{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article1423","name":"Secure Linear Algebra Using Linearly Recurrent Sequences","abstract":"In this work we present secure two-party protocols for\r\nvarious core problems in linear algebra.\r\nOur main building block is a protocol to obliviously decide singularity\r\nof an encrypted matrix:\r\nBob holds an $n \times n$ matrix $M$, encrypted with Alice's secret\r\nkey, and wants to learn whether\r\nthe matrix is singular or not (and nothing beyond that).\r\nWe give an interactive protocol between Alice and Bob that solves the\r\nabove problem\r\nwith optimal communication complexity while at the same time achieving\r\nlow round complexity.\r\nMore precisely, the number of communication rounds in our protocol\r\nis $polylog(n)$ and\r\nthe overall communication is roughly $O(n^2)$ (note that the input size is $n^2$).\r\nAt the core of our protocol we exploit some nice mathematical\r\nproperties of linearly recurrent sequences and their\r\nrelation to the characteristic polynomial of the matrix $M$, following [Wiedemann, 1986].\r\nWith our new techniques we are able to improve the round complexity of\r\nthe communication efficient solution of [Nissim and Weinreb, 2006] from $n^{0.275}$ to $polylog(n)$.\r\n\r\nBased on our singularity protocol we further\r\nextend our result to the problems of securely computing the rank of an\r\nencrypted matrix and solving systems of linear equations.","keywords":["Secure Linear Algebra","Linearly Recurrent Sequences","Wiedemann's Algorithm"],"author":[{"@type":"Person","name":"Kiltz, Eike","givenName":"Eike","familyName":"Kiltz"},{"@type":"Person","name":"Weinreb, Enav","givenName":"Enav","familyName":"Weinreb"}],"position":16,"pageStart":1,"pageEnd":19,"dateCreated":"2006-11-20","datePublished":"2006-11-20","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kiltz, Eike","givenName":"Eike","familyName":"Kiltz"},{"@type":"Person","name":"Weinreb, Enav","givenName":"Enav","familyName":"Weinreb"}],"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06111.16","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume601","volumeNumber":6111,"name":"Dagstuhl Seminar Proceedings, Volume 6111","dateCreated":"2006-10-09","datePublished":"2006-10-09","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article1423","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume601"}}}