{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:52:45Z","timestamp":1760233965037,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,3,15]],"date-time":"2021-03-15T00:00:00Z","timestamp":1615766400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002347","name":"Bundesministerium f\u00fcr Bildung und Forschung","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 McEliece cryptosystem is a promising candidate for post-quantum public-key encryption. In this work, we propose q-ary codes over Gaussian integers for the McEliece system and a new channel model. With this one Mannheim error channel, errors are limited to weight one. We investigate the channel capacity of this channel and discuss its relation to the McEliece system. The proposed codes are based on a simple product code construction and have a low complexity decoding algorithm. For the one Mannheim error channel, these codes achieve a higher error correction capability than maximum distance separable codes with bounded minimum distance decoding. This improves the work factor regarding decoding attacks based on information-set decoding.<\/jats:p>","DOI":"10.3390\/cryptography5010011","type":"journal-article","created":{"date-parts":[[2021,3,15]],"date-time":"2021-03-15T22:16:54Z","timestamp":1615846614000},"page":"11","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["A New Class of Q-Ary Codes for the McEliece Cryptosystem"],"prefix":"10.3390","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5913-4981","authenticated-orcid":false,"given":"J\u00fcrgen","family":"Freudenberger","sequence":"first","affiliation":[{"name":"Institute for System Dynamics (ISD), HTWG Konstanz, University of Applied Sciences, 78462 Konstanz, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4318-2660","authenticated-orcid":false,"given":"Johann-Philipp","family":"Thiers","sequence":"additional","affiliation":[{"name":"Institute for System Dynamics (ISD), HTWG Konstanz, University of Applied Sciences, 78462 Konstanz, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,15]]},"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, Santa Fe, NM, USA."},{"key":"ref_2","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_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.","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","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 2006 IEEE International Symposium on Information Theory, Seattle, WA, USA.","DOI":"10.1109\/ISIT.2006.261651"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Preneel, B. (2009). Reducing key length of the McEliece cryptosystem. Progress in Cryptology\u2014AFRICACRYPT, Springer.","DOI":"10.1007\/978-3-642-02384-2"},{"key":"ref_7","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_8","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_9","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_10","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_11","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_12","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_13","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, Springer.","DOI":"10.1007\/3-540-45961-8"},{"key":"ref_14","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_15","unstructured":"Lee, D.H., and Wang, X. (2011). Decoding random linear codes in O(20.054n). Advances in Cryptology\u2014ASIACRYPT 2011, Springer."},{"key":"ref_16","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_17","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1109\/18.272484","article-title":"Codes over Gaussian integers","volume":"40","author":"Huber","year":"1994","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1512","DOI":"10.1109\/18.412699","article-title":"Groups of complex integers used as QAM signals","volume":"41","author":"Rifa","year":"1995","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1848","DOI":"10.1109\/18.705563","article-title":"Groups of algebraic integers used for coding QAM signals","volume":"44","author":"Dong","year":"1998","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_20","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_21","first-page":"153","article-title":"An asymmetric cryptography using Gaussian integers","volume":"47","author":"Juraphanthong","year":"2020","journal-title":"Eng. Appl. Sci. Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2420","DOI":"10.1109\/TCOMM.2015.2441691","article-title":"New Four-Dimensional Signal Constellations From Lipschitz Integers for Transmission Over the Gaussian Channel","volume":"63","author":"Freudenberger","year":"2015","journal-title":"IEEE Trans. Commun."},{"key":"ref_23","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 2018 IEEE International Symposium on Information Theory (ISIT), Vail, CO, USA.","DOI":"10.1109\/ISIT.2018.8437456"},{"key":"ref_24","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_25","doi-asserted-by":"crossref","unstructured":"Ivanov, F., Kabatiansky, G., Krouk, E., and Rumenko, N. (2020, January 9\u201310). A new code-based cryptosystem. Proceedings of the 8th International Workshop on Code-Based Cryptography, CBCrypto, Zagreb, Croatia.","DOI":"10.1007\/978-3-030-54074-6_3"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3042","DOI":"10.1109\/TIT.2007.903126","article-title":"Perfect Codes for Metrics Induced by Circulant Graphs","volume":"53","author":"Martinez","year":"2007","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Safieh, M., and Freudenberger, J. (2021). Montgomery Reduction for Gaussian Integers. Cryptography, 5.","DOI":"10.3390\/cryptography5010006"},{"key":"ref_28","unstructured":"Bossert, M. (1999). Channel Coding for Telecommunications, Wiley."},{"key":"ref_29","unstructured":"Jiang, Y. (2010). A Practical Guide to Error-Control Coding Using Matlab, Artech House."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1049\/iet-cds.2014.0278","article-title":"Decoder Architecture for Generalized Concatenated Codes","volume":"9","author":"Spinner","year":"2015","journal-title":"IET Circuits Devices Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1049\/iet-cds.2017.0347","article-title":"Soft input decoder for high-rate generalised concatenated codes","volume":"12","author":"Spinner","year":"2018","journal-title":"IET Circuits Devices Syst."},{"key":"ref_32","unstructured":"Gallager, R.G. (1968). Information Theory and Reliable Communication, John Wiley & Sons, Inc."},{"key":"ref_33","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_34","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"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Huber, K. (1994). Codes over Eisenstein-Jacobi Integers. Contemp. 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