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Codes Cryptogr."],"published-print":{"date-parts":[[2024,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Circulant Column Parity Mixers (CCPMs) are a particular type of linear maps, used as the mixing layer in permutation-based cryptographic primitives like <jats:sc>Keccak<\/jats:sc>-<jats:italic>f<\/jats:italic> (SHA3) and <jats:sc>Xoodoo<\/jats:sc>. Although being successfully applied, not much is known regarding their algebraic properties. They are limited to invertibility of CCPMs, and that the set of invertible CCPMs forms a group. A possible explanation is due to the complexity of describing CCPMs in terms of linear algebra. In this paper, we introduce a new approach to studying CCPMs using module theory from commutative algebra. We show that many interesting algebraic properties can be deduced using this approach, and that known results regarding CCPMs resurface as trivial consequences of module theoretic concepts. We also show how this approach can be used to study the linear layer of <jats:sc>Xoodoo<\/jats:sc>, and other linear maps with a similar structure which we call DCD-compositions. Using this approach, we prove that every DCD-composition where the underlying vector space with the same dimension as that of <jats:sc>Xoodoo<\/jats:sc> has a low order. This provides a solid mathematical explanation for the low order of the linear layer of <jats:sc>Xoodoo<\/jats:sc>, which equals 32. We design a DCD-composition using this module-theoretic approach, but with a higher order using a different dimension.<\/jats:p>","DOI":"10.1007\/s10623-024-01476-w","type":"journal-article","created":{"date-parts":[[2024,8,23]],"date-time":"2024-08-23T06:55:32Z","timestamp":1724396132000},"page":"4057-4083","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An algebraic approach to circulant column parity mixers"],"prefix":"10.1007","volume":"92","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5534-2655","authenticated-orcid":false,"given":"Robert Christian","family":"Subroto","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,8,21]]},"reference":[{"key":"1476_CR1","doi-asserted-by":"publisher","DOI":"10.1201\/9780429493638","volume-title":"Introduction to Commutative Algebra","author":"M Atiyah","year":"2018","unstructured":"Atiyah M.: Introduction to Commutative Algebra. CRC Press, Boca Raton, FL (2018)."},{"key":"1476_CR2","doi-asserted-by":"publisher","unstructured":"Beierle C., Canteaut A., Leander G., Rotella Y.: Proving resistance against invariant attacks: how to choose the round constants. In: Katz J., Shacham H. (eds.) Advances in Cryptology\u2014CRYPTO 2017\u201437th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 20\u201324, 2017, Proceedings, Part II. Lecture Notes in Computer Science, vol. 10402, pp. 647\u2013678. Springer (2017). https:\/\/doi.org\/10.1007\/978-3-319-63715-0_22.","DOI":"10.1007\/978-3-319-63715-0_22"},{"key":"1476_CR3","unstructured":"Bertoni G., Daemen J., Peeters M., Assche G.V.: Keccak. 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