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However, it is computationally overloaded for lightweight devices arisen in quantity with the dawn of the things of internet (IoT) era. As an important form of cloud computing services, secure computation outsourcing has become a popular topic. In this paper, we design an efficient outsourcing scheme that enables the resource-constrained client to find a solution of the<jats:inline-formula><jats:alternatives><jats:tex-math>$\\mathcal {LMSLE}$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u2112<\/mml:mi><mml:mi>\u2133<\/mml:mi><mml:mi>S<\/mml:mi><mml:mi>\u2112<\/mml:mi><mml:mi>E<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>with the assistance of a public cloud server. By utilizing affine transformation based on sparse unimodular matrices, our scheme has three merits compared with previous work: 1) Our scheme is efficiency\/security-adjustable. Our encryption method is dynamic, and it can balance the security and efficiency to match different application scenarios by skillfully control the number of unimodular matrices. 2) Our scheme is versatile. It is suit for generic<jats:italic>m<\/jats:italic>-by-<jats:italic>n<\/jats:italic>coefficient matrix<jats:bold>A<\/jats:bold>, no matter it is square or not. 3) Our scheme satisfies public verifiability and achieves the optimal verification probability. It enables any verifier which is not necessarily the client to verify the correctness of the results returned from the cloud server with probability 1. Finally, theoretical analysis and comprehensive experimental results confirm our scheme\u2019s security and high efficiency.<\/jats:p>","DOI":"10.1186\/s13677-019-0145-8","type":"journal-article","created":{"date-parts":[[2019,12,27]],"date-time":"2019-12-27T17:02:58Z","timestamp":1577466178000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Publicly verifiable and efficiency\/security-adjustable outsourcing scheme for solving large-scale modular system of linear equations"],"prefix":"10.1186","volume":"8","author":[{"given":"Panpan","family":"Meng","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2474-910X","authenticated-orcid":false,"given":"Chengliang","family":"Tian","sequence":"additional","affiliation":[]},{"given":"Xiangguo","family":"Cheng","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,12,27]]},"reference":[{"issue":"2","key":"145_CR1","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1109\/MCE.2016.2640718","volume":"6","author":"KJ Singh","year":"2017","unstructured":"Singh KJ, Kapoor DS (2017) Create your own internet of things: A survey of iot platforms. 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