{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T21:19:21Z","timestamp":1775251161371,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2018,10,15]],"date-time":"2018-10-15T00:00:00Z","timestamp":1539561600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Cryptography"],"abstract":"<jats:p>We propose a rank metric codes based encryption based on the hard problem of rank syndrome decoding problem. We propose a new encryption with a public key matrix by considering the adding of a random distortion matrix over F q m of full column rank n. We show that IND-CPA security is achievable for our encryption under assumption of the Decisional Rank Syndrome Decoding problem. Furthermore, we also prove some bounds for the number of matrices of a fixed rank with entries over a finite field. Our proposal allows the choice of the error terms with rank up to r 2 , where r is the error-correcting capability of a code. Our encryption based on Gabidulin codes has public key size of 13 . 68 KB, which is 82 times smaller than the public key size of McEliece Cryptosystem based on Goppa codes. For similar post-quantum security level of 2 140 bits, our encryption scheme has a smaller public key size than the key size suggested by LOI17 Encryption.<\/jats:p>","DOI":"10.3390\/cryptography2040032","type":"journal-article","created":{"date-parts":[[2018,10,16]],"date-time":"2018-10-16T02:52:53Z","timestamp":1539658373000},"page":"32","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A New Technique in Rank Metric Code-Based Encryption"],"prefix":"10.3390","volume":"2","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1218-9133","authenticated-orcid":false,"given":"Terry","family":"Lau","sequence":"first","affiliation":[{"name":"Temasek Laboratories, National University of Singapore, T-Lab Building, 5A, Engineering Drive 1, #09-02, Singapore 117411, Singapore"}]},{"given":"Chik","family":"Tan","sequence":"additional","affiliation":[{"name":"Temasek Laboratories, National University of Singapore, T-Lab Building, 5A, Engineering Drive 1, #09-02, Singapore 117411, Singapore"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,15]]},"reference":[{"key":"ref_1","first-page":"114","article-title":"A public-key cryptosystem based on algebraic coding theory","volume":"4244","author":"McEliece","year":"1978","journal-title":"Coding Thv."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1515\/dma.1992.2.4.439","article-title":"On insecurity of cryptosystems based on generalied Reed-Solomon codes","volume":"2","author":"Sidelnikov","year":"1992","journal-title":"Discret. Math. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Baldi, M., and Chiaraluce, F. (2007, January 24\u201329). Cryptanalysis of a new instance of McEliece cryptosystem based on QC-LDPC codes. Proceedings of the IEEE International Symposium on Information Theory, Nice, France.","DOI":"10.1109\/ISIT.2007.4557609"},{"key":"ref_4","first-page":"3","article-title":"Theory of codes with maximum rank distance","volume":"21","author":"Gabidulin","year":"1985","journal-title":"Probl. Pereda. Inf."},{"key":"ref_5","unstructured":"Gabidulin, E.M., Paramonov, A.V., and Tretjakov, O.V. (1991, January 8\u201311). Ideals over a non-commutative ring and their application in cryptology. Proceedings of the Worshop on the Theory and Application of Cryptographic Techniques, Brighton, UK."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/BF01390769","article-title":"Severely denting the Gabidulin version of the McEliece public-key cryptosystem","volume":"6","author":"Gibson","year":"1995","journal-title":"Des. Codes Cryptogr."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/S1571-0653(04)00168-4","article-title":"Modified GPT PKC with right scrambler","volume":"6","author":"Gabidulin","year":"2001","journal-title":"Electron. Notes Discret. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/S0166-218X(02)00446-8","article-title":"Column Scrambler for the GPT cryptosystem","volume":"128","author":"Ourivski","year":"2003","journal-title":"Discret. Appl. Math."},{"key":"ref_9","unstructured":"Overbeck, R. (2005, January 14\u201318). Extending Gibson\u2019s attacks on the GPT cryptosystem. Proceedings of the International Workshop on Coding and Cryptography, Bergen, Norway."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1007\/s00145-007-9003-9","article-title":"Structural attacks for public key cryptosystems based on Gabidulin codes","volume":"21","author":"Overbeck","year":"2008","journal-title":"J. Cryptol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/s10623-007-9160-8","article-title":"Attacks and counter-attacks on the GPT public key cryptosystem","volume":"48","author":"Gabidulin","year":"2008","journal-title":"Des. Codes Cryptogr."},{"key":"ref_12","unstructured":"Gabidulin, E.M., Rashwan, H., and Honary, B. (July, January 28). On improving security of GPT cryptosystems. Proceedings of the IEEE International Symposium on Information Theory, Seoul, South Korea."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Loidreau, P. (2010, January 25\u201328). Designing a rank metric based McEliece cryptosystem. Proceedings of the 3rd International Workshop on Post-Quantum Cryptography, Darmstadt, Germany.","DOI":"10.1007\/978-3-642-12929-2_11"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Rashwan, H., Gabidulin, E.M., and Honary, B. (2010, January 13\u201318). A smart approach for GPT cryptosystem based on rank codes. Proceedings of the IEEE International Symposium on Information Theory, Austin, TX, USA.","DOI":"10.1109\/ISIT.2010.5513549"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/s10623-017-0343-7","article-title":"Extension of Overbeck\u2019s Attack for Gabidulin Based Cryptosystems","volume":"86","author":"Marshall","year":"2018","journal-title":"Des. Codes Cryptogr."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1983","DOI":"10.1007\/s10623-017-0434-5","article-title":"Improved Cryptanalysis of Rank Metric Schemes Based on Gabidulin Codes","volume":"86","author":"Otmani","year":"2018","journal-title":"Des. Codes Cryptogr."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1006","DOI":"10.1109\/TIT.2015.2511786","article-title":"On the complexity of the rank syndrome decoding problem","volume":"62","author":"Gaborit","year":"2016","journal-title":"IEEE Trans. Inf. Theor."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Horlemann-Trautmann, A., Marshall, K., and Rosenthal, J. (2016, January 10\u201315). Considerations for Rank-based Cryptosystems. Proceedings of the IEEE International Symposium on Information Theory, Barcelona, Spain.","DOI":"10.1109\/ISIT.2016.7541758"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1023\/A:1020369320078","article-title":"New technique for decoding codes in the rank metric and its cryptography applications","volume":"38","author":"Ourivski","year":"2002","journal-title":"Probl. Inf. Transm."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Gaborit, P., Hauteville, A., Phan, D.H., and Tillich, J.P. (2017, January 20\u201324). Identity-based Encryption from Codes with Rank Metric. Proceedings of the 37th Annual International Cryptology Conference, Santa Barbara, CA, USA.","DOI":"10.1007\/978-3-319-63697-9_7"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Loidreau, P. (2017, January 26\u201328). A New Rank Metric Codes Based Encryption Scheme. Proceedings of the 8th International Workshop on Post-Quantum Cryptography, Utrecht, The Netherlands.","DOI":"10.1007\/978-3-319-59879-6_1"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"7245","DOI":"10.1109\/TIT.2016.2616127","article-title":"On the hardness of the decoding and the minimum distance problems for rank codes","volume":"62","author":"Gaborit","year":"2016","journal-title":"IEEE Trans. Inf. Theor."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Aragon, A., Gaborit, P., Hauteville, A., and Tillich, J.P. (2018, January 17\u201322). A New Algorithm for Solving the Rank Syndrome Decoding Problem. Proceedings of the 2018 IEEE International Symposium on Information Theory, Vail, CO, USA.","DOI":"10.1109\/ISIT.2018.8437464"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Goubin, L., and Courtois, N.T. (2000, January 3\u20137). Cryptanalysis of the TTM cryptosystem. Proceedings of the 6th International Conference on the Theory and Application of Cryptology and Information Security, Kyoto, Japan.","DOI":"10.1007\/3-540-44448-3_4"},{"key":"ref_25","first-page":"87","article-title":"\u00dcber eine Azahibestimmung und eine damit zusammeng\u00e4ngende Reihe","volume":"111","author":"Landsberg","year":"1893","journal-title":"J. Reine Angew. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"19","DOI":"10.7146\/math.scand.a-12477","article-title":"Counting Matrices with Coordinates in Finite Fields and of Fixed Rank","volume":"74","author":"Laksov","year":"1994","journal-title":"Math. Scand."},{"key":"ref_27","unstructured":"Shoup, V. (2018, October 07). Sequences of Games: A Tool for Taming Complexity in Security Proofs. Available online: http:\/\/www.shoup.net\/papers\/games.pdf."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Bernstein, D.J. (2010, January 25\u201328). Grover vs. McEliece. Proceedings of the 3rd International Workshop on Post-Quantum Cryptography, Darmstadt, Germany.","DOI":"10.1007\/978-3-642-12929-2_6"}],"container-title":["Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2410-387X\/2\/4\/32\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T20:08:51Z","timestamp":1775246931000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2410-387X\/2\/4\/32"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,15]]},"references-count":28,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,12]]}},"alternative-id":["cryptography2040032"],"URL":"https:\/\/doi.org\/10.3390\/cryptography2040032","relation":{},"ISSN":["2410-387X"],"issn-type":[{"value":"2410-387X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,10,15]]}}}