{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T17:21:37Z","timestamp":1740158497078,"version":"3.37.3"},"reference-count":32,"publisher":"Walter de Gruyter GmbH","issue":"0","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1318716"],"award-info":[{"award-number":["DMS-1318716"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,4,17]]},"abstract":"<jats:title>Abstract<\/jats:title>\n            <jats:p>We investigate security properties of two secret-sharing protocols\nproposed by Fine, Moldenhauer, and Rosenberger\nin Sections 4 and 5 of [B. Fine, A. Moldenhauer and G. Rosenberger,\nCryptographic protocols based on Nielsen transformations,\nJ. Comput. Comm. 4 2016, 63\u2013107]\n(Protocols I and II resp.).\nFor both protocols, we consider a one missing share challenge.\nWe show that Protocol I can be reduced to a system of polynomial equations\nand (for most randomly generated instances)\nsolved by the computer algebra system Singular.\nProtocol II is approached using the technique of Stallings\u2019 graphs.\nWe show that knowledge of <jats:inline-formula id=\"j_gcc-2018-0001_ineq_9999\">\n                  <jats:alternatives>\n                     \n                     <jats:inline-graphic xlink:href=\"graphic\/j_gcc-2018-0001_eq_0652.png\"\/>\n                     <jats:tex-math>{m-1}<\/jats:tex-math>\n                  <\/jats:alternatives>\n               <\/jats:inline-formula> shares reduces the space of possible values\nof a secret to a set of polynomial size.<\/jats:p>","DOI":"10.1515\/gcc-2018-0001","type":"journal-article","created":{"date-parts":[[2018,4,17]],"date-time":"2018-04-17T22:16:33Z","timestamp":1524003393000},"source":"Crossref","is-referenced-by-count":0,"title":["Analysis of secret sharing schemes based on Nielsen transformations"],"prefix":"10.1515","volume":"0","author":[{"given":"Matvei","family":"Kotov","sequence":"first","affiliation":[]},{"given":"Dmitry","family":"Panteleev","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Ushakov","sequence":"additional","affiliation":[]}],"member":"374","reference":[{"journal-title":"Preprint","article-title":"Singular 4-0-2 \u2013 A computer algebra system for polynomial computations","year":"2015","key":"ref201"},{"key":"ref111","first-page":"125","article-title":"Cryptanalysis of a combinatorial public key cryptosystem","volume":"9","year":"2017","journal-title":"Groups Complex. Cryptol."},{"key":"ref281","first-page":"125","article-title":"Cryptanalysis of a combinatorial public key cryptosystem","volume":"9","year":"2017","journal-title":"Groups Complex. Cryptol."},{"key":"ref171","doi-asserted-by":"crossref","first-page":"634","DOI":"10.1090\/S0002-9939-1973-0330295-8","article-title":"An inverse function theorem for free groups","volume":"41","year":"1973","journal-title":"Proc. Amer. Math. Soc."},{"key":"ref131","doi-asserted-by":"crossref","first-page":"612","DOI":"10.1145\/359168.359176","article-title":"How to share a secret","volume":"22","year":"1979","journal-title":"Commun. ACM"},{"volume-title":"Non-Commutative Cryptography and Complexity of Group-Theoretic Problems","year":"2011","key":"ref71"},{"key":"ref41","doi-asserted-by":"crossref","first-page":"63","DOI":"10.4236\/jcc.2016.412004","article-title":"Cryptographic protocols based on Nielsen transformations","volume":"4","year":"2016","journal-title":"J. Comput. Comm."},{"key":"ref151","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1070\/RM1994v049n02ABEH002233","article-title":"Primitive elements of free groups","volume":"49","year":"1994","journal-title":"Russian Math. Surveys"},{"key":"ref191","first-page":"51","article-title":"On secret sharing protocols","year":"2016","journal-title":"Algebra and Computer Science"},{"journal-title":"Preprint","article-title":"Secret sharing protocols based on the closest vector theorem and Nielsen transformation","year":"2014","key":"ref251"},{"key":"ref11","first-page":"313","article-title":"Safeguarding cryptographic keys","year":"1979","journal-title":"Proceedings of the 1979 AFIPS National Computer Conference"},{"journal-title":"Cryptographic protocols based on inner product spaces and group theory with a special focus on use of Nielsen transformations","year":"2016","key":"ref91"},{"key":"ref221","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1006\/jabr.2001.9033","article-title":"Stallings foldings and subgroups of free groups","volume":"248","year":"2002","journal-title":"J. Algebra"},{"journal-title":"Preprint","key":"ref231","article-title":"Implementation of attacks on secret sharing schemes based on Nielsen transformations"},{"key":"ref321","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1070\/RM1994v049n02ABEH002233","article-title":"Primitive elements of free groups","volume":"49","year":"1994","journal-title":"Russian Math. Surveys"},{"key":"ref211","doi-asserted-by":"crossref","first-page":"63","DOI":"10.4236\/jcc.2016.412004","article-title":"Cryptographic protocols based on Nielsen transformations","volume":"4","year":"2016","journal-title":"J. Comput. Comm."},{"key":"ref301","doi-asserted-by":"crossref","first-page":"612","DOI":"10.1145\/359168.359176","article-title":"How to share a secret","volume":"22","year":"1979","journal-title":"Commun. ACM"},{"journal-title":"Preprint","article-title":"A secret sharing scheme using groups","year":"2010","key":"ref101"},{"journal-title":"Preprint","article-title":"A secret sharing scheme using groups","year":"2010","key":"ref271"},{"volume-title":"Essays in Algebra and Cryptology. Solvable Groups","year":"2017","key":"ref291"},{"key":"ref141","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1007\/BF02095993","article-title":"Topology of finite graphs","volume":"71","year":"1983","journal-title":"Invent. Math."},{"key":"ref51","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1006\/jabr.2001.9033","article-title":"Stallings foldings and subgroups of free groups","volume":"248","year":"2002","journal-title":"J. Algebra"},{"key":"ref01","doi-asserted-by":"crossref","first-page":"634","DOI":"10.1090\/S0002-9939-1973-0330295-8","article-title":"An inverse function theorem for free groups","volume":"41","year":"1973","journal-title":"Proc. Amer. Math. Soc."},{"volume-title":"Non-Commutative Cryptography and Complexity of Group-Theoretic Problems","year":"2011","key":"ref241"},{"journal-title":"Cryptographic protocols based on inner product spaces and group theory with a special focus on use of Nielsen transformations","year":"2016","key":"ref261"},{"volume-title":"Essays in Algebra and Cryptology. Solvable Groups","year":"2017","key":"ref121"},{"key":"ref311","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1007\/BF02095993","article-title":"Topology of finite graphs","volume":"71","year":"1983","journal-title":"Invent. Math."},{"key":"ref21","first-page":"51","article-title":"On secret sharing protocols","year":"2016","journal-title":"Algebra and Computer Science"},{"journal-title":"Preprint","article-title":"Singular 4-0-2 \u2013 A computer algebra system for polynomial computations","year":"2015","key":"ref31"},{"key":"ref181","first-page":"313","article-title":"Safeguarding cryptographic keys","year":"1979","journal-title":"Proceedings of the 1979 AFIPS National Computer Conference"},{"journal-title":"Preprint","article-title":"Secret sharing protocols based on the closest vector theorem and Nielsen transformation","year":"2014","key":"ref81"},{"journal-title":"Preprint","key":"ref61","article-title":"Implementation of attacks on secret sharing schemes based on Nielsen transformations"}],"container-title":["Groups Complexity Cryptology"],"original-title":[],"link":[{"URL":"http:\/\/www.degruyter.com\/view\/j\/gcc.ahead-of-print\/gcc-2018-0001\/gcc-2018-0001.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/www.degruyter.com\/view\/j\/gcc.ahead-of-print\/gcc-2018-0001\/gcc-2018-0001.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,27]],"date-time":"2021-02-27T03:08:48Z","timestamp":1614395328000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/gcc-2018-0001\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,4,17]]},"references-count":32,"journal-issue":{"issue":"0"},"URL":"https:\/\/doi.org\/10.1515\/gcc-2018-0001","relation":{},"ISSN":["1867-1144","1869-6104"],"issn-type":[{"type":"print","value":"1867-1144"},{"type":"electronic","value":"1869-6104"}],"subject":[],"published":{"date-parts":[[2018,4,17]]}}}