\u00a0<jats:italic>n<\/jats:italic>) that admit ideal secret sharing schemes. For the non-ideal (<jats:italic>k<\/jats:italic>,\u00a0<jats:italic>n<\/jats:italic>)-consecutive access structures, we give both upper and lower bounds on the information ratio of the best secret sharing schemes that can realize them. Some of these bounds are obtained after proving relations between the information ratios of access structures in the two considered families.<\/jats:p>","DOI":"10.1007\/s10623-025-01651-7","type":"journal-article","created":{"date-parts":[[2025,5,24]],"date-time":"2025-05-24T04:00:53Z","timestamp":1748059253000},"page":"3543-3563","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["(k,\u00a0n)-Consecutive access structures"],"prefix":"10.1007","volume":"93","author":[{"given":"Javier","family":"Herranz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Germ\u00e1n","family":"S\u00e1ez","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,24]]},"reference":[{"key":"1651_CR1","doi-asserted-by":"crossref","unstructured":"Applebaum B., Beimel A., Farr\u00e0s O., Nir O., Peter N.: Secret-sharing schemes for general and uniform access structures. In: Proceedings of EUROCRYPT\u20192019 (Part III). Lecture Notes in Computer Science, vol. 11478 , pp. 441\u2013471 (2019).","DOI":"10.1007\/978-3-030-17659-4_15"},{"key":"1651_CR2","doi-asserted-by":"crossref","unstructured":"Applebaum B., Nir O.: Upslices, downslices, and secret-sharing with complexity of 1.5$$^n$$. In: Proceedings of CRYPTO\u20192021. Lecture Notes in Computer Science, vol. 12827 , pp. 627\u2013655 (2021).","DOI":"10.1007\/978-3-030-84252-9_21"},{"key":"1651_CR3","unstructured":"Beimel A.: Lower bounds for secret-sharing schemes for $$k$$-hypergraphs. In: Proceedings of ITC\u20192023, Schloss Dagstuhl. LIPIcs Series, vol. 267, pp. 16:1\u201316:13 (2013)."},{"key":"1651_CR4","doi-asserted-by":"crossref","unstructured":"Beimel A., Farr\u00e0s O.: The share size of secret-sharing schemes for almost all access structures and graphs. In: Proceedings of TCC\u20192020 (Part III). Lecture Notes in Computer Science, vol. 12552 , pp. 499\u2013529 (2020).","DOI":"10.1007\/978-3-030-64381-2_18"},{"issue":"2","key":"1651_CR5","doi-asserted-by":"publisher","first-page":"336","DOI":"10.1007\/s00145-014-9195-8","volume":"29","author":"A Beimel","year":"2016","unstructured":"Beimel A., Farr\u00e0s O., Mintz Y.: Secret-sharing schemes for very dense graphs. J. Cryptol. 29(2), 336\u2013362 (2016).","journal-title":"J. Cryptol."},{"issue":"1","key":"1651_CR6","doi-asserted-by":"publisher","first-page":"360","DOI":"10.1137\/S0895480104445654","volume":"22","author":"A Beimel","year":"2008","unstructured":"Beimel A., Tassa T., Weinreb E.: Characterizing ideal weighted threshold secret sharing. SIAM J. Discret. Math. 22(1), 360\u2013397 (2008).","journal-title":"SIAM J. Discret. Math."},{"key":"1651_CR7","doi-asserted-by":"crossref","unstructured":"Blakley G.R.: Safeguarding cryptographic keys. In: AFIPS. Conf. Proc., vol. 48, pp. 313\u2013317 (1979).","DOI":"10.1109\/MARK.1979.8817296"},{"issue":"2","key":"1651_CR8","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1023\/A:1008216403325","volume":"11","author":"C Blundo","year":"1997","unstructured":"Blundo C., De Santis A., De Simone R., Vaccaro U.: Tight bounds on the information ratio of secret sharing schemes. Des. Codes Cryptogr. 11(2), 107\u2013122 (1997).","journal-title":"Des. Codes Cryptogr."},{"issue":"1","key":"1651_CR9","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1007\/BF00204801","volume":"8","author":"C Blundo","year":"1995","unstructured":"Blundo C., De Santis A., Stinson D.R., Vaccaro U.: Graph decompositions and secret sharing schemes. J. Cryptol. 8(1), 39\u201364 (1995).","journal-title":"J. Cryptol."},{"key":"1651_CR10","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1007\/BF00196772","volume":"4","author":"EF Brickell","year":"1991","unstructured":"Brickell E.F., Davenport D.M.: On the classification of ideal secret sharing schemes. J. Cryptol. 4, 123\u2013134 (1991).","journal-title":"J. Cryptol."},{"issue":"4","key":"1651_CR11","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1007\/s001459900029","volume":"10","author":"L Csirmaz","year":"1997","unstructured":"Csirmaz L.: The size of a share must be large. J. Cryptol. 10(4), 223\u2013231 (1997).","journal-title":"J. Cryptol."},{"issue":"3","key":"1651_CR12","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1007\/s12095-018-0338-x","volume":"11","author":"L Csirmaz","year":"2019","unstructured":"Csirmaz L., Ligeti P.: Secret sharing on large girth graphs. Cryptogr. Commun. 11(3), 399\u2013410 (2019).","journal-title":"Cryptogr. Commun."},{"issue":"4","key":"1651_CR13","doi-asserted-by":"publisher","first-page":"2527","DOI":"10.1109\/TIT.2012.2236958","volume":"59","author":"L Csirmaz","year":"2013","unstructured":"Csirmaz L., Tardos G.: Optimal information ratio of secret sharing schemes on trees. IEEE Trans. Inf. Theory 59(4), 2527\u20132530 (2013).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"5","key":"1651_CR14","doi-asserted-by":"publisher","first-page":"928","DOI":"10.1016\/j.dam.2008.04.001","volume":"157","author":"G Di Crescenzo","year":"2009","unstructured":"Di Crescenzo G., Galdi C.: Hypergraph decomposition and secret sharing. Discret. Appl. Math. 157(5), 928\u2013946 (2009).","journal-title":"Discret. Appl. Math."},{"issue":"11","key":"1651_CR15","doi-asserted-by":"publisher","first-page":"7088","DOI":"10.1109\/TIT.2020.3005706","volume":"66","author":"O Farr\u00e0s","year":"2020","unstructured":"Farr\u00e0s O., Kaced T., Mart\u00edn S., Padr\u00f3 C.: Improving the linear programming technique in the search for lower bounds in secret sharing. IEEE Trans. Inf. Theory 66(11), 7088\u20137100 (2020).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"3","key":"1651_CR16","doi-asserted-by":"publisher","first-page":"434","DOI":"10.1007\/s00145-011-9101-6","volume":"25","author":"O Farr\u00e0s","year":"2012","unstructured":"Farr\u00e0s O., Mart\u00ed-Farr\u00e9 J., Padr\u00f3 C.: Ideal multipartite secret sharing schemes. J. Cryptol. 25(3), 434\u2013463 (2012).","journal-title":"J. Cryptol."},{"issue":"5","key":"1651_CR17","doi-asserted-by":"publisher","first-page":"3273","DOI":"10.1109\/TIT.2011.2182034","volume":"58","author":"O Farr\u00e0s","year":"2012","unstructured":"Farr\u00e0s O., Padr\u00f3 C.: Ideal hierarchical secret sharing schemes. IEEE Trans. Inf. Theory 58(5), 3273\u20133286 (2012).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"1","key":"1651_CR18","doi-asserted-by":"publisher","first-page":"722","DOI":"10.1109\/TIT.2023.3322601","volume":"70","author":"M Gyarmati","year":"2024","unstructured":"Gyarmati M.: Information ratio of unicyclic graphs. IEEE Trans. Inf. Theory 70(1), 722\u2013734 (2024).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1651_CR19","doi-asserted-by":"crossref","unstructured":"Herranz, J., S\u00e1ez, G.: Secret sharing schemes for $$(k, n)-$$consecutive access structures Proceedings of CANS\u20192018. Lecture Notes in Computer Science, vol. 11124 , pp. 463\u2013480 (2018).","DOI":"10.1007\/978-3-030-00434-7_23"},{"issue":"1","key":"1651_CR20","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/BF01388562","volume":"4","author":"WA Jackson","year":"1994","unstructured":"Jackson W.A., Martin K.M.: Geometric secret sharing schemes and their duals. Des. Codes Cryptogr. 4(1), 83\u201395 (1994).","journal-title":"Des. Codes Cryptogr."},{"issue":"3","key":"1651_CR21","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1023\/A:1027328505246","volume":"9","author":"WA Jackson","year":"1996","unstructured":"Jackson W.A., Martin K.M.: Perfect secret sharing schemes on five participants. Des. Codes Cryptogr. 9(3), 267\u2013286 (1996).","journal-title":"Des. Codes Cryptogr."},{"key":"1651_CR22","unstructured":"Juels A., Pappu R., Parno B.: Unidirectional key distribution across time and space with applications to RFID security. In: Proceedings of the USENIX Security Symposium, vol. 2008, pp. 75\u201390 (2008)."},{"key":"1651_CR23","unstructured":"KIRA Network: Documentation about \u201cConsensus nodes ranks & statuses\u201d, November 2024. https:\/\/ipfs.kira.network\/ipfs\/bafybeigwa4mtipb2kpncdd4rbumvgakdbdua74klwvodd2ft5dbldd3viq\/consensus-nodes-ranks-statuses.html."},{"issue":"1","key":"1651_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.37236\/1825","volume":"11","author":"J Mart\u00ed-Farr\u00e9","year":"2004","unstructured":"Mart\u00ed-Farr\u00e9 J., Padr\u00f3 C.: Secret sharing schemes on sparse homogeneous access structures with rank three. Electron. J. Combin. 11(1), 1\u201316 (2004).","journal-title":"Electron. J. Combin."},{"issue":"1","key":"1651_CR25","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1007\/s10623-003-4192-1","volume":"34","author":"J Mart\u00ed-Farr\u00e9","year":"2005","unstructured":"Mart\u00ed-Farr\u00e9 J., Padr\u00f3 C.: Secret sharing schemes with three or four minimal qualified subsets. Des. Codes Cryptogr. 34(1), 17\u201334 (2005).","journal-title":"Des. Codes Cryptogr."},{"issue":"3","key":"1651_CR26","doi-asserted-by":"publisher","first-page":"552","DOI":"10.1016\/j.dam.2005.09.003","volume":"154","author":"J Mart\u00ed-Farr\u00e9","year":"2006","unstructured":"Mart\u00ed-Farr\u00e9 J., Padr\u00f3 C.: Secret sharing schemes on access structures with intersection number equal to one. Discret. Appl. Math. 154(3), 552\u2013563 (2006).","journal-title":"Discret. Appl. Math."},{"issue":"2","key":"1651_CR27","first-page":"1356","volume":"170","author":"A Mart\u00edn","year":"2005","unstructured":"Mart\u00edn A., Pereira J., Rodr\u00edguez G.: A secret sharing scheme based on cellular automata. Appl. Math. Comput. 170(2), 1356\u20131364 (2005).","journal-title":"Appl. Math. Comput."},{"issue":"7","key":"1651_CR28","doi-asserted-by":"publisher","first-page":"2596","DOI":"10.1109\/18.887867","volume":"46","author":"C Padr\u00f3","year":"2000","unstructured":"Padr\u00f3 C., S\u00e1ez G.: Secret sharing schemes with bipartite access structure. IEEE Trans. Inf. Theory 46(7), 2596\u20132604 (2000).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1651_CR29","doi-asserted-by":"publisher","first-page":"612","DOI":"10.1145\/359168.359176","volume":"22","author":"A Shamir","year":"1979","unstructured":"Shamir A.: How to share a secret. Commun. ACM 22, 612\u2013613 (1979).","journal-title":"Commun. ACM"},{"issue":"1","key":"1651_CR30","doi-asserted-by":"publisher","first-page":"118","DOI":"10.1109\/18.272461","volume":"40","author":"DR Stinson","year":"1994","unstructured":"Stinson D.R.: Decomposition constructions for secret sharing schemes. IEEE Trans. Inf. Theory 40(1), 118\u2013125 (1994).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"2","key":"1651_CR31","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1007\/s00145-006-0334-8","volume":"20","author":"T Tassa","year":"2007","unstructured":"Tassa T.: Hierarchical threshold secret sharing. J. Cryptol. 20(2), 237\u2013264 (2007).","journal-title":"J. Cryptol."}],"container-title":["Designs, Codes and Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-025-01651-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10623-025-01651-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-025-01651-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,15]],"date-time":"2025-09-15T17:30:45Z","timestamp":1757957445000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10623-025-01651-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,24]]},"references-count":31,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2025,9]]}},"alternative-id":["1651"],"URL":"https:\/\/doi.org\/10.1007\/s10623-025-01651-7","relation":{},"ISSN":["0925-1022","1573-7586"],"issn-type":[{"type":"print","value":"0925-1022"},{"type":"electronic","value":"1573-7586"}],"subject":[],"published":{"date-parts":[[2025,5,24]]},"assertion":[{"value":"22 November 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 April 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 May 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 May 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors did not receive support from any organization for the submitted work. The authors have no relevant financial or non-financial interests to disclose. This submission is the extended and revisited version of the paper [], presented in the conference CANS 2018 and published in the proceedings of that conference, in Lecture Notes in Computer Science (Springer). This submission has, therefore, a significant overlap with that paper of 2018: large parts of Sects. , , and  in this submission already appeared in []. In contrast, all the material and results in Sects.  and  of this submission are new. Also some results in Sect. , specifically Lemma  and Theorem , are new. We want to stress that some of the new results lead to improvements with respect to the results in the conference version [].","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}