{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,17]],"date-time":"2026-06-17T14:18:56Z","timestamp":1781705936145,"version":"3.54.5"},"reference-count":24,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,2,3]],"date-time":"2024-02-03T00:00:00Z","timestamp":1706918400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,2,3]],"date-time":"2024-02-03T00:00:00Z","timestamp":1706918400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Open Project of Henan Key Laboratory of Network Cryptography Technology","award":["LNCT2021-A09"],"award-info":[{"award-number":["LNCT2021-A09"]}]},{"name":"the Advanced Discipline Construction Project of Beijing Universities","award":["0210101Z0401"],"award-info":[{"award-number":["0210101Z0401"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cybersecurity"],"abstract":"Abstract<\/jats:title>In MILCOM 2015, Kelly et al. proposed the authentication encryption algorithm MK-3, which applied the 16-bit S-box. This paper aims to implement the 16-bit S-box with less circuit area. First, we classified the irreducible polynomials over$$\\mathbb {F}_{2^n}$$<\/jats:tex-math>F<\/mml:mi>2<\/mml:mn>n<\/mml:mi><\/mml:msup><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>into three kinds. Then we compared the logic gates required for multiplication over the finite field constructed by the three types of irreducible polynomials. According to the comparison result, we constructed the composite fields,$$\\mathbb {F}_{(2^4)^2}$$<\/jats:tex-math>F<\/mml:mi>(<\/mml:mo>2<\/mml:mn>4<\/mml:mn><\/mml:msup>)<\/mml:mo><\/mml:mrow>2<\/mml:mn><\/mml:msup><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and$$\\mathbb {F}_{(2^8)^2}$$<\/jats:tex-math>F<\/mml:mi>(<\/mml:mo>2<\/mml:mn>8<\/mml:mn><\/mml:msup>)<\/mml:mo><\/mml:mrow>2<\/mml:mn><\/mml:msup><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Based on the isomorphism of finite fields, the operations over$$\\mathbb {F}_{2^{16}}$$<\/jats:tex-math>F<\/mml:mi>2<\/mml:mn>16<\/mml:mn><\/mml:msup><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>can be conducted over$$\\mathbb {F}_{(2^8)^2}$$<\/jats:tex-math>F<\/mml:mi>(<\/mml:mo>2<\/mml:mn>8<\/mml:mn><\/mml:msup>)<\/mml:mo><\/mml:mrow>2<\/mml:mn><\/mml:msup><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Similarly, elements over$$\\mathbb {F}_{2^8}$$<\/jats:tex-math>F<\/mml:mi>2<\/mml:mn>8<\/mml:mn><\/mml:msup><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>can be mapped to the corresponding elements over$$\\mathbb {F}_{(2^4)^2}$$<\/jats:tex-math>F<\/mml:mi>(<\/mml:mo>2<\/mml:mn>4<\/mml:mn><\/mml:msup>)<\/mml:mo><\/mml:mrow>2<\/mml:mn><\/mml:msup><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Next, the SAT solver was used to optimize the operations over smaller field$$\\mathbb {F}_{2^4}$$<\/jats:tex-math>F<\/mml:mi>2<\/mml:mn>4<\/mml:mn><\/mml:msup><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. At last, the architecture of the optimized MK-3\u00a0S-box was worked out. Compared with the implementation proposed by the original designer, the circuit area of the MK-3\u00a0S-box in this paper is reduced by at least 55.9%.<\/jats:p>","DOI":"10.1186\/s42400-024-00207-x","type":"journal-article","created":{"date-parts":[[2024,2,3]],"date-time":"2024-02-03T01:29:55Z","timestamp":1706923795000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A circuit area optimization of MK-3\u00a0S-box"],"prefix":"10.1186","volume":"7","author":[{"given":"Yanjun","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Weiguo","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-6332-3731","authenticated-orcid":false,"given":"Yiping","family":"Lin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jian","family":"Zou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jian","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,2,3]]},"reference":[{"key":"207_CR1","first-page":"168","volume":"2008","author":"J Boyar","year":"2008","unstructured":"Boyar J, Matthews P, Peralta R (2008) On the shortest linear straight-line program for computing linear forms. Math Found Comput Sci 2008:168\u2013179","journal-title":"Math Found Comput Sci"},{"key":"207_CR2","doi-asserted-by":"publisher","first-page":"280","DOI":"10.1007\/s00145-012-9124-7","volume":"26","author":"Joan Boyar","year":"2013","unstructured":"Boyar Joan, Matthews Philip, Peralta Ren\u00e9 (2013) Logic minimization techniques with applications to cryptology. J Cryptol 26:280\u2013312","journal-title":"J Cryptol"},{"key":"207_CR3","doi-asserted-by":"crossref","unstructured":"Boyar J, Peralta R (2010) A new combinational logic minimization technique with applications to cryptology. Exp Algorithm, pp 178\u2013189","DOI":"10.1007\/978-3-642-13193-6_16"},{"key":"207_CR4","doi-asserted-by":"crossref","unstructured":"Canright D (2005) A very compact S-Box for AES. Cryptographic hardware and embedded systems\u2014CHES 2005, pp 441\u2013455","DOI":"10.1007\/11545262_32"},{"key":"207_CR5","doi-asserted-by":"crossref","unstructured":"Daemen J, Rijmen V (1998) The block Cipher Rijndael. Smart card research and advanced application conference","DOI":"10.1007\/BFb0052343"},{"key":"207_CR6","doi-asserted-by":"publisher","first-page":"1179","DOI":"10.1007\/s10623-020-00741-y","volume":"88","author":"X Dong","year":"2020","unstructured":"Dong X, Dong B, Wang X (2020) Quantum attacks on some feistel block Ciphers. Des Codes Crypt 88:1179\u20131203","journal-title":"Des Codes Crypt"},{"key":"207_CR7","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1016\/0890-5401(88)90024-7","volume":"78","author":"T Itoh","year":"1988","unstructured":"Itoh T, Tsujii S (1988) A fast algorithm for computing multiplicative inverses in GF (2m) using normal bases. Inf Comput 78:171\u2013177","journal-title":"Inf Comput"},{"key":"207_CR8","first-page":"207","volume":"2016","author":"M Kaplan","year":"2016","unstructured":"Kaplan M, Leurent G, Leverrier A, Naya-Plasencia M (2016) Breaking symmetric cryptosystems using quantum period finding. Adv Cryptol 2016:207\u2013237","journal-title":"Adv Cryptol"},{"key":"207_CR9","doi-asserted-by":"crossref","unstructured":"Kelly M, Kaminsky A, Kurdziel MT, Lukowiak M, Radziszowski SP (2015) Customizable sponge-based authenticated encryption using 16-bit s-boxes. MILCOM 2015\u20132015 IEEE military communications conference, pp 43\u201348","DOI":"10.1109\/MILCOM.2015.7357416"},{"key":"207_CR10","doi-asserted-by":"crossref","unstructured":"Kuwakado H, Morii M (2010) Quantum distinguisher between the 3-round Feistel cipher and the Random Permutation. 2010 IEEE international symposium on information theory, pp 2682\u20132685","DOI":"10.1109\/ISIT.2010.5513654"},{"key":"207_CR11","unstructured":"Kuwakado H, Morii M (2012) Security on the quantum-type even-Mansour Cipher. 2012 international symposium on information theory and its applications, pp 312\u2013316"},{"key":"207_CR12","doi-asserted-by":"crossref","unstructured":"Maximov A, Ekdahl P (2019) New circuit minimization techniques for smaller and faster AES SBoxes. IACR Trans Cryptograph Hardware Embedded Syst, pp 91\u2013125","DOI":"10.46586\/tches.v2019.i4.91-125"},{"key":"207_CR13","unstructured":"NIST A (2001) Specification of the advanced encryption standard (AES). Federal information processing standards publication 197"},{"key":"207_CR14","doi-asserted-by":"crossref","unstructured":"Paar C (1995) Some remarks on efficient inversion in finite fields. In: Proceedings of 1995 IEEE international symposium on information theory vol 58","DOI":"10.1109\/ISIT.1995.531160"},{"key":"207_CR15","doi-asserted-by":"crossref","unstructured":"Paar C (1997) Optimized arithmetic for reed-solomon encoders. Proceedings of IEEE international symposium on information theory, vol 250","DOI":"10.1109\/ISIT.1997.613165"},{"key":"207_CR16","doi-asserted-by":"publisher","first-page":"856","DOI":"10.1109\/12.508323","volume":"45","author":"C Paar","year":"1996","unstructured":"Paar C (1996) A new architecture for a parallel finite field multiplier with low complexity based on composite fields. IEEE Trans Comput 45:856\u2013861","journal-title":"IEEE Trans Comput"},{"key":"207_CR17","doi-asserted-by":"crossref","unstructured":"Reyhani-Masoleh A, Taha M, Ashmawy D (2018) Smashing the implementation records of AES S-box. IACR transactions on cryptographic hardware and embedded systems, pp 298\u2013336","DOI":"10.46586\/tches.v2018.i2.298-336"},{"key":"207_CR18","unstructured":"Rijmen V (2000) Efficient implementation of the Rijndael S-box. Katholieke Universiteit Leuven, Dept. ESAT. Belgium"},{"key":"207_CR19","doi-asserted-by":"crossref","unstructured":"Satoh A, Morioka S, Takano K, Munetoh S (2001) A compact Rijndael hardware architecture with S-box optimization. Advances in cryptology-ASIACRYPT 2001: Proceedings 7th international conference on the theory and application of cryptology and information security gold coast, Australia, December 9\u201313, pp 239\u2013254","DOI":"10.1007\/3-540-45682-1_15"},{"key":"207_CR20","doi-asserted-by":"publisher","first-page":"46","DOI":"10.1109\/CC.2018.8300271","volume":"15","author":"T Shi","year":"2018","unstructured":"Shi T, Jin C, Guan J (2018) Collision attacks against AEZ-PRF for authenticated encryption AEZ. China Commun 15:46\u201353","journal-title":"China Commun"},{"key":"207_CR21","doi-asserted-by":"publisher","first-page":"1474","DOI":"10.1137\/S0097539796298637","volume":"26","author":"DR Simon","year":"1997","unstructured":"Simon DR (1997) On the power of quantum computation. SIAM J Comput 26:1474\u20131483","journal-title":"SIAM J Comput"},{"key":"207_CR22","doi-asserted-by":"crossref","unstructured":"Stoffelen K (2016) Optimizing S-box implementations for several criteria using SAT solvers. In: Fast software encryption: 23rd international conference, FSE 2016, Bochum, Germany, March 20-23, 2016, Revised Selected Papers 23, pp 140\u2013160","DOI":"10.1007\/978-3-662-52993-5_8"},{"key":"207_CR23","doi-asserted-by":"crossref","unstructured":"Wood CA, Radziszowski SP, Lukowiak M (2015) Constructing large S-boxes with area minimized implementations. MILCOM 2015\u20132015 IEEE military communications conference, pp 49\u201354","DOI":"10.1109\/MILCOM.2015.7357417"},{"key":"207_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11128-021-03036-w","volume":"20","author":"Y Xu","year":"2021","unstructured":"Xu Y, Liu W, Yu W (2021) Quantum forgery attacks on COPA, AES-COPA and marble authenticated encryption algorithms. Quant Inf Process 20:1\u201321","journal-title":"Quant Inf Process"}],"container-title":["Cybersecurity"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s42400-024-00207-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1186\/s42400-024-00207-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s42400-024-00207-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,10]],"date-time":"2024-11-10T00:35:47Z","timestamp":1731198947000},"score":1,"resource":{"primary":{"URL":"https:\/\/cybersecurity.springeropen.com\/articles\/10.1186\/s42400-024-00207-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,3]]},"references-count":24,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["207"],"URL":"https:\/\/doi.org\/10.1186\/s42400-024-00207-x","relation":{},"ISSN":["2523-3246"],"issn-type":[{"value":"2523-3246","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,2,3]]},"assertion":[{"value":"25 October 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 January 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 February 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"17"}}