{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,8]],"date-time":"2025-05-08T04:12:43Z","timestamp":1746677563715,"version":"3.40.5"},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2024,12,22]],"date-time":"2024-12-22T00:00:00Z","timestamp":1734825600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,12,22]],"date-time":"2024-12-22T00:00:00Z","timestamp":1734825600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100005276","name":"National Board for Higher Mathematics","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100005276","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2025,5]]},"DOI":"10.1007\/s10623-024-01548-x","type":"journal-article","created":{"date-parts":[[2024,12,22]],"date-time":"2024-12-22T09:31:14Z","timestamp":1734859874000},"page":"1283-1293","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A class of permutations on $${\\mathbb {Z}}_{p}$$ with differential uniformity at most 3"],"prefix":"10.1007","volume":"93","author":[{"given":"Prachi","family":"Gupta","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"P. R.","family":"Mishra","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Atul","family":"Gaur","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,12,22]]},"reference":[{"key":"1548_CR1","unstructured":"Apostol, T. M.: Introduction to Analytic Number Theory. Springer (2013)"},{"key":"1548_CR2","first-page":"65","volume-title":"Advances in Cryptology, LNCS 765","author":"T Beth","year":"1994","unstructured":"Beth T., Ding C.: On almost perfect nonlinear permutations. In: Helleseth T. (ed.) Advances in Cryptology, LNCS 765, pp. 65\u201376. Springer, Berlin, Heidelberg (1994)."},{"issue":"1","key":"1548_CR3","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/BF00630563","volume":"4","author":"E Biham","year":"1991","unstructured":"Biham E., Shamir A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3\u201372 (1991).","journal-title":"J. Cryptol."},{"issue":"2","key":"1548_CR4","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1016\/j.disc.2007.12.025","volume":"309","author":"L Budaghyan","year":"2009","unstructured":"Budaghyan L., Pott A.: On differential uniformity and nonlinearity of functions. Discrete Math. 309(2), 371\u2013384 (2009).","journal-title":"Discrete Math."},{"key":"1548_CR5","doi-asserted-by":"publisher","first-page":"3320","DOI":"10.1016\/j.dam.2009.06.029","volume":"157","author":"K Drakakis","year":"2009","unstructured":"Drakakis K., Gow R., McGuire G.: APN permutations on $${\\mathbb{Z} }_{n}$$ and costas arrays. Discrete Appl. Math. 157, 3320\u20133326 (2009).","journal-title":"Discrete Appl. Math."},{"issue":"5","key":"1548_CR6","doi-asserted-by":"publisher","first-page":"841","DOI":"10.1007\/s12095-020-00437-z","volume":"12","author":"P Felke","year":"2020","unstructured":"Felke P.: The multivariate method strikes again: new power functions with low differential uniformity in odd characteristic. Cryptogr. Commun. 12(5), 841\u2013857 (2020).","journal-title":"Cryptogr. Commun."},{"key":"1548_CR7","doi-asserted-by":"crossref","unstructured":"Gupta, P., Mishra, P. R., Gaur, A.: On differential uniformity & nonlinearity of permutations on $${\\mathbb{Z}}_n$$. In: Proceedings of the Seventh International Conference on Mathematics and Computing, Adv. Intell. Syst. Comput., vol. 1412, pp. 627\u2013636. Springer, Singapore (2022)","DOI":"10.1007\/978-981-16-6890-6_47"},{"key":"1548_CR8","doi-asserted-by":"publisher","first-page":"859","DOI":"10.1007\/s00200-022-00594-z","volume":"35","author":"P Gupta","year":"2024","unstructured":"Gupta P., Mishra P.R., Gaur A.: Bounds on maximum nonlinearity of permutations on the rings $${\\mathbb{Z} }_p$$ & $${\\mathbb{Z} }_{2p}$$. Appl. Algebra Eng. Commun. Comput. 35, 859\u2013874 (2024).","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"issue":"4","key":"1548_CR9","doi-asserted-by":"publisher","first-page":"855","DOI":"10.1016\/j.ffa.2012.05.001","volume":"18","author":"RM Hakala","year":"2012","unstructured":"Hakala R.M.: An upper bound for the linearity of exponential welch costas functions. Finite Fields Appl. 18(4), 855\u2013862 (2012).","journal-title":"Finite Fields Appl."},{"key":"1548_CR10","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1007\/s00200-016-0307-6","volume":"28","author":"Y Kumar","year":"2017","unstructured":"Kumar Y., Mishra P.R., Pillai N.R., Sharma R.K.: Affine equivalence and non-linearity of permutations over $${\\mathbb{Z} }_n$$. Appl. Algebra Eng. Commun. Comput. 28, 257\u2013279 (2017).","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"key":"1548_CR11","first-page":"386","volume-title":"Advances in Cryptology, LNCS 765","author":"M Matsui","year":"1994","unstructured":"Matsui M.: Linear cryptanalysis method for DES cipher. In: Helleseth T. (ed.) Advances in Cryptology, LNCS 765, pp. 386\u2013397. Springer, Berlin, Heidelberg (1994)."},{"issue":"1","key":"1548_CR12","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1080\/01611194.2017.1325788","volume":"42","author":"PR Mishra","year":"2018","unstructured":"Mishra P.R., Kumar Y., Pillai N.R., Sharma R.K.: On non-linearity and affine equivalence of permutations over an arbitrary finite commutative ring with unity. Cryptologia 42(1), 81\u201394 (2018).","journal-title":"Cryptologia"},{"issue":"3","key":"1548_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2023.102305","volume":"92","author":"PR Mishra","year":"2023","unstructured":"Mishra P.R., Gupta P., Gaur A.: Construction of highly nonlinear permutations on $${\\mathbb{Z} }_{2p}$$ with differential uniformity at most $$8$$8. Finite Fields Appl. 92(3), 102305 (2023).","journal-title":"Finite Fields Appl."},{"key":"1548_CR14","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1007\/s00200-021-00503-w","volume":"34","author":"PR Mishra","year":"2023","unstructured":"Mishra P.R., Gupta P., Gaur A.: On full differential uniformity of permutations on the ring of integers modulo $$n$$. Appl. Algebra Eng. Commun. Comput. 34, 301\u2013319 (2023).","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"key":"1548_CR15","first-page":"55","volume-title":"Advances in Cryptology, LNCS 765","author":"K Nyberg","year":"1994","unstructured":"Nyberg K.: Differentially uniform mappings for cryptography. In: Helleseth T. (ed.) Advances in Cryptology, LNCS 765, pp. 55\u201364. Springer, Berlin, Heidelberg (1994)."},{"key":"1548_CR16","doi-asserted-by":"crossref","unstructured":"Paul, G., Maitra, S.: RC4 Stream Cipher and its Variants. CRC Press (2012)","DOI":"10.1201\/b11310"},{"key":"1548_CR17","doi-asserted-by":"publisher","first-page":"897","DOI":"10.1007\/s00200-021-00528-1","volume":"34","author":"RK Sharma","year":"2023","unstructured":"Sharma R.K., Mishra P.R., Kumar Y.: A new class of differential 4-uniform permutations from exponential permutation. Appl. Algebra Eng. Commun. Comput. 34, 897\u2013912 (2023).","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"key":"1548_CR18","doi-asserted-by":"crossref","unstructured":"Zoltak, B.: VMPC one-way function and stream cipher. In: Fast Software Encryption. LNCS 3017, pp. 210\u2013225, Springer, Berlin, Heidelberg (2004)","DOI":"10.1007\/978-3-540-25937-4_14"}],"container-title":["Designs, Codes and Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-024-01548-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10623-024-01548-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-024-01548-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,7]],"date-time":"2025-05-07T17:46:52Z","timestamp":1746640012000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10623-024-01548-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,22]]},"references-count":18,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,5]]}},"alternative-id":["1548"],"URL":"https:\/\/doi.org\/10.1007\/s10623-024-01548-x","relation":{},"ISSN":["0925-1022","1573-7586"],"issn-type":[{"type":"print","value":"0925-1022"},{"type":"electronic","value":"1573-7586"}],"subject":[],"published":{"date-parts":[[2024,12,22]]},"assertion":[{"value":"5 September 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 November 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 December 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 December 2024","order":4,"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 no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}