{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T01:34:46Z","timestamp":1769045686551,"version":"3.49.0"},"reference-count":33,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,11,29]],"date-time":"2021-11-29T00:00:00Z","timestamp":1638144000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002347","name":"Federal Ministry of Education and Research","doi-asserted-by":"publisher","award":["16ES1045"],"award-info":[{"award-number":["16ES1045"]}],"id":[{"id":"10.13039\/501100002347","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Cryptography"],"abstract":"<jats:p>The code-based McEliece and Niederreiter cryptosystems are promising candidates for post-quantum public-key encryption. Recently, q-ary concatenated codes over Gaussian integers were proposed for the McEliece cryptosystem, together with the one-Mannheim error channel, where the error values are limited to the Mannheim weight one. Due to the limited error values, the codes over Gaussian integers achieve a higher error correction capability than maximum distance separable (MDS) codes with bounded minimum distance decoding. This higher error correction capability improves the work factor regarding decoding attacks based on information-set decoding. The codes also enable a low complexity decoding algorithm for decoding beyond the guaranteed error correction capability. In this work, we extend this coding scheme to codes over Eisenstein integers. These codes have advantages for the Niederreiter system. Additionally, we propose an improved code construction based on generalized concatenated codes. These codes extend to the rate region, where the work factor is beneficial compared to MDS codes. Moreover, generalized concatenated codes are more robust against structural attacks than ordinary concatenated codes.<\/jats:p>","DOI":"10.3390\/cryptography5040033","type":"journal-article","created":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T03:12:56Z","timestamp":1638328376000},"page":"33","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Generalized Concatenated Codes over Gaussian and Eisenstein Integers for Code-Based Cryptography"],"prefix":"10.3390","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4318-2660","authenticated-orcid":false,"given":"Johann-Philipp","family":"Thiers","sequence":"first","affiliation":[{"name":"Institute for System Dynamics (ISD), HTWG Konstanz, University of Applied Sciences, 78462 Konstanz, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5913-4981","authenticated-orcid":false,"given":"J\u00fcrgen","family":"Freudenberger","sequence":"additional","affiliation":[{"name":"Institute for System Dynamics (ISD), HTWG Konstanz, University of Applied Sciences, 78462 Konstanz, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,29]]},"reference":[{"key":"ref_1","unstructured":"Shor, P.W. (1994, January 20\u201322). Algorithms for quantum computation: Discrete logarithms and factoring. Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Washington, DC, USA."},{"key":"ref_2","first-page":"317","article-title":"Shor\u2019s Discrete Logarithm Quantum Algorithm for Elliptic Curves","volume":"3","author":"Proos","year":"2003","journal-title":"Quantum Inf. Comput."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Alagic, G., Alperin-Sheriff, J., Apon, D., Cooper, D., Dang, Q., Kelsey, J., Liu, Y.K., Miller, C., Moody, D., and Peralta, R. (2020). Status Report on the Second Round of the NIST Post-Quantum Cryptography Standardization Process, National Institute of Standards and Technology. Nistir 8309.","DOI":"10.6028\/NIST.IR.8240"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1109\/TIT.1978.1055873","article-title":"On the inherent intractability of certain coding problems","volume":"24","author":"Berlekamp","year":"1978","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_5","first-page":"114","article-title":"A public-key cryptosystem based on algebraic coding theory","volume":"42\u201344","author":"McEliece","year":"1978","journal-title":"DSN Prog. Rep."},{"key":"ref_6","first-page":"159","article-title":"Knapsack-type cryptosystems and algebraic coding theory","volume":"15","author":"Niederreiter","year":"1986","journal-title":"Probl. Control Inf. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1109\/TIT.1962.1057777","article-title":"The use of information sets in decoding cyclic codes","volume":"8","author":"Prange","year":"1962","journal-title":"IRE Trans. Inf. Theory"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Barstow, D., Brauer, W., Brinch Hansen, P., Gries, D., Luckham, D., Moler, C., Pnueli, A., Seegm\u00fcller, G., Stoer, J., and Wirth, N. (1988). An Observation on the Security of McEliece\u2019s Public-Key Cryptosystem. Advances in Cryptology\u2014EUROCRYPT \u201988, Proceedings of the Workshop on the Theory and Application of Cryptographic Techniques, Davos, Switzerland, 25\u201327 May 1988, Springer.","DOI":"10.1007\/3-540-45961-8"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Cohen, G., and Wolfmann, J. (1989). A method for finding codewords of small weight. Coding Theory and Applications, Springer.","DOI":"10.1007\/BFb0019841"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Buchmann, J., and Ding, J. (2008). Attacking and Defending the McEliece Cryptosystem. Post-Quantum Cryptography, Springer.","DOI":"10.1007\/978-3-540-88403-3"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Lee, D.H., and Wang, X. (2011). Decoding Random Linear Codes in O(20.054n). Advances in Cryptology\u2014ASIACRYPT 2011, Proceedings of the 17th International Conference on the Theory and Application of Cryptology and Information Security, Seoul, South Korea, 4\u20138 December 2011, Springer.","DOI":"10.1007\/978-3-642-25385-0"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Boyd, C. (2001). How to Achieve a McEliece-Based Digital Signature Scheme. Advances in Cryptology\u2014ASIACRYPT 2001, Proceedings of the 7th International Conference on the Theory and Application of Cryptology and Information Security, Gold Coast, Australia, 9\u201313 December 2001, Springer.","DOI":"10.1007\/3-540-45682-1"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Wieschebrink, C. (2006, January 9\u201314). Two NP-complete Problems in Coding Theory with an Application in Code Based Cryptography. Proceedings of the IEEE International Symposium on Information Theory, Seattle, WA, USA.","DOI":"10.1109\/ISIT.2006.261651"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Preneel, B. (2009). Reducing Key Length of the McEliece Cryptosystem. Progress in Cryptology\u2014AFRICACRYPT, Proceedings of the Second International Conference on Cryptology in Africa, Gammarth, Tunisia, 21\u201325 June 2009, Springer.","DOI":"10.1007\/978-3-642-02384-2"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Le Van, T., and Hoan, P.K. (2015, January 28\u201330). McEliece cryptosystem based identification and signature scheme using chained BCH codes. Proceedings of the International Conference on Communications, Management and Telecommunications (ComManTel), DaNang, Vietnam.","DOI":"10.1109\/ComManTel.2015.7394272"},{"key":"ref_16","unstructured":"Monico, C., Rosenthal, J., and Shokrollahi, A. (2000, January 25\u201330). Using low density parity check codes in the McEliece cryptosystem. Proceedings of the 2000 IEEE International Symposium on Information Theory, Sorrento, Italy."},{"key":"ref_17","unstructured":"Shooshtari, M.K., Ahmadian, M., and Payandeh, A. (2009, January 15\u201318). Improving the security of McEliece-like public key cryptosystem based on LDPC codes. Proceedings of the 11th International Conference on Advanced Communication Technology, Gangwon-Do, Korea."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Baldi, M., Bianchi, M., Maturo, N., and Chiaraluce, F. (2013, January 7\u201310). Improving the efficiency of the LDPC code-based McEliece cryptosystem through irregular codes. Proceedings of the IEEE Symposium on Computers and Communications (ISCC), Split, Croatia.","DOI":"10.1109\/ISCC.2013.6754945"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"714","DOI":"10.1109\/LCOMM.2016.2640271","article-title":"A New Variant of the McEliece Cryptosystem Based on QC-LDPC and QC-MDPC Codes","volume":"21","author":"Moufek","year":"2017","journal-title":"IEEE Commun. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Hooshmand, R., Shooshtari, M.K., Eghlidos, T., and Aref, M.R. (2014, January 3\u20134). Reducing the key length of McEliece cryptosystem using polar codes. Proceedings of the 11th International ISC Conference on Information Security and Cryptology, Tehran, Iran.","DOI":"10.1109\/ISCISC.2014.6994031"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1515\/dma.1992.2.4.439","article-title":"On insecurity of cryptosystems based on generalized Reed-Solomon codes","volume":"2","author":"Sidelnikov","year":"1992","journal-title":"Discret. Math. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/s002000050104","article-title":"On the Concatenated Structure of a Linear Code","volume":"9","author":"Sendrier","year":"1998","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Freudenberger, J., and Thiers, J.P. (2021). A new class of q-ary codes for the McEliece cryptosystem. Cryptography, 5.","DOI":"10.3390\/cryptography5010011"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1109\/18.272484","article-title":"Codes over Gaussian integers","volume":"Volume 40","author":"Huber","year":"1994","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3114","DOI":"10.1109\/TCOMM.2013.061913.120742","article-title":"New Coding Techniques for Codes over Gaussian Integers","volume":"61","author":"Freudenberger","year":"2013","journal-title":"IEEE Trans. Commun."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Kotsireas, I.S., and Mart\u00ednez-Moro, E. (2017). Code-Based Cryptosystems Using Generalized Concatenated Codes. Applications of Computer Algebra, Springer International Publishing.","DOI":"10.1007\/978-3-319-56932-1"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1090\/conm\/168\/01696","article-title":"Codes over Eisenstein-Jacobi Integers","volume":"Volume 168","author":"Huber","year":"1994","journal-title":"Contemporary Mathematics"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Conway, J., and Sloane, N. (1999). Sphere Packings, Lattices and Groups, Springer. [3rd ed.].","DOI":"10.1007\/978-1-4757-6568-7"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Rohweder, D., Freudenberger, J., and Shavgulidze, S. (2018, January 17\u201322). Low-Density Parity-Check Codes over Finite Gaussian Integer Fields. Proceedings of the IEEE International Symposium on Information Theory (ISIT), Vail, CO, USA.","DOI":"10.1109\/ISIT.2018.8437456"},{"key":"ref_30","unstructured":"Gallager, R.G. (1968). Information Theory and Reliable Communication, John Wiley & Sons, Inc."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Neubauer, A., Freudenberger, J., and K\u00fchn, V. (2007). Coding Theory: Algorithms, Architectures and Applications, John Wiley & Sons.","DOI":"10.1002\/9780470519837"},{"key":"ref_32","unstructured":"Bossert, M. (1999). Channel Coding for Telecommunications, Wiley."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Fabsic, T., Hromada, V., Stankovski, P., Zajac, P., Guo, Q., and Johansson, T. (2017, January 26\u201328). A reaction attack on the QC-LDPC McEliece cryptosystem. Proceedings of the Post-Quantum Cryptography\u20148th International Workshop (PQCrypto), Utrecht, The Netherlands.","DOI":"10.1007\/978-3-319-59879-6_4"}],"container-title":["Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2410-387X\/5\/4\/33\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:37:18Z","timestamp":1760168238000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2410-387X\/5\/4\/33"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,29]]},"references-count":33,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["cryptography5040033"],"URL":"https:\/\/doi.org\/10.3390\/cryptography5040033","relation":{},"ISSN":["2410-387X"],"issn-type":[{"value":"2410-387X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,29]]}}}