{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T14:25:39Z","timestamp":1760711139578,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,2]],"date-time":"2023-01-02T00:00:00Z","timestamp":1672617600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Ningxia","award":["2021AAC03068","2021AAC03102","2021BEB04065","2021BEG03071"],"award-info":[{"award-number":["2021AAC03068","2021AAC03102","2021BEB04065","2021BEG03071"]}]},{"name":"Key R&amp;D Program of Ningxia","award":["2021AAC03068","2021AAC03102","2021BEB04065","2021BEG03071"],"award-info":[{"award-number":["2021AAC03068","2021AAC03102","2021BEB04065","2021BEG03071"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Permutations with low differential uniformity, high nonlinearity and high algebraic degree over F22k are preferred substitution boxes in modern block ciphers. In this paper, we study the bijectivity and the difference uniformity of piecewise function with the help of permutation group theory. Based on our results, We found many at least differentially 6-uniform and differentially 4-uniform permutations over F22k, which can be chosen as the substitution boxes.<\/jats:p>","DOI":"10.3390\/sym15010131","type":"journal-article","created":{"date-parts":[[2023,1,3]],"date-time":"2023-01-03T02:05:53Z","timestamp":1672711553000},"page":"131","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["New Differentially 4-Uniform Piecewise Permutations over F22k from the Inverse Function"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8871-3272","authenticated-orcid":false,"given":"Shuai","family":"Li","sequence":"first","affiliation":[{"name":"School of Information Engineering, Ningxia University, Yinchuan 750021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Li","family":"Miao","sequence":"additional","affiliation":[{"name":"School of Information Engineering, Ningxia University, Yinchuan 750021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"8065","DOI":"10.1109\/TII.2022.3161631","article-title":"Permissioned blockchain and deep-learning for secure and efficient data sharing in industrial healthcare systems","volume":"18","author":"Kumar","year":"2022","journal-title":"IEEE Trans. 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