{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T16:35:08Z","timestamp":1774370108505,"version":"3.50.1"},"reference-count":0,"publisher":"Universitatsbibliothek der Ruhr-Universitat Bochum","license":[{"start":{"date-parts":[[2021,6,11]],"date-time":"2021-06-11T00:00:00Z","timestamp":1623369600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["ToSC"],"abstract":"<jats:p>At FSE 2004, Lipmaa et al. studied the additive differential probability adp\u2295(\u03b1,\u03b2 \u2192 \u03b3) of exclusive-or where differences \u03b1,\u03b2,\u03b3 \u2208 Fn2 are expressed using addition modulo 2n. This probability is used in the analysis of symmetric-key primitives that combine XOR and modular addition, such as the increasingly popular Addition-Rotation-XOR (ARX) constructions. The focus of this paper is on maximal differentials, which are helpful when constructing differential trails. We provide the missing proof for Theorem 3 of the FSE 2004 paper, which states that max\u03b1,\u03b2adp\u2295(\u03b1,\u03b2 \u2192 \u03b3) = adp\u2295(0,\u03b3 \u2192 \u03b3) for all \u03b3. Furthermore, we prove that there always exist either two or eight distinct pairs \u03b1,\u03b2 such that adp\u2295( \u03b1,\u03b2 \u2192 \u03b3) = adp\u2295(0,\u03b3 \u2192 \u03b3), and we obtain recurrence formulas for calculating adp\u2295. To gain insight into the range of possible differential probabilities, we also study other properties such as the minimum value of adp\u2295(0,\u03b3 \u2192 \u03b3), and we find all \u03b3 that satisfy this minimum value.<\/jats:p>","DOI":"10.46586\/tosc.v2021.i2.292-313","type":"journal-article","created":{"date-parts":[[2021,6,11]],"date-time":"2021-06-11T12:03:04Z","timestamp":1623412984000},"page":"292-313","source":"Crossref","is-referenced-by-count":6,"title":["Maximums of the Additive Differential Probability of Exclusive-Or"],"prefix":"10.46586","author":[{"given":"Nicky","family":"Mouha","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nikolay","family":"Kolomeec","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Danil","family":"Akhtiamov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ivan","family":"Sutormin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Matvey","family":"Panferov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kseniya","family":"Titova","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tatiana","family":"Bonich","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Evgeniya","family":"Ishchukova","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Natalia","family":"Tokareva","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bulat","family":"Zhantulikov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25480","published-online":{"date-parts":[[2021,6,11]]},"container-title":["IACR Transactions on Symmetric Cryptology"],"original-title":[],"link":[{"URL":"https:\/\/tosc.iacr.org\/index.php\/ToSC\/article\/download\/8912\/8488","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/tosc.iacr.org\/index.php\/ToSC\/article\/download\/8912\/8488","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,11]],"date-time":"2021-06-11T12:03:05Z","timestamp":1623412985000},"score":1,"resource":{"primary":{"URL":"https:\/\/tosc.iacr.org\/index.php\/ToSC\/article\/view\/8912"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,11]]},"references-count":0,"URL":"https:\/\/doi.org\/10.46586\/tosc.v2021.i2.292-313","relation":{},"ISSN":["2519-173X"],"issn-type":[{"value":"2519-173X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,11]]}}}