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Commun."],"published-print":{"date-parts":[[2023,11]]},"abstract":"Abstract<\/jats:title>To evaluate the security of a cryptographic primitive, investigating its resistance against differential and linear cryptanalysis is required. Many modern cryptographic primitives repeatedly apply similar round functions alternated with the addition of round keys or constants. A round function usually consists of a non-linear mapping and a number of linear mappings. The non-linear mapping $$\\chi$$<\/jats:tex-math>\n \u03c7<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is used in different cryptographic primitives such as Keccak<\/jats:sc> and Subterranean<\/jats:sc>. An alternative version of $$\\chi$$<\/jats:tex-math>\n \u03c7<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is used in Ascon<\/jats:sc> and the non-linear layer of Simon<\/jats:sc> has the same differential and linear properties of $$\\chi$$<\/jats:tex-math>\n \u03c7<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The mapping $$\\chi$$<\/jats:tex-math>\n \u03c7<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> can be applied to strings with different lengths. For instance, it can be parallelly applied to small-length strings as in Keccak<\/jats:sc>, where it works on 5-bit strings, or it can be applied to big-length strings as in Subterranean<\/jats:sc>, where it works on a string of length 257. Investigating the differential and linear properties of $$\\chi$$<\/jats:tex-math>\n \u03c7<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> working on alternative lengths of strings, provides useful information to designers to make a better choice for the non-linear layer. Some differential properties of $$\\chi$$<\/jats:tex-math>\n \u03c7<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> have been analyzed in\u00a0[8] and in this work we provide a revised presentation of them. We then extend this study and we analyze linear propagation properties of $$\\chi$$<\/jats:tex-math>\n \u03c7<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Thanks to these additional results, we extend the comparison between the application of parallel instances of $$\\chi$$<\/jats:tex-math>\n \u03c7<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> on small-length strings and the application of a single instance of $$\\chi$$<\/jats:tex-math>\n \u03c7<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> on a big-length string. We show how we can apply the results of this study also to the non-linear layers of Ascon<\/jats:sc> and Simon<\/jats:sc> thanks to their affine-equivalence with $$\\chi$$<\/jats:tex-math>\n \u03c7<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s12095-023-00639-1","type":"journal-article","created":{"date-parts":[[2023,4,26]],"date-time":"2023-04-26T12:05:09Z","timestamp":1682510709000},"page":"1087-1116","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Differential and Linear properties of vectorial boolean functions based on chi"],"prefix":"10.1007","volume":"15","author":[{"given":"Silvia","family":"Mella","sequence":"first","affiliation":[]},{"given":"Alireza","family":"Mehrdad","sequence":"additional","affiliation":[]},{"given":"Joan","family":"Daemen","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,4,26]]},"reference":[{"key":"639_CR1","unstructured":"Beaulieu, R., Shors, D., Smith, J., Treatman-Clark, S., Weeks, B., Wingers, L.: The SIMON and SPECK families of lightweight block ciphers. 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Cambridge University Press (2010)","DOI":"10.1017\/CBO9780511780448.012"},{"key":"639_CR5","unstructured":"Daemen, J.: Cipher and hash function design, strategies based on linear and differential cryptanalysis, PhD Thesis. K.U.Leuven (1995). http:\/\/jda.noekeon.org\/"},{"issue":"4","key":"639_CR6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.46586\/tosc.v2018.i4.1-38","volume":"2018","author":"J Daemen","year":"2018","unstructured":"Daemen, J., Hoffert, S., Assche, G.V., Keer, R.V.: The design of Xoodoo and Xoofff. IACR Trans. Symmetric Cryptol. 2018(4), 1\u201338 (2018)","journal-title":"IACR Trans. Symmetric Cryptol."},{"issue":"S1","key":"639_CR7","doi-asserted-by":"publisher","first-page":"262","DOI":"10.46586\/tosc.v2020.iS1.262-294","volume":"2020","author":"J Daemen","year":"2020","unstructured":"Daemen, J., Massolino, P.M.C., Mehrdad, A., Rotella, Y.: The Subterranean 2.0 Cipher Suite. IACR Trans. Symmetric Cryptol. 2020(S1), 262\u2013294 (2020)","journal-title":"IACR Trans. Symmetric Cryptol."},{"key":"639_CR8","doi-asserted-by":"crossref","unstructured":"Daemen, J., Mehrdad, A., Mella, S.: Computing the distribution of differentials over the non-linear mapping $$\\chi$$. In: Security, Privacy, and Applied Cryptography Engineering - 11th International Conference, SPACE 2021, Kolkata, India, December 10-13, 2021, Proceedings, Lecture Notes in Computer Science, vol. 13162, pp. 3\u201321. Springer (2021)","DOI":"10.1007\/978-3-030-95085-9_1"},{"key":"639_CR9","unstructured":"Daemen, J., Mella, S., Assche, G.V.: Tighter trail bounds for xoodoo. Cryptology ePrint Archive, Paper 2022\/1088 (2022). https:\/\/eprint.iacr.org\/2022\/1088"},{"key":"639_CR10","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04722-4","volume-title":"The Design of Rijndael: AES - The Advanced Encryption Standard","author":"J Daemen","year":"2002","unstructured":"Daemen, J., Rijmen, V.: The Design of Rijndael: AES - The Advanced Encryption Standard. Springer, Information Security and Cryptography (2002)"},{"key":"639_CR11","unstructured":"Dobraunig, C., Eichlseder, M., Mendel, F., Schl\u00e4ffer, M.: Ascon v1.2 (2014). https:\/\/ascon.iaik.tugraz.at\/"},{"issue":"3","key":"639_CR12","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/s00145-021-09398-9","volume":"34","author":"C Dobraunig","year":"2021","unstructured":"Dobraunig, C., Eichlseder, M., Mendel, F., Schl\u00e4ffer, M.: Ascon v1.2: Lightweight authenticated encryption and hashing. J. Cryptol. 34(3), 33 (2021)","journal-title":"J. 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Symmetric Cryptol."},{"issue":"2","key":"639_CR15","doi-asserted-by":"publisher","first-page":"253","DOI":"10.46586\/tosc.v2022.i2.253-288","volume":"2022","author":"A Mehrdad","year":"2022","unstructured":"Mehrdad, A., Mella, S., Grassi, L., Daemen, J.: Differential trail search in cryptographic primitives with big-circle chi: Application to subterranean. IACR Trans. Symmetric Cryptol. 2022(2), 253\u2013288 (2022)","journal-title":"IACR Trans. Symmetric Cryptol."},{"issue":"1","key":"639_CR16","doi-asserted-by":"publisher","first-page":"329","DOI":"10.46586\/tosc.v2017.i1.329-357","volume":"2017","author":"S Mella","year":"2017","unstructured":"Mella, S., Daemen, J., Van Assche, G.: New techniques for trail bounds and application to differential trails in Keccak. IACR Trans. Symmetric Cryptol. 2017(1), 329\u2013357 (2017)","journal-title":"IACR Trans. Symmetric Cryptol."},{"key":"639_CR17","unstructured":"Nyberg, K.: Differentially uniform mappings for cryptography. 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