{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T12:42:05Z","timestamp":1781008925220,"version":"3.54.1"},"reference-count":20,"publisher":"Springer Science and Business Media LLC","issue":"10","license":[{"start":{"date-parts":[[2020,6,23]],"date-time":"2020-06-23T00:00:00Z","timestamp":1592870400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,6,23]],"date-time":"2020-06-23T00:00:00Z","timestamp":1592870400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100006254","name":"Ruhr-Universit\u00e4t Bochum","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006254","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2020,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Division property is a cryptanalysis method that proves to be very efficient on block ciphers. Computer-aided techniques such as MILP have been widely and successfully used to study various cryptanalysis techniques, and it especially led to many new results for the division property. Nonetheless, we claim that the previous techniques do not consider the full search space. We show that even if the previous techniques fail to find a distinguisher based on the division property over a given function, we can potentially find a relevant distinguisher over a linearly equivalent function. We show that the representation of the block cipher heavily influences the propagation of the division property, and exploiting this, we give an algorithm to efficiently search for such linear mappings. As a result, we exhibit a new distinguisher over 10 rounds of , while the previous best was over 9 rounds, and rule out such a distinguisher over more than 9 rounds of . We also give some insight about the construction of an S-box to strengthen a block cipher against our technique. We prove that using an S-box satisfying a certain criterion is optimal in term of resistance against classical division property. Accordingly, we exhibit stronger variants of  and , improving the resistance against division property based distinguishers by 2 rounds.<\/jats:p>","DOI":"10.1007\/s10623-020-00773-4","type":"journal-article","created":{"date-parts":[[2020,6,23]],"date-time":"2020-06-23T17:02:45Z","timestamp":1592931765000},"page":"2207-2231","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Linearly equivalent S-boxes and the division property"],"prefix":"10.1007","volume":"88","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3040-4503","authenticated-orcid":false,"given":"Baptiste","family":"Lambin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Patrick","family":"Derbez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pierre-Alain","family":"Fouque","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,6,23]]},"reference":[{"issue":"4","key":"773_CR1","doi-asserted-by":"crossref","first-page":"99","DOI":"10.46586\/tosc.v2017.i4.99-129","volume":"2017","author":"A Abdelkhalek","year":"2017","unstructured":"Abdelkhalek A., Sasaki Y., Todo Y., Tolba M., Youssef A.M.: MILP modeling for (large) S-boxes to optimize probability of differential characteristics. IACR Trans. Symmetric Cryptol. 2017(4), 99\u2013129 (2017).","journal-title":"IACR Trans. Symmetric Cryptol."},{"key":"773_CR2","doi-asserted-by":"crossref","unstructured":"Beierle C., Jean J., K\u00f6lbl S., Leander G., Moradi A., Peyrin T., Sasaki Y., Sasdrich P., Sim S.M.: The SKINNY family of block ciphers and its low-latency variant MANTIS. In: Advances in Cryptology\u2014CRYPTO 2016\u201436th Annual International Cryptology Conference, Santa Barbara, CA, August 14\u201318, 2016, Proceedings, Part II, pp. 123\u2013153 (2016).","DOI":"10.1007\/978-3-662-53008-5_5"},{"key":"773_CR3","doi-asserted-by":"crossref","unstructured":"Bogdanov A., Knudsen L.R., Leander G., Paar C., Poschmann A., Robshaw M.J.B., Seurin Y., Vikkelsoe C.: PRESENT: an ultra-lightweight block Cipher. In: Cryptographic Hardware and Embedded Systems\u2014CHES 2007, 9th International Workshop, Vienna, Austria, September 10\u201313, 2007, Proceedings, pp. 450\u2013466 (2007).","DOI":"10.1007\/978-3-540-74735-2_31"},{"key":"773_CR4","doi-asserted-by":"crossref","unstructured":"Boura C., Canteaut A.: Another view of the division property. In: Advances in Cryptology\u2014CRYPTO 2016\u201436th Annual International Cryptology Conference, Santa Barbara, CA, August 14\u201318, 2016, Proceedings, Part I, pp. 654\u2013682 (2016).","DOI":"10.1007\/978-3-662-53018-4_24"},{"key":"773_CR5","unstructured":"Daemen J., Rijmen V.: AES proposal: Rijndael (1999)."},{"key":"773_CR6","doi-asserted-by":"crossref","unstructured":"Derbez P., Fouque P.: Exhausting Demirci-Sel\u00e7uk meet-in-the-middle attacks against reduced-round AES. In: Fast Software Encryption\u201420th International Workshop, FSE 2013, Singapore, March 11\u201313, 2013. 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